$V:=\delta V+V_{0}^{+}$. Then we find $\tilde{V}$ by solving the linear system
(3.79) in $\mathbb{D}$ with the boundary conditions (3.85) and (3.89), to
determine $(\tilde{w},\tilde{p})$ and then determine $u_{1}$ from (3.71) and
$\rho$ from (3.37) (which holds in the $\mathbf{z}$–coordinates without
change), and the boundary condition (3.88). The final step is to use solution
$(\delta u_{1},\delta\rho,\delta w,\delta p)$ and $U^{-}_{\hat{\sigma}}$
defined by (3.77) on the right-hand side of (3.80) to update
$\delta\hat{\sigma}^{\prime}$. This defines the iteration map $\mathcal{Q}$
from $V$ to $\tilde{V}$, except that we discuss below how the boundary value
problem for (3.79) with the boundary conditions (3.85) and (3.89) is solved in
$\mathbb{D}$.

As we discussed above, we perform the iteration in the spaces from (3.69) for
Problem 3.2(ST) and similar norms with appropriate growths for Problem
3.2(WT), expressed in the $\mathbf{z}$–coordinates (3.76). We focus below on
the case of Problem 3.2(WT), since the other case is similar.

For $\tau>0$, define

$\begin{split}\Sigma^{\tau}_{1}&=\\{v\,:\;\|v\|^{(-\alpha;\mathcal{L}_{1})}_{2,\alpha;(0,1+\beta);\mathbb{D}}+\|v_{z_{1}}\|^{(1-\alpha;\mathcal{L}_{1})}_{2,\alpha;(1+\beta,1);\mathbb{D}}\leq\tau\\},\\\\[2.84526pt]
\Sigma^{\tau}_{2}&=\\{v\,:\;\|v\|^{(-\alpha;O)(-1-\alpha;\mathcal{L}_{1})}_{2,\alpha;(1+\beta,0);\mathbb{D}}\leq\tau\\},\qquad\Sigma^{\tau}_{3}=\\{v\,:\;\|v\|^{(-\alpha;0)}_{2,\alpha;(1+\beta);\mathbb{R}^{+}}\leq\tau\\},\\\\[2.84526pt]
\Sigma^{\tau}&=\\{(\delta U_{1},\delta
U_{2},\delta\hat{\sigma}^{\prime})\,:\;\delta
U_{1}\in\Sigma^{\tau}_{1}\times\Sigma^{\tau}_{1},\;\;\delta
U_{2}\in\Sigma^{\tau}_{2}\times\Sigma^{\tau}_{2},\;\;\delta\hat{\sigma}^{\prime}\in\Sigma^{\tau}_{3}\\}.\end{split}$
(3.90)

The condition on $v_{z_{1}}$ in $\Sigma^{\tau}_{1}$ is added for technical
reasons.

It remains to discuss how we find
$(\delta\tilde{w},\delta\tilde{p})\in\Sigma^{C_{0}\varepsilon}_{2}\times\Sigma^{C_{0}\varepsilon}_{2}$
that solves (3.79) in $\mathbb{D}$ with the boundary conditions (3.85) and
(3.89). From system (3.79), we obtain

$\displaystyle(\delta\tilde{p})_{z_{1}}=\frac{\lambda_{R}-\delta\hat{\sigma}^{\prime}}{h\lambda_{I}}(\delta\tilde{w})_{z_{1}}+\frac{(\lambda_{R}-\delta\hat{\sigma}^{\prime})^{2}+\lambda_{I}^{2}}{h\lambda_{I}}(\delta\tilde{w})_{z_{2}},$
(3.91)
$\displaystyle(\delta\tilde{p})_{z_{2}}=-\frac{1}{h\lambda_{I}}(\delta\tilde{w})_{z_{1}}-\frac{\lambda_{R}-\delta\hat{\sigma}^{\prime}}{h\lambda_{I}}(\delta\tilde{w})_{z_{2}}.$
(3.92)

Now, differentiating and subtracting the equations, we eliminate
$\delta\tilde{p}$ to obtain a second-order PDE for $\delta\tilde{w}$ of the
form:

$\sum_{i,j=1}^{2}(a_{ij}(\delta\tilde{w})_{z_{j}})_{z_{i}}=0,$ (3.93)

where the coefficients are computed explicitly from (3.91)–(3.92). Note that,
at the subsonic background solution (3.60), we have

$\lambda_{R0}=0,\quad\lambda_{I0}>0,\quad h_{0}>0,$

where the left-hand sides are constants and $\delta\hat{\sigma}_{0}=0$. Then,
computing the coefficients at the background solution, equation (3.93) becomes

$\frac{1}{\lambda_{I0}}(\delta\tilde{w})_{z_{1}z_{1}}+\lambda_{I0}(\delta\tilde{w})_{z_{2}z_{2}}=0,$

that is, the equation is uniformly elliptic. Then, for the coefficients
computed at $(U^{+}_{10}+\delta U_{1},U^{+}_{20}+\delta
U_{2},\delta\hat{\sigma}^{\prime})$ for $(\delta U_{1},\delta
U_{2},\delta\hat{\sigma}^{\prime})\in\Sigma^{C_{0}\varepsilon}$, equation
(3.93) is uniformly elliptic if $\varepsilon$ is small. This allows us to
obtain the unique solution $\delta\tilde{w}\in\Sigma^{C_{0}\varepsilon}_{2}$
of (3.93) in $\mathbb{D}$ with the boundary conditions (3.85) and (3.89). Note
that the inclusion $\delta\tilde{w}\in\Sigma^{C_{0}\varepsilon}_{2}$ involves
the asymptotic condition at infinity, which makes the boundary value problem
well-defined and allows us to prove the uniqueness. After $\delta\tilde{w}$ is
determined, we determine $\delta\tilde{p}$ by the $z_{2}$–integration from
(3.92) with the initial condition (3.85), where it can be shown that
$b_{1}\neq 0$. Then we show that $\delta p\in\Sigma^{C_{0}\varepsilon}_{2}$.
This completes the definition of the iteration map.

The iteration set for Problem 3.2(WT) is $\Sigma^{C_{0}\varepsilon}$. We first
show that
$\mathcal{Q}(\Sigma^{C_{0}\varepsilon})\subset\Sigma^{C_{0}\varepsilon}$ when
$\varepsilon$ is small, and then obtain a fixed point by the Schauder fixed
point theorem, via showing that $\Sigma^{C_{0}\varepsilon}$ is a compact
subset in the Banach space defined by replacing $\alpha$ via
$\alpha^{\prime}\in(0,\alpha)$ in the norms used in the definition of
$\Sigma^{\tau}$ and showing that map $\mathcal{Q}$ is continuous in this norm.

3\. Asymptotic limit of the fixed point in the $\mathbf{y}$–coordinates. Let
$(\delta U_{1},\delta
U_{2},\delta\hat{\sigma}^{\prime})\in\Sigma^{C_{0}\varepsilon}$ be a fixed
point of the iteration map, and let
$(U_{1},U_{2},\hat{\sigma}^{\prime})=(U^{+}_{10}+\delta
U_{1},U^{+}_{20}+\delta U_{2},\delta\hat{\sigma}^{\prime})$.

We change from the $\mathbf{z}$– to $\mathbf{y}$–coordinates by inverting
(3.76):

$\mathbf{z}:=(z_{1},z_{2})\,\to\,\mathbf{y}:=(y_{1},y_{2})=(z_{1},z_{2}+\delta\hat{\sigma}(z_{1})).$

Since $\delta\hat{\sigma}^{\prime}\in\Sigma^{C_{0}\varepsilon}_{3}$, then both
(3.76) and its inverse are close to the identity map in
$C^{2,\alpha}(\mathbb{D}_{\hat{\sigma}};\mathbb{R}^{2})$ and
$C^{2,\alpha}(\mathbb{D};\mathbb{R}^{2})$, respectively. Then it follows that,
in the $\mathbf{y}$–coordinates, $(\delta U_{1},\delta
U_{2},\delta\hat{\sigma}^{\prime})\in\tilde{\Sigma}^{2C_{0}\varepsilon}$ if
$\varepsilon$ is small, where

$\begin{split}\tilde{\Sigma}^{\tau}_{1}&=\\{v\,:\;\|v\|^{(-\alpha;\mathcal{L}_{1})}_{2,\alpha;(0,1+\beta);\mathbb{D}_{\hat{\sigma}}}+\|v_{z_{1}}\|^{(1-\alpha;\mathcal{L}_{1})}_{2,\alpha;(1+\beta,1);\mathbb{D}_{\hat{\sigma}}}\leq\tau\\},\quad\tilde{\Sigma}^{\tau}_{2}=\\{v\,:\;\|v\|^{(-\alpha;O)(-1-\alpha;\mathcal{L}_{1})}_{2,\alpha;(1+\beta,0);\mathbb{D}_{\hat{\sigma}}}\leq\tau\\},\\\\[2.84526pt]
\tilde{\Sigma}^{\tau}&=\\{(\delta U_{1},\delta
U_{2},\delta\hat{\sigma}^{\prime})\,:\;\delta
U_{1}\in\tilde{\Sigma}^{\tau}_{1}\times\tilde{\Sigma}^{\tau}_{1},\;\;\delta
U_{2}\in\tilde{\Sigma}^{\tau}_{2}\times\tilde{\Sigma}^{\tau}_{2},\;\;\delta\hat{\sigma}^{\prime}\in\Sigma^{\tau}_{3}\\}.\end{split}$
(3.94)

In particular, this leads to the estimates of the second and third terms on
the left-hand side of (3.68).

Note that, if $v\in\tilde{\Sigma}^{\tau}_{2}$, then $v\to 0$ as
$|\mathbf{y}|\to\infty$ in $\mathbb{D}_{\hat{\sigma}}$, with decay rate
$|\mathbf{y}|^{-(\beta+1)}$. However, for $v\in\tilde{\Sigma}^{\tau}_{1}$, no
asymptotic limit as $|\mathbf{y}|\to\infty$ in $\mathbb{D}_{\hat{\sigma}}$ is
defined.

Then, from (3.60)–(3.61), it follows that $U_{2}=(w,p)\to(0,p_{0}^{+})$ as
$|\mathbf{y}|\to\infty$ in $\mathbb{D}$; however, for $U_{1}=(u_{1},\rho)$,
the limit is not determined by space $\Sigma^{\tau}_{1}$, and $(u_{1},\rho)$
does not converge to $(u_{10}^{+},\rho_{0}^{+})$ in general, as we see below.
Thus, we have to determine the limiting profiles
$(u_{1}^{\infty}(y_{2}),\rho^{\infty}(y_{2}))$.

To determine $\rho^{\infty}(y_{2})$, we first obtain (3.37) from (3.7)–(3.10),
which implies (3.38). Since $\hat{\sigma}(y_{1})$ is determined, then
$A(y_{2})$ in (3.38) is determined by the upstream state $U^{-}(\mathbf{y})$
from the Rankine-Hugoniot conditions (3.28)–(3.31). Then, noting that $p\to
p_{0}^{\infty}$, we obtain formally

$\rho\to\rho^{\infty}(y_{2})=\Big{(}\frac{p_{0}^{+}}{A(y_{2})}\Big{)}^{\frac{1}{\gamma}}\qquad\mbox{as
$|\mathbf{y}|\to\infty$ in $\mathbb{D}_{\hat{\sigma}}$}.$

Similarly, we use (3.71) to obtain

$u_{1}\to u_{1}^{\infty}(y_{2})=\sqrt{2\Big{(}B(y_{2})-\frac{\gamma
p_{0}^{+}}{(\gamma-1)\rho^{\infty}(y_{2})}\Big{)}}\qquad\mbox{as
$|\mathbf{y}|\to\infty$ in $\mathbb{D}_{\hat{\sigma}}$}.$

Defining
$\mathcal{U}^{\infty}(y_{2})=(u_{1}^{\infty}(y_{2}),0,p_{0}^{+},\rho^{\infty}(y_{2}))$,
we can show that the estimates of the first and the last terms on the left-
hand side of (3.68) hold. This completes the argument for case (i) of Theorem
3.3.

Case (ii) is handled similarly. Note that the slower decay at infinity for
case (ii), i.e., $|\mathbf{y}|^{-\beta}$, is from the elliptic estimates, even
if the faster decay at infinity in (3.26) is required. The reason for the
difference in the rates for cases (i) and (ii) is (3.87).

4\. Return to the $\mathbf{x}$-coordinates. We obtain Theorem 3.2 directly
from Theorem 3.3 by changing the coordinates. Recall that, when the Lagrangian
coordinates are defined for Theorem 3.3, we have used the rotated coordinates
$\mathbf{x}$ in (3.6); see the discussion in the paragraph before Theorem 3.3.

From the estimates in Theorem 3.3, it follows that, in the Lagrangian
coordinates, $|U-U_{0}^{+}|\leq C\varepsilon$ in $\mathbb{D}_{\hat{\sigma}}$,
where $C$ depends only on $(U_{0}^{-},U_{0}^{+})$. Thus, the same is true in
the $\mathbf{x}$–coordinates in $\Omega$. Then it follows from (3.5)–(3.6) and
(3.60) for positive $u_{10}^{+}$ and $\rho_{0}^{+}$ that the change of
coordinates $\mathbf{x}\to\mathbf{y}$ given by (3.6) is bi-Lipschitz. Then
(3.67) follows from (3.69) directly.

Similarly, the estimates of the second and third terms on the left-hand side
of (3.65) follow from (3.68) directly. In order to obtain the estimates of the
remaining terms on the left-hand side of (3.65), we need to identify
$U^{\infty}(x_{2})$.

Note that, on shock $\mathcal{S}$, using (3.6) and the estimate of the third
term on the left-hand side of (3.65), we see that, for small $\varepsilon$,

$\partial_{{\bm{\tau}}_{\mathcal{S}}}\psi=\rho\mathbf{u}\cdot{\bm{\nu}}_{\mathcal{S}}\geq\rho\mathbf{u}_{0}^{+}\cdot{\bm{\nu}}_{\mathcal{S}_{0}}-C\varepsilon\geq\frac{1}{2}\rho\mathbf{u}_{0}^{+}\cdot{\bm{\nu}}_{\mathcal{S}_{0}}>0.$

Recall also that $\psi({\bf 0})=0$ by (3.5). Then, for each $y_{2}>0$, there
exists a unique $\mathbf{x}^{\rm in}(y_{2})=(x_{1}^{\rm in}(y_{2}),x_{2}^{\rm
in}(y_{2}))\in\mathcal{S}$ such that $\psi(\mathbf{x}^{\rm in}(y_{2}))=y_{2}$
and

$\|\mathbf{x}^{\rm in}\|_{C^{2,\alpha}([0,\infty))}\leq
C,\qquad\,\,(\mathbf{x}^{\rm in})^{\prime}\geq C^{-1}>0\;\;\mbox{ on
$[0,\infty)$}.$

From this and (3.5), it follows that, for each $y_{2}>0$,

$\Omega\cap\\{\mathbf{x}\,:\,\psi(\mathbf{x})=y_{2}\\}=\\{(x_{1},x_{2}^{*}(x_{1};y_{2}))\;:\;x_{1}>x_{1}^{\rm
in}(y_{2})\\},$

where $x_{2}^{*}(\cdot;y_{2})$ is the solution of the initial value problem
for the differential equation:

$\begin{cases}\partial_{x_{1}}x_{2}^{*}(x_{1};y_{2})=w(x_{1},x_{2}^{*}(x_{1};y_{2})),\\\\[2.84526pt]
x_{2}^{*}(x_{1}^{\rm in}(y_{2});y_{2})=x_{2}^{\rm in}(y_{2}),\end{cases}$
(3.95)

where $w=\frac{u_{2}}{u_{1}}$ (cf. (3.61)). Since we have obtained the
estimate of the second term on the left-hand side of (3.25), using (3.60), we
have

$|D^{k}w(\mathbf{x})|\leq
C_{0}\varepsilon(1+|\mathbf{x}|)^{-1-\beta}\qquad\mbox{in $\Omega,\;\;$ for
$k=0,1,2$}.$ (3.96)

In particular, for each $y_{2}\geq 0$ and $k=0,1,2$,

$\int_{x_{1}^{\rm
in}(y_{2})}^{\infty}|D^{k}w(x_{1},x_{2}^{*}(x_{1};y_{2}))|\,{\rm d}x_{1}\leq
C_{0}\varepsilon\int_{0}^{\infty}(1+x_{1})^{-1-\beta}\,{\rm d}x_{1}\leq
C\varepsilon.$ (3.97)

Applying this with $k=1$, we conclude that
$\lim_{x_{1}\to\infty}x_{2}^{*}(x_{1};y_{2})$ exists for each $y_{2}\geq 0$,
which is denoted as $x_{2}^{\infty}(y_{2})$.

Recall the structure of $\Omega$ in (3.63), where $b_{\rm rot}(x_{1})\to 0$
and $\tilde{\sigma}(x_{1})\to\infty$ as $x_{1}\to\infty$ by (3.64) and the
estimate of the third term in (3.65). Differentiating (3.95) twice with
respect to $y_{2}$ and using the $C^{2}$–estimate of $\mathbf{x}^{\rm in}$ and
(3.97), we obtain that $\|x_{2}^{*}(x_{1};\cdot)\|_{C^{2}([b_{\rm
rot}(x_{1}),\tilde{\sigma}(x_{1})])}\leq C$ for each $x_{1}>0$. From this, we
have

$\mbox{$x_{2}^{*}(x_{1};\cdot)\to x_{2}^{\infty}(\cdot)$ in $C^{1}$ on compact
subsets on $[0,\infty)$ as $x_{1}\to\infty$},$ (3.98)

with $\|x_{2}^{\infty}\|_{C^{2}([0,\infty))}\leq C$. Also, by a similar
argument, using the $C^{2,\alpha}$–regularity of $\mathbf{x}^{\rm in}$ and the
estimate of $w$ in the second term in (3.65), we obtain that
$x_{2}^{\infty}\in C^{2,\alpha}([0,\infty))$.

Furthermore, for the background solution, the potential functions
$\psi_{0}^{-}$ of $U_{0}^{-}$, $\psi_{0}^{+}$ of $U_{0}^{+}$, and $\psi_{0}$
of the transonic shock solution $(U_{0}^{-},U_{0}^{+})$ in
$\\{x_{1}>0,x_{2}>0\\}$ are:

$\psi_{0}^{-}(\mathbf{x})=\rho^{-}_{0}u^{-}_{10}(x_{1}-x_{2}\tan\theta_{\rm
w}),\quad\psi_{0}^{+}(\mathbf{x})=\rho^{+}_{0}u^{+}_{10}x_{1},\quad\psi_{0}(\mathbf{x})=\begin{cases}\psi_{0}^{-}(\mathbf{x})\;\;\;\mbox{
if }x_{2}<\tilde{s}x_{1},\\\\[1.42262pt] \psi_{0}^{+}(\mathbf{x})\;\;\;\mbox{
if }x_{2}>\tilde{s}x_{1},\end{cases}$

by using (3.60), where $\psi_{0}$ is Lipschitz. Then, estimating
$\psi-\psi_{0}^{-}$ in $\Omega^{-}$ via (3.64) (where the polynomial decay is
of degree $-(1+\beta)$ so that the calculations similar to (3.97) can be used)
and using the Rankine-Hugoniot conditions on $\mathcal{S}$, we obtain

$|(\mathbf{x}^{\rm in})^{\prime}-(\mathbf{x}^{\rm in}_{0})^{\prime}|\leq
C\varepsilon\qquad\mbox{ on $[0,\infty)$},$

where $\mathbf{x}^{\rm
in}_{0}(y_{2})=\frac{y_{2}}{\rho^{+}_{0}u^{+}_{10}}(1,\tilde{s}_{0})$ that is
the corresponding function $\mathbf{x}^{\rm in}$ of the background solution.

Denote by $x^{*}_{20}(x_{1};y_{2})$ the corresponding function
$x_{2}^{*}(x_{1};y_{2})$ of the background solution. Then

$x^{*}_{20}(x_{1};y_{2})=\frac{y_{2}}{\rho^{+}_{0}u^{+}_{10}}\qquad\,\mbox{on
$x_{1}>\frac{y_{2}}{\rho^{+}_{0}u^{+}_{10}\tilde{s}_{0}}$ for each $y_{2}\geq
0$}.$

Thus, $x^{*}_{20}(x_{1};y_{2})$ is independent of $x_{1}$, so that
$x^{*}_{20}(x_{1};y_{2})=x^{*}_{20}(y_{2})$. Then, denoting

$g(x_{1};y_{2})=x^{*}_{2}(x_{1};y_{2})-x^{*}_{20}(y_{2}),$

we see that $g$ satisfies

$\begin{cases}\partial_{x_{1}}g(x_{1};y_{2})=w(x_{1},x_{2}^{*}(x_{1};y_{2})),\\\\[2.84526pt]
|g(x_{1}^{\rm in}(y_{2});y_{2})|\leq C\varepsilon.\end{cases}$ (3.99)

From this and (3.97)–(3.98), we obtain that
$|(x_{2}^{\infty})^{\prime}-(x_{20}^{\infty})^{\prime}|\leq C\varepsilon$,
where
$(x_{20}^{\infty})^{\prime}(y_{2})=(x^{*}_{20})^{\prime}(y_{2})=\frac{1}{\rho^{+}_{0}u^{+}_{10}}$.
Therefore, we have

$(x_{2}^{\infty})^{\prime}\geq\frac{1}{2\rho^{+}_{0}u^{+}_{10}}\quad\mbox{ on
$[0,\infty)$},$

if $\varepsilon$ is small. In particular, noting that $x_{2}^{\infty}(0)=0$
since $\partial W$ is a streamline corresponding to $\psi=0$ and
$\lim_{x_{1}\to\infty}b_{\rm rot}(x_{1})=0$ by (3.64), we obtain
$x_{2}^{\infty}([0,\infty))=[0,\infty)$. Then there exists the inverse
function $y_{2}^{*}:[0,\infty)\to[0,\infty)$ to $x_{2}^{\infty}(\cdot)$ such
that $y_{2}^{*}\in C^{2,\alpha}([0,\infty))$ with $y_{2}^{*}(0)=0$ and
$(y_{2}^{*})^{\prime}\geq\frac{1}{C}>0$.

Finally, defining $U^{\infty}(x_{2})=\mathcal{U}^{\infty}(y_{2}^{*}(x_{2}))$,
we obtain (3.65) from (3.68).

For more details, see Chen-Chen-Feldman [25].

######  Remark 3.1.

For the global stability of weak transonic shocks for the $3$-D wedge problem,
see [28, 26]; also see the instability phenomenon for strong transonic shocks
for the $3$-D wedge problem in [77]. For the global stability of conical
shocks for the M-D conic problem, see [27] for the transonic shock case and
[38, 49, 82] for the supersonic shock case.

##  4\. Two-Dimensional Transonic Shocks and Free Boundary Problems for the
Self-Similar Euler Equations for Potential Flow

In §2–§3, we have discussed the free boundary problems for steady transonic
shock solutions of the compressible Euler equations. Now we discuss free
boundary problems for time-dependent solutions.

Time-dependent solutions with shocks of the Cauchy problem for the
compressible Euler system may exhibit non-uniqueness in general; see [50, 71]
and the references cited therein. On the other hand, many fundamental physical
phenomena, including shock reflection/diffraction, are determined by the time-
dependent solutions of self-similar structure; moreover, the uniqueness can be
established in a carefully chosen class of self-similar solutions with shocks.
In this section, we focus on this case; more precisely, we describe transonic
shocks and free boundary problems for self-similar shock
reflection/diffraction for the Euler equations for potential flow.

The two-dimensional compressible potential flow is governed by the
conservation law of mass and the Bernoulli law for the density function $\rho$
and the velocity potential $\Phi$ (i.e., $\mathbf{u}=\nabla\Phi$):

$\displaystyle\partial_{t}\rho+\nabla_{\bf x}\cdot(\rho\nabla_{\bf x}\Phi)=0,$
(4.1) $\displaystyle\partial_{t}\Phi+\frac{1}{2}|\nabla_{\bf
x}\Phi|^{2}+h(\rho)=B$ (4.2)

for $t\in\mathbb{R}^{+}:=(0,\infty)$ and ${\bf x}\in\mathbb{R}^{2}$, where $B$
is the Bernoulli constant, and $h(\rho)$ is given by

$h(\rho)=\frac{\rho^{\gamma-1}-1}{\gamma-1}\qquad\,\,\mbox{for the adiabatic
exponent $\gamma>1$.}$ (4.3)

By (4.2)–(4.3), $\rho$ can be expressed as

$\rho(\partial_{t}\Phi,\nabla_{\bf
x}\Phi)=h^{-1}(B-\partial_{t}\Phi-\frac{1}{2}|\nabla_{\bf x}\Phi|^{2}).$ (4.4)

Then system (4.1)–(4.2) can be rewritten as the following second-order
nonlinear wave equation:

$\partial_{t}\rho(\partial_{t}\Phi,\nabla_{\bf x}\Phi)+\nabla_{\bf
x}\cdot\big{(}\rho(\partial_{t}\Phi,\nabla_{\bf x}\Phi)\nabla_{\bf
x}\Phi\big{)}=0$ (4.5)

with $\rho(\partial_{t}\Phi,\nabla_{\bf x}\Phi)$ determined by (4.4).

Note that equation (4.4) is invariant under the self-similar scaling:

$(t,{\bf x})\rightarrow(\alpha t,\alpha{\bf
x}),\quad\Phi\rightarrow\frac{\Phi}{\alpha}\qquad\quad\text{for}\;\;\alpha\neq
0,$ (4.6)

and thus it admits self-similar solutions in the form of

$\Phi(t,{\bf
x})=t\phi({\bm{\xi}})\qquad\quad\text{for}\;\;{\bm{\xi}}=\frac{{\bf x}}{t}.$
(4.7)

Then the pseudo-potential function

$\varphi({\bm{\xi}})=\phi({\bm{\xi}})-\frac{1}{2}|{\bm{\xi}}|^{2}$

satisfies the following equation:

${\rm
div}(\rho(|D\varphi|^{2},\varphi)D\varphi)+2\rho(|D\varphi|^{2},\varphi)=0$
(4.8)

for

$\rho(|D\varphi|^{2},\varphi)=\bigl{(}B_{0}-(\gamma-1)(\frac{1}{2}|D\varphi|^{2}+\varphi)\bigr{)}^{\frac{1}{\gamma-1}},$
(4.9)

where $B_{0}=(\gamma-1)B+1$, and the divergence ${\rm div}$ and gradient $D$
are with respect to ${\bm{\xi}}\in\mathbb{R}^{2}$.

Equation (4.8) written in the non-divergence form is

$(c^{2}-\varphi_{{\xi_{1}}}^{2})\varphi_{{\xi_{1}}{\xi_{1}}}-2\varphi_{\xi_{1}}\varphi_{\xi_{2}}\varphi_{{\xi_{1}}{\xi_{2}}}+(c^{2}-\varphi_{{\xi_{2}}}^{2})\varphi_{{\xi_{2}}{\xi_{2}}}+2c^{2}-|D\varphi|^{2}=0,$
(4.10)

where the sonic speed $c=c(|D\varphi|^{2},\varphi)$ is determined by

$c^{2}(|D\varphi|^{2},\varphi)=\rho^{\gamma-1}(|D\varphi|^{2},\varphi)=B_{0}-(\gamma-1)\big{(}\frac{1}{2}|D\varphi|^{2}+\varphi\big{)}.$
(4.11)

Another form of (4.10), which uses both the potential $\phi$ and the pseudo-
potential $\varphi$, is

$(c^{2}-\varphi_{\xi_{1}}^{2})\phi_{\xi_{1}\xi_{1}}-2\varphi_{\xi_{1}}\varphi_{\xi_{2}}\phi_{\xi_{1}\xi_{2}}+(c^{2}-\varphi_{\xi_{2}}^{2})\phi_{\xi_{2}\xi_{2}}=0.$
(4.12)

Equation (4.8) is a nonlinear PDE of mixed elliptic-hyperbolic type. It is
elliptic at ${\bm{\xi}}$ if and only if

$|D\varphi|<c(|D\varphi|^{2},\varphi)\qquad\,\,\mbox{at ${\bm{\xi}}$},$ (4.13)

and is hyperbolic if the opposite inequality holds. This can be seen more
clearly from the rotational invariance of (4.10), by fixing ${\bm{\xi}}$ and
choosing coordinates $(\xi_{1},\xi_{2})$ so that $\xi_{1}$ is along the
direction of $D\varphi({\bm{\xi}})$.

Moreover, from (4.10)–(4.11), equation (4.8) satisfies the Galilean invariance
property: If $\varphi({\bm{\xi}})$ is a solution, then its shift
$\varphi({\bm{\xi}}-{\bm{\xi}}_{0})$ for any constant vector ${\bm{\xi}}_{0}$
is also a solution. Furthermore, $\varphi({\bm{\xi}})+{\it const.}$ is a
solution of (4.8) with adjusted constant $B$ correspondingly in (4.9) and
(4.11).

One class of solutions of (4.8) is that of constant states that are the
solutions with constant velocity $\mathbf{v}\in\mathbb{R}^{2}$. This implies
that the pseudo-potential of a constant state satisfies
$D\varphi=\mathbf{v}-{\bm{\xi}}$ so that

$\varphi({\bm{\xi}})=-\frac{1}{2}|{\bm{\xi}}|^{2}+\mathbf{v}\cdot{\bm{\xi}}+C,$
(4.14)

where $C$ is a constant. For such $\varphi$, the expressions in (4.9) and
(4.11) imply that the density and sonic speed are positive constants $\rho$
and $c$, i.e., independent of ${\bm{\xi}}$. Then, from (2.4) and (4.14), the
ellipticity condition for the constant state is

$|{\bm{\xi}}-\mathbf{v}|<c.$

Thus, for a constant state $\mathbf{v}$, equation (4.8) is elliptic inside the
sonic circle, with center $\mathbf{v}$ and radius $c$, and hyperbolic outside
this circle.

We also note that, if density $\rho$ is a constant, then the solution is a
constant state; that is, the corresponding pseudo-potential $\varphi$ is of
form (4.14).

Since the problem involves transonic shocks, we have to consider weak
solutions of equation (4.8), which admit shocks. As in [33], the weak
solutions are defined in the distributional sense in a domain $\Lambda$ in the
${\bm{\xi}}$–coordinates.

######  Definition 4.1.

A function $\varphi\in W^{1,1}_{\rm loc}(\Lambda)$ is called a weak solution
of (4.8) if

  1. (i)

$B_{0}-(\gamma-1)(\frac{1}{2}|D\varphi|^{2}+\varphi)\geq 0\quad\text{a.e. in
}\Lambda$,

  2. (ii)

$(\rho(|D\varphi|^{2},\varphi),\rho(|D\varphi|^{2},\varphi)|D\varphi|)\in(L^{1}_{\rm
loc}(\Lambda))^{2}$,

  3. (iii)

For every $\zeta\in C^{\infty}_{\rm c}(\Lambda)$,

$\int_{\Lambda}\big{(}\rho(|D\varphi|^{2},\varphi)D\varphi\cdot
D\zeta-2\rho(|D\varphi|^{2},\varphi)\zeta\big{)}\mathrm{d}{\bm{\xi}}=0.$
(4.15)

A shock is a curve across which $D\varphi$ is discontinuous. If $\Lambda^{+}$
and $\Lambda^{-}(:=\Lambda\setminus\overline{\Lambda^{+}})$ are two nonempty
open subsets of a domain $\Lambda\subset\mathbb{R}^{2}$, and
$\mathcal{S}:=\partial\Lambda^{+}\cap\Lambda$ is a $C^{1}$-curve where
$D\varphi$ has a jump, then $\varphi\in
C^{1}(\Lambda^{\pm}\cup\mathcal{S})\cap C^{2}(\Lambda^{\pm})$ is a global weak
solution of (4.8) in $\Lambda$ if and only if $\varphi$ is in
$W^{1,\infty}_{\rm loc}(\Lambda)$ and satisfies equation (4.8) and the
Rankine-Hugoniot condition on $\mathcal{S}$:

$\rho(|D\varphi|^{2},\varphi)D\varphi\cdot{\bm{\nu}}|_{\Lambda^{+}\cap\mathcal{S}}=\rho(|D\varphi|^{2},\varphi)D\varphi\cdot{\bm{\nu}}|_{\Lambda^{-}\cap\mathcal{S}}.$
(4.16)

Note that the condition $\varphi\in W^{1,\infty}_{\rm loc}(\Lambda)$ requires
that

$\varphi_{\Lambda^{+}\cap\mathcal{S}}=\varphi_{\Lambda^{-}\cap\mathcal{S}},$
(4.17)

which is consistent with ${\rm curl}(D\varphi)=0$ in the distributional sense.

A piecewise smooth solution with the discontinuities is called an entropy
solution of (4.8) if it satisfies the entropy condition: density $\rho$
increases in the pseudo-flow direction of
$D\varphi_{\Lambda^{+}\cap\mathcal{S}}$ across any discontinuity. Then such a
discontinuity is called a shock.

###  4.1. The von Neumann Problem for Shock Reflection-Diffraction

We now describe the von Neumann problem for shock reflection-diffraction,
proposed for mathematical analysis first in [105, 106, 107]. When a vertical
planar shock perpendicular to the flow direction $x_{1}$ and separating two
uniform states (0) and (1), with constant velocities $\mathbf{u}_{0}=(0,0)$
and $\mathbf{u}_{1}=(u_{1},0)$ and constant densities $\rho_{0}<\rho_{1}$
(state (0) is ahead or to the right of the shock, and state (1) is behind the
shock), hits a symmetric wedge:

$W:=\big{\\{}(x_{1},x_{2})\,:\,|x_{2}|<x_{1}\tan\theta_{\rm
w},x_{1}>0\big{\\}}$

head-on at time $t=0$, a reflection-diffraction process takes place when
$t>0$. Then a fundamental question is what types of wave patterns of
reflection-diffraction configurations may be formed around the wedge. The
complexity of reflection-diffraction configurations was first reported by
Ernst Mach [89] in 1878, who first observed two patterns of reflection-
diffraction configurations: Regular reflection (two-shock configuration; see
e.g. Figs. 4.2–4.2) and Mach reflection (three-shock/one-vortex-sheet
configuration); also see [8, 35, 52, 103]. The issues remained dormant until
the 1940s when John von Neumann [105, 106, 107], as well as other
mathematical/experimental scientists (cf. [8, 35, 52, 63, 103] and the
references cited therein), began extensive research into all aspects of shock
reflection-diffraction phenomena, due to its importance in applications. It
has been found that the situations are much more complicated than what Mach
originally observed: The Mach reflection can be further divided into more
specific sub-patterns, and various other patterns of shock reflection-
diffraction configurations may occur such as the double Mach reflection, the
von Neumann reflection, and the Guderley reflection; see [8, 35, 52, 63, 65,
103] and the references cited therein. Then the fundamental scientific issues
include:

  * (i)

Structures of the shock reflection-diffraction configurations;

  * (ii)

Transition criteria among the different patterns of shock reflection-
diffraction configurations;

  * (iii)

Dependence of the patterns upon the physical parameters such as the wedge
angle $\theta_{\rm w}$, the incident-shock-wave Mach number, and the adiabatic
exponent $\gamma>1$.

In particular, several transition criteria among the different patterns of
shock reflection-diffraction configurations have been proposed, including the
sonic conjecture and the detachment conjecture by von Neumann [105, 106, 107].

A careful asymptotic analysis has been made for various reflection-diffraction
configurations in Lighthill [83, 84], Keller-Blank [69], Hunter-Keller [68],
Harabetian [67], Morawetz [93], and the references cited therein; also see
Glimm-Majda [63]. Large or small scale numerical simulations have been also
performed; cf. [8, 63, 110] and the references cited therein. However, most of
the fundamental issues for shock reflection-diffraction phenomena have not
been understood, especially the global structure and transition between the
different patterns of shock reflection-diffraction configurations. This is
partially because physical and numerical experiments are hampered by many
difficulties and have not yielded clear transition criteria between the
different patterns. In particular, numerical dissipation or physical viscosity
smear the shocks and cause boundary layers that interact with the reflection-
diffraction patterns and can cause spurious Mach steams; cf. [110].
Furthermore, some different patterns occur when the wedge angles are only
fractions of a degree apart, a resolution as yet unreachable even by
sophisticated experiments (cf. [8, LD]). For this reason, it is impossible to
distinguish experimentally between the sonic and detachment criteria clearly,
as pointed out in [8]. In this regard, the necessary approach to understand
fully the shock reflection-diffraction phenomena, especially the transition
criteria, is via rigorous mathematical analysis. To achieve this, it is
essential to formulate the shock reflection-diffraction problem as a free
boundary problem and establish the global existence, regularity, and
structural stability of its solution.

Mathematically, the shock reflection-diffraction problem is a two-dimensional
lateral Riemann problem in domain $\mathbb{R}^{2}\setminus\overline{W}$.

######  Problem 4.1 (Two-Dimensional Lateral Riemann Problem).

Piecewise constant initial data, consisting of state $(0)$ with velocity
$\mathbf{u}_{0}=(0,0)$ on $\\{x_{1}>0\\}\setminus\overline{W}$ and state $(1)$
with velocity $\mathbf{u}_{1}=(u_{1},0)$ on $\\{x_{1}<0\\}$ connected by a
shock at $x_{1}=0$, are prescribed at $t=0$. Seek a solution of the Euler
system (4.1)–(4.2) for $t\geq 0$ subject to these initial data and the
boundary condition $\nabla\Phi\cdot{\bm{\nu}}=0$ on $\partial W$.

In order to define the notion of weak solutions of Problem 4.1, it is noted
that the boundary condition can be written as
$\rho\nabla\Phi\cdot{\bm{\nu}}=0$ on $\partial W$, which is spatial conormal
to equation (4.5). Then we have

######  Definition 4.2 (Weak Solutions of Problem 4.1).

A function $\Phi\in W^{1,1}_{\rm
loc}(\mathbb{R}_{+}\times({\mathbb{R}}^{2}\setminus W))$ is called a weak
solution of Problem 4.1 if $\Phi$ satisfies the following properties:

  1. (i)

$B-\left(\partial_{t}\Phi+\frac{1}{2}|\nabla_{\mathbf{x}}\Phi|^{2}\right)\geq
h(0+)$ a.e. in $\mathbb{R}_{+}\times(\mathbb{R}^{2}\setminus W)$.

  2. (ii)

For $\rho(\partial_{t}\Phi,\nabla_{\mathbf{x}}\Phi)$ determined by (4.4),

$(\rho(\partial_{t}\Phi,|\nabla_{\mathbf{x}}\Phi|^{2}),\rho(\partial_{t}\Phi,|\nabla_{\mathbf{x}}\Phi|^{2})|\nabla_{\mathbf{x}}\Phi|)\in(L^{1}_{\rm
loc}(\overline{\mathbb{R}}_{+}\times\overline{\mathbb{R}^{2}\setminus
W}))^{2}.$

  3. (iii)

For every $\zeta\in
C^{\infty}_{c}(\overline{\mathbb{R}_{+}}\times{\mathbb{R}}^{2})$,

$\displaystyle\int_{0}^{\infty}\int_{\mathbb{R}^{2}\setminus
W}\Big{(}\rho(\partial_{t}\Phi,|\nabla_{\mathbf{x}}\Phi|^{2})\partial_{t}\zeta+\rho(\partial_{t}\Phi,|\nabla_{\mathbf{x}}\Phi|^{2})\nabla\Phi\cdot\nabla\zeta\Big{)}\,{\rm
d}\mathbf{x}{\rm d}t+\int_{\mathbb{R}^{2}\setminus
W}\rho(0,\mathbf{x})\zeta(0,\mathbf{x}){\rm d}\mathbf{x}=0,$

where

$\rho|_{t=0}=\begin{cases}\rho_{0}\qquad\,\,&\mbox{for
$|x_{2}|>x_{1}\tan\theta_{\rm w}$ and $x_{1}>0$},\\\
\rho_{1}\qquad\,\,&\mbox{for}\,\,x_{1}<0.\end{cases}$

######  Remark 4.3.

Since $\zeta$ does not need to be zero on $\partial\Lambda$, the integral
identity in Definition 4.2 is a weak form of equation (4.5) and the boundary
condition $\rho\nabla\Phi\cdot{\bm{\nu}}=0$ on $\partial W$.

######  Remark 4.4.

A weak solution is called an entropy solution if it satisfies the entropy
condition that is consistent with the second law of thermodynamics $($cf. [35,
52, 53, 73]$)$. In particular, a piecewise smooth solution is an entropy
solution if the discontinuities are all shocks.

Notice that Problem 4.1 is invariant under scaling (4.6), so it admits self-
similar solutions determined by equation (4.8) with (4.9), along with the
appropriate boundary conditions, through (4.7). We now show how such solutions
in self-similar coordinates ${\bm{\xi}}=(\xi_{1},\xi_{2})=\frac{{\bf x}}{t}$
can be constructed.

First, by the symmetry of the problem with respect to the $\xi_{1}$–axis, we
consider only the upper half-plane $\\{\xi_{2}>0\\}$ and prescribe the
boundary condition: $\varphi_{{\bm{\nu}}}=0$ on the symmetry line
$\\{\xi_{2}=0\\}$. Note that state (1) satisfies this condition. Then Problem
4.1 is reformulated as a boundary value problem in the unbounded domain:

$\Lambda:=\mathbb{R}^{2}_{+}\setminus\\{{\bm{\xi}}\,:\,|\xi_{2}|\leq\xi_{1}\tan\theta_{\rm
w},\xi_{1}>0\\}$

in the self-similar coordinates ${\bm{\xi}}=(\xi_{1},\xi_{2})$, where
$\mathbb{R}^{2}_{+}:=\mathbb{R}^{2}\cap\\{\xi_{2}>0\\}$. The incident shock in
the self-similar coordinates is the half-line
$\mathcal{S}_{0}=\\{\xi_{1}=\xi_{1}^{0}\\}\cap\Lambda$, where

$\xi_{1}^{0}=\rho_{1}\sqrt{\frac{2(c_{1}^{2}-c_{0}^{2})}{(\gamma-1)(\rho_{1}^{2}-\rho_{0}^{2})}}=\frac{\rho_{1}u_{1}}{\rho_{1}-\rho_{0}},$
(4.18)

which is determined by the Rankine-Hugoniot conditions between states (0) and
(1) on $\mathcal{S}_{0}$. Then Problem 4.1 for self-similar solutions becomes
the following problem:

######  Problem 4.2 (Boundary Value Problem).

Seek a solution $\varphi$ of equation (4.8)–(4.9) in the self-similar domain
$\Lambda$ with the slip boundary condition
$D\varphi\cdot{\bm{\nu}}|_{\partial\Lambda}=0$ and the asymptotic boundary
condition at infinity:

$\varphi\to\bar{\varphi}=\begin{cases}\varphi_{0}\qquad\mbox{for}\,\,\,\xi_{1}>\xi_{1}^{0},\xi_{2}>\xi_{1}\tan\theta_{\rm
w},\\\
\varphi_{1}\qquad\mbox{for}\,\,\,\xi_{1}<\xi_{1}^{0},\;\xi_{2}>0,\end{cases}\qquad\mbox{when
$|{\bm{\xi}}|\to\infty$,}$

where $\varphi_{0}=-\frac{1}{2}|{\bm{\xi}}|^{2}$ and
$\varphi_{1}=-\frac{1}{2}|{\bm{\xi}}|^{2}+u_{1}(\xi_{1}-\xi^{0}_{1})$.

A weak solution of Problem 4.2 is obtained by the following modification of
Definition 4.1: (4.15) is now required to hold for all $\zeta\in
C^{\infty}_{c}({\mathbb{R}}^{2})$. As discussed in Remark 4.3, with such a
choice of function $\zeta$, the integral identity (4.15) includes both
equation (4.8) and the boundary condition of conormal form: $\rho
D\varphi\cdot{\bm{\nu}}=0$ on $\partial\Lambda$. A weak solution is called
entropy solution if it satisfies the entropy condition: density $\rho$
increases in the pseudo-flow direction of
$D\varphi|_{\Lambda^{+}\cap\mathcal{S}}$ across any discontinuity curve (i.e.,
shock).

Figure 4.1. Supersonic regular shock reflection-diffraction configuration

Figure 4.2. Subsonic regular shock reflection-diffraction configuration

Now we describe the more detailed structure of the regular reflection-
diffraction configurations as shown in Figs. 4.2–4.2. If a solution has one of
the regular shock reflection-diffraction configurations, and if its pseudo-
potential $\varphi$ is $C^{1}$ in the subregion $\widehat{\Omega}$ between the
wedge and the reflected shock, then, at $P_{0}$, it should satisfy both the
slip boundary condition on the wedge and the Rankine-Hugoniot conditions with
state $(1)$ across the flat shock
$\mathcal{S}_{1}=\\{\varphi_{1}=\varphi_{2}\\}$, which passes through point
$P_{0}$ where the incident shock meets the wedge boundary. Define the uniform
state (2) with pseudo-potential $\varphi_{2}({\bm{\xi}})$ such that

$\varphi_{2}(P_{0})=\varphi(P_{0}),\qquad D\varphi_{2}({P_{0}})=\lim_{P\to
P_{0},\;P\in\widehat{\Omega}}D\varphi(P).$

Then the constant density $\rho_{2}$ of state (2) is equal to
$\rho(|D\varphi|^{2},\varphi)({P_{0}})$ defined by (4.8):

$\rho_{2}=\rho(|D\varphi_{2}|^{2},\varphi_{2})({P_{0}}).$

From the properties of $\varphi$ discussed above, it follows that
$D\varphi_{2}\cdot{\bm{\nu}}=0$ on the wedge boundary and the Rankine-Hugoniot
conditions (4.16)–(4.17) hold on the flat shock
$\mathcal{S}_{1}=\\{\varphi_{1}=\varphi_{2}\\}$ between states (1) and (2),
which passes through ${P_{0}}$. In particular, $\varphi_{2}$ satisfies the
following three conditions at $P_{0}$:

$D\varphi_{2}\cdot{\bm{\nu}}_{\rm
w}=0,\,\,\,\,\varphi_{2}=\varphi_{1},\,\,\,\,\rho(|D\varphi_{2}|^{2},\varphi_{2})D\varphi_{2}\cdot{\bm{\nu}}_{\mathcal{S}_{1}}=\rho_{1}D\varphi_{1}\cdot{\bm{\nu}}_{\mathcal{S}_{1}}\qquad\mbox{for
${\bm{\nu}}_{\mathcal{S}_{1}}=\frac{D(\varphi_{1}-\varphi_{2})}{|D(\varphi_{1}-\varphi_{2})|}$}.$
(4.19)

where ${\bm{\nu}}_{\rm w}$ is the outward normal to the wedge boundary.

The entropy solution $\varphi$, correspondingly state (2), can be either
supersonic or subsonic at ${P_{0}}$. This determines the supersonic or
subsonic type of regular shock reflection-diffraction configurations. The
regular reflection solution in the supersonic region is expected to consist of
the constant states separated by straight shocks (cf. [98, Theorem 4.1]).
Then, when state (2) is supersonic at ${P_{0}}$, it can be shown that the
constant state (2), extended up to arc $P_{1}P_{4}$ of the sonic circle of
state (2) between the wall and the straight shock
$P_{0}P_{1}\subset\mathcal{S}_{1}$ separating it from state (1), as shown in
Fig. 4.2, satisfies equation (4.8) in the region, the Rankine-Hugoniot
condition (4.16)–(4.17) on the straight shock $P_{0}P_{1}$, and the slip
boundary condition: $D\varphi_{2}\cdot{\bm{\nu}}_{\rm w}=0$ on the wedge
$P_{0}P_{4}$, and is expected to be a part of the regular shock reflection-
diffraction configuration. Then the supersonic regular shock reflection-
diffraction configuration on Fig. 4.2 consists of three uniform states (0),
(1), (2), and a non-uniform state in domain $\Omega=P_{1}P_{2}P_{3}P_{4}$
where equation (4.8) is elliptic. The reflected shock ${P_{0}}{P_{1}}{P_{2}}$
has a straight part ${P_{0}}{P_{1}}$. The elliptic domain $\Omega$ is
separated from the hyperbolic region ${P_{0}}{P_{1}}{P_{4}}$ of state (2) by
the sonic arc ${P_{1}}{P_{4}}$ which lies on the sonic circle of state (2),
and the ellipticity in $\Omega$ degenerates on the sonic arc $P_{1}P_{4}$. The
subsonic regular shock reflection-diffraction configuration as shown in Fig.
4.2 consists of two uniform states (0) and (1), and a non-uniform state in
domain $\Omega=P_{0}P_{2}P_{3}$, where the equation is elliptic, and
$\varphi_{|\Omega}({P_{0}})=\varphi_{2}({P_{0}})$ and
$D(\varphi_{|\Omega})({P_{0}})=D\varphi_{2}({P_{0}})$.

For the supersonic regular shock reflection-diffraction configurations in Fig.
4.2, we use $\Gamma_{\rm sonic}$, $\Gamma_{\rm shock}$, $\Gamma_{\rm wedge}$,
and $\Gamma_{\rm sym}$ for the sonic arc $P_{1}P_{4}$, the curved part of the
reflected shock $P_{1}P_{2}$, the wedge boundary $P_{3}P_{4}$, and the
symmetry line segment $P_{2}P_{3}$, respectively.

For the subsonic regular shock reflection-diffraction configurations in Fig.
4.2, $\Gamma_{\rm shock}$, $\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$ denote
$P_{0}P_{2}$, $P_{0}P_{3}$, and $P_{2}P_{3}$, respectively. We unify the
notations with the supersonic reflection case by introducing points ${P_{1}}$
and ${P_{4}}$ for the subsonic reflection case as

${P_{1}}:={P_{0}},\quad{P_{4}}:={P_{0}},\quad\overline{\Gamma}_{\rm
sonic}:=\\{{P_{0}}\\}.$ (4.20)

The corresponding solution for $\theta_{\rm w}=\frac{\pi}{2}$ is called normal
reflection. In this case, the incident shock normally reflects from the flat
wall; see Fig. 4.3. The reflected shock is also a plane
$\\{\xi_{1}=\bar{\xi}_{1}\\}$, where $\bar{\xi}_{1}<0$.

Figure 4.3. Normal reflection configuration (cf. [35])

From the discussion above, it follows that a necessary condition for the
existence of a regular reflection solution is the existence of the uniform
state (2) with pseudo-potential $\varphi_{2}$ determined by the boundary
condition $D\varphi_{2}\cdot{\bm{\nu}}=0$ on the wedge and the Rankine-
Hugoniot conditions (4.16)–(4.17) across the flat shock
$\mathcal{S}_{1}=\\{\varphi_{1}=\varphi_{2}\\}$ separating it from state (1),
and satisfying the entropy condition: $\rho_{2}>\rho_{1}$. These conditions
lead to the system of algebraic equations (4.19) for the constant velocity
$\mathbf{u}_{2}$ and density $\rho_{2}$ of state (2). System (4.19) has
solutions for some but not all of the wedge angles. More specifically, for any
fixed densities $0<\rho_{0}<\rho_{1}$ of states (0) and (1), there exist a
sonic angle $\theta_{\rm w}^{\rm s}$ and a detachment angle $\theta_{\rm
w}^{\rm d}$ satisfying

$0<\theta_{\rm w}^{\rm d}<\theta_{\rm w}^{\rm s}<\frac{\pi}{2}$

such that the algebraic system (4.19) has two solutions for each $\theta_{\rm
w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, which become equal when
$\theta_{\rm w}=\theta_{\rm w}^{\rm d}$. Thus, for each $\theta_{\rm
w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, there exist two states (2),
called weak and strong, with densities $\rho_{2}^{\rm weak}<\rho_{2}^{\rm
strong}$. The weak state (2) is supersonic at the reflection point
${P_{0}}(\theta_{\rm w})$ for $\theta_{\rm w}\in(\theta_{\rm w}^{\rm
s},\frac{\pi}{2})$, sonic for $\theta_{\rm w}=\theta_{\rm w}^{\rm s}$, and
subsonic for $\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\hat{\theta}_{\rm
w}^{\rm s})$ for some $\hat{\theta}_{\rm w}^{\rm s}\in(\theta_{\rm w}^{\rm
d},\theta_{\rm w}^{\rm s}]$. The strong state (2) is subsonic at
${P_{0}}(\theta_{\rm w})$ for all $\theta_{\rm w}\in(\theta_{\rm w}^{\rm
d},\frac{\pi}{2})$.

There had been a long debate to determine which of the two states (2) for
$\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, weak or strong, is
physical for the local theory; see [8, 35, 52] and the references cited
therein. It was conjectured that the strong shock reflection-diffraction
configuration would be non-physical; indeed, it is shown in Chen-Feldman [33,
35] that the weak shock reflection-diffraction configuration tends to the
unique normal reflection in Fig. 4.3, but the strong reflection-diffraction
configuration does not, when the wedge angle $\theta_{\rm w}$ tends to
$\frac{\pi}{2}$. The entropy condition and the definition of weak and strong
states (2) imply that $0<\rho_{1}<\rho_{2}^{\rm weak}<\rho_{2}^{\rm strong}$,
which shows that the strength of the corresponding reflected shock near
$P_{0}$ in the weak shock reflection-diffraction configuration is relatively
weak, compared to the other shock given by the strong state (2).

If the weak state (2) is supersonic, the propagation speeds of the solution
are finite, and state (2) is completely determined by the local information:
state (1), state (0), and the location of point $P_{0}$. That is, any
information from the reflection-diffraction region, particularly the
disturbance at corner $P_{3}$, cannot travel towards the reflection point
$P_{0}$. However, if it is subsonic, the information can reach $P_{0}$ and
interact with it, potentially altering the subsonic reflection-diffraction
configuration. This argument motivated the following conjecture by von Neumann
in [105, 106]:

The Sonic Conjecture: There exists a supersonic regular shock reflection-
diffraction configuration when $\theta_{\rm w}\in(\theta_{\rm w}^{\rm
s},\frac{\pi}{2})$ for $\theta_{\rm w}^{\rm s}>\theta_{\rm w}^{\rm d}$. That
is, the supersonicity of the weak state (2) implies the existence of a
supersonic regular reflection solution, as shown in Fig. 4.2.

Another conjecture is that the global regular shock reflection-diffraction
configuration is possible whenever the local regular reflection at the
reflection point is possible:

The von Neumanm Detachment Conjecture: There exists a regular shock
reflection-diffraction configuration for any wedge angle $\theta_{\rm
w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$. That is, the existence of state
(2) implies the existence of a regular reflection solution, as shown in Figs.
4.2–4.2.

It is clear that the supersonic/subsonic regular shock reflection-diffraction
configurations are not possible without a local two-shock configuration at the
reflection point on the wedge, so the detachment conjecture is the weakest
possible criterion for the existence of supersonic/subsonic regular shock
reflection-diffraction configurations.

From now on, for the given wedge angle $\theta_{\rm w}\in(\theta_{\rm w}^{\rm
d},\frac{\pi}{2})$, state (2) represents the unique weak state (2) and
$\varphi_{2}$ is its pseudo-potential. We now show how the solutions of
regular shock reflection-diffraction configurations can be constructed. This
provides a solution to the von Neumann conjectures for potential flow. Note
that state (2) is obtained from the algebraic conditions described above,
which determine line $\mathcal{S}_{1}$ and the sonic arc $P_{1}P_{4}$ when
state (2) is supersonic at $P_{0}$, and the slope of $\Gamma_{\rm shock}$ at
$P_{0}$ (arc $P_{1}P_{4}$ on the boundary of $\Omega$ becomes a corner point
$P_{0}$) when state (2) is subsonic at $P_{0}$. Thus, the unknowns are domain
$\Omega$ (or equivalently, the curved part of the reflected shock $\Gamma_{\rm
shock}$) and the pseudo-potential $\varphi$ in $\Omega$. Then, from
(4.16)–(4.17), in order to construct a solution of Problem 4.2 of the
supersonic or subsonic regular shock reflection-diffraction configuration, it
suffices to solve the following problem:

######  Problem 4.3 (Free Boundary Problem).

For $\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, find a free
boundary $($curved reflected shock$)$ $\Gamma_{\rm
shock}\subset\Lambda\cap\\{{\xi_{1}}<\xi_{1{P_{1}}}\\}$ ($\Gamma_{\rm
shock}=P_{1}P_{2}$ on Fig. 4.2 and $\Gamma_{\rm shock}=P_{0}P_{2}$ on Fig.
4.2$)$ and a function $\varphi$ defined in region $\Omega$ as shown in Figs.
4.2–4.2 such that

  * (i)

Equation (4.8) is satisfied in $\Omega$, and the equation is strictly elliptic
for $\varphi$ in $\overline{\Omega}\setminus\overline{\Gamma}_{\rm sonic}$,

  * (ii)

$\varphi=\varphi_{1}$ and $\rho D\varphi\cdot{\bm{\nu}}_{\rm
s}=\rho_{1}D\varphi_{1}\cdot{\bm{\nu}}_{\rm s}$ on the free boundary
$\Gamma_{\rm shock}$,

  * (iii)

$\varphi=\varphi_{2}$ and $D\varphi=D\varphi_{2}$ on $P_{1}P_{4}$ in the
supersonic case as shown in Fig. 4.2 and at $P_{0}$ in the subsonic case as
shown in Fig. 4.2,

  * (iv)

$D\varphi\cdot{\bm{\nu}}_{\rm w}=0$ on $\Gamma_{\rm wedge}$, and
$D\varphi\cdot{\bm{\nu}}_{\rm sym}=0$ on $\Gamma_{\rm sym}$,

where ${\bm{\nu}}_{\rm s}$, ${\bm{\nu}}_{\rm w}$, and ${\bm{\nu}}_{\rm sym}$
are the interior unit normals to $\Omega$ on $\Gamma_{\rm shock}$,
$\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$, respectively.

Indeed, if $\varphi$ is a solution of Problem 4.3, we define its extension
from $\Omega$ to $\Lambda$ by setting:

$\varphi=\begin{cases}\,\varphi_{0}\qquad\,\mbox{for}\,\,\xi_{1}>\xi_{1}^{0}\mbox{
and }\xi_{2}>\xi_{1}\tan\theta_{\rm w},\\\\[2.84526pt]
\,\varphi_{1}\qquad\,\mbox{for}\,\,\xi_{1}<\xi_{1}^{0}\mbox{ and above
curve}\,\,P_{0}{P_{1}}{P_{2}},\\\\[2.84526pt] \,\varphi_{2}\qquad\,\mbox{in
region}\,\,P_{0}{P_{1}}{P_{4}},\end{cases}$ (4.21)

where we have used the notational convention (4.20) for the subsonic
reflection case, in which region $P_{0}{P_{1}}{P_{4}}$ is one point and curve
$P_{0}{P_{1}}{P_{2}}$ is $P_{0}{P_{2}}$; see Figs. 4.2–4.2. Also,
$\xi_{1}^{0}$ used in (4.21) is the location of the incident shock (cf.
(4.18)), and the extension by (4.21) is well-defined because of the
requirement that $\Gamma_{\rm
shock}\subset\Lambda\cap\\{{\xi_{1}}<\xi_{1{P_{1}}}\\}$ in Problem 4.3.

Note that the conditions in Problem 4.3(ii) are the Rankine-Hugoniot
conditions (4.16)–(4.17) on $\Gamma_{\rm shock}$ between $\varphi_{|\Omega}$
and $\varphi_{1}$. Since $\Gamma_{\rm shock}$ is a free boundary and equation
(4.8) is strictly elliptic for $\varphi$ in
$\overline{\Omega}\setminus\overline{\Gamma}_{\rm sonic}$, then two conditions
— the Dirichlet and oblique derivative conditions — on $\Gamma_{\rm shock}$
are consistent with one-phase free boundary problems for nonlinear elliptic
PDEs of second order (cf. [1, 3]).

In the supersonic case, the conditions in Problem 4.3(iii) are the Rankine-
Hugoniot conditions on $\Gamma_{\rm sonic}$ between $\varphi_{|\Omega}$ and
$\varphi_{2}$. Indeed, since state (2) is sonic on $\Gamma_{\rm sonic}$, then
it follows from (4.16)–(4.17) that no gradient jump occurs on $\Gamma_{\rm
sonic}$. Then, if $\varphi$ is a solution of Problem 4.3, its extension by
(4.21) is a weak solution of Problem 4.2. From now on, we consider a solution
of Problem 4.3 to be a function defined in $\Lambda$ by extension via (4.21).

Since $\Gamma_{\rm sonic}$ is not a free boundary (its location is fixed), it
is not possible in general to prescribe two conditions given in Problem
4.3(iii) on $\Gamma_{\rm sonic}$ for a second-order elliptic PDE. In the
iteration problem, we prescribe the condition: $\varphi=\varphi_{2}$ on
$\Gamma_{\rm sonic}$, and then prove that $D\varphi=D\varphi_{2}$ on
$\Gamma_{\rm sonic}$ by exploiting the elliptic degeneracy on $\Gamma_{\rm
sonic}$, as we describe below.

We observe that the key obstacle to prove the existence of regular shock
reflection-diffraction configurations as conjectured by von Neumann [105, 106]
is an additional possibility that, for some wedge angle $\theta_{\rm w}^{\rm
a}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, shock ${P_{0}}{P_{2}}$ may
attach to the wedge vertex ${P_{3}}$, as observed by experimental results (cf.
[103, Fig. 238]). To describe the conditions of such an attachment, we note
that

$\rho_{1}>\rho_{0},\qquad
u_{1}=(\rho_{1}-\rho_{0})\sqrt{\frac{2(\rho_{1}^{\gamma-1}-\rho_{0}^{\gamma-1})}{\rho_{1}^{2}-\rho_{0}^{2}}},\,\,\qquad
c_{1}=\rho_{1}^{\frac{\gamma-1}{2}}.$

Then it follows from the explicit expressions above that, for each $\rho_{0}$,
there exists $\rho^{\rm c}>\rho_{0}$ such that

$\displaystyle u_{1}\leq c_{1}\quad\mbox{if $\rho_{1}\in(\rho_{0},\rho^{\rm
c}]$};\,\,\qquad u_{1}>c_{1}\quad\mbox{if $\rho_{1}\in(\rho^{\rm
c},\infty)$}.$

If $u_{1}\leq c_{1}$, we can rule out the solution with a shock attached to
the wedge vertex. This is based on the fact that, if $u_{1}\leq c_{1}$, then
the wedge vertex $P_{3}=(0,0)$ lies within the sonic circle
$\overline{B_{c_{1}}((u_{1},0))}$ of state (1), and $\Gamma_{\rm shock}$ does
not intersect $\overline{B_{c_{1}}((u_{1},0))}$, as we show below.

If $u_{1}>c_{1}$, there would be a possibility that the reflected shock could
be attached to the wedge vertex as the experiments show $($e.g., [103, Fig.
238]$)$.

Thus, in [33, 35], we have obtained the following results:

######  Theorem 4.1.

There are two cases:

  1. (i)

If $\rho_{0}$ and $\rho_{1}$ are such that $u_{1}\leq c_{1}$, then the
supersonic/subsonic regular reflection solution exists for each wedge angle
$\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$. That is, for each
$\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$, there exists a
solution $\varphi$ of Problem 4.3 such that

$\Phi(t,{\bf x})=t\,\varphi(\frac{\bf x}{t})+\frac{|{\bf
x}|^{2}}{2t}\qquad\mbox{for}\,\,\frac{\bf x}{t}\in\Lambda,\,t>0$

with

$\rho(t,{\bf
x})=\Big{(}\rho_{0}^{\gamma-1}-(\gamma-1)\big{(}\Phi_{t}+\frac{1}{2}|\nabla_{\bf
x}\Phi|^{2}\big{)}\Big{)}^{\frac{1}{\gamma-1}}$

is a global weak solution of Problem 4.1 in the sense of Definition 4.2
satisfying the entropy condition; that is, $\Phi(t,{\bf x})$ is an entropy
solution.

  2. (ii)

If $\rho_{0}$ and $\rho_{1}$ are such that $u_{1}>c_{1}$, then there exists
$\theta_{\rm w}^{\rm a}\in[\theta_{\rm w}^{\rm d},\frac{\pi}{2})$ so that the
regular reflection solution exists for each wedge angle $\theta_{\rm
w}\in(\theta_{\rm w}^{\rm a},\frac{\pi}{2})$, and the solution is of self-
similar structure described in (i) above. Moreover, if $\theta_{\rm w}^{\rm
a}>\theta_{\rm w}^{\rm d}$, then, for the wedge angle $\theta_{\rm
w}=\theta_{\rm w}^{\rm a}$, there exists an attached solution, i.e., $\varphi$
is a solution of Problem 4.3 with ${P_{2}}={P_{3}}$.

The type of regular shock reflection-diffraction configurations $($supersonic
as in Fig. 4.2 or subsonic as in Fig. 4.2$)$ is determined by the type of
state (2) at ${P_{0}}$:

  1. (a)

For the supersonic and sonic reflection case, the reflected shock
${P_{0}}{P_{2}}$ is $C^{2,\alpha}$–smooth for some $\alpha\in(0,1)$ and its
curved part $P_{1}P_{2}$ is $C^{\infty}$ away from $P_{1}$. The solution
$\varphi$ is in $C^{1,\alpha}(\overline{\Omega})\cap C^{\infty}(\Omega)$, and
is $C^{1,1}$ across the sonic arc which is optimal; that is, $\varphi$ is not
$C^{2}$ across sonic arc.

  2. (b)

For the subsonic reflection case $($Fig. 4.2$)$, the reflected shock
${P_{0}}{P_{2}}$ and solution $\varphi$ in $\Omega$ is in $C^{1,\alpha}$ near
$P_{0}$ and $P_{3}$ for some $\alpha\in(0,1)$, and $C^{\infty}$ away from
$\\{P_{0},P_{3}\\}$.

Moreover, the regular reflection solution tends to the unique normal
reflection $($as in Fig. 4.3) when the wedge angle $\theta_{\rm w}$ tends to
$\frac{\pi}{2}$. In addition, for both supersonic and subsonic reflection
cases,

$\varphi_{2}<\varphi<\varphi_{1}\qquad\mbox{in $\Omega$}.$ (4.22)

Furthermore, $\varphi$ is an admissible solution in the sense of Definition
4.8 below, so that $\varphi$ satisfies further properties listed in Definition
4.8.

Theorem 4.1 is proved by solving Problem 4.3. The first results on the
existence of global solutions of the free boundary problem (Problem 4.3) were
obtained for the wedge angles sufficiently close to $\frac{\pi}{2}$ in Chen-
Feldman [33]. Later, in Chen-Feldman [35], these results were extended up to
the detachment angle as stated in Theorem 4.1. For this extension, the
techniques developed in [33], notably the estimates near the sonic arc, were
the starting point.

Case I: The wedge angles close to $\frac{\pi}{2}$. Let us first discuss the
techniques in [33], where we employ the approach of Chen-Feldman [29] to
develop an iteration scheme for constructing a global solution of Problem 4.3,
when the wedge angle $\theta_{\rm w}$ is close to $\frac{\pi}{2}$. For this
case, the solutions are of the supersonic regular shock reflection-diffraction
configuration as in Fig. 4.2. The general procedure is similar to the one
described in §2.2, which can be presented in the following four steps:

1\. Fix $\theta_{\rm w}$ sufficiently close to $\frac{\pi}{2}$ so that various
constants in the argument can be controlled. The iteration set consists of
functions defined on a region ${\mathcal{D}}$, where ${\mathcal{D}}$ contains
all possible $\Omega$ for the fixed $\theta_{\rm w}$. Specifically, an
important property of the regular shock reflection-diffraction configurations
is (4.22), which implies that $\Omega\subset\\{\varphi_{2}<\varphi_{1}\\}$;
that is, $\Omega$ lies below line $\mathcal{S}_{1}$ passing through $P_{0}$
and $P_{1}$ on Fig. 4.2. Note that, when $\theta_{\rm w}$ is close to
$\frac{\pi}{2}$, this line is close to the vertical reflected shock of normal
reflection on Fig. 4.3. Then ${\mathcal{D}}$ is defined as a region bounded by
$\mathcal{S}_{1}$, $\Gamma_{\rm sonic}=P_{1}P_{4}$, $\Gamma_{\rm
wedge}=P_{3}P_{4}$, and the symmetry line $\xi_{2}=0$. The iteration set is a
set of functions $\varphi$ on ${\mathcal{D}}$, defined by
$\varphi\geq\varphi_{2}$ on ${\mathcal{D}}$ and the bound of norm of
$\varphi-\varphi_{2}$ on ${\mathcal{D}}$ in the scaled and weighted
$C^{2,\alpha}$ space defined in (4.38) below. Such functions satisfy

$\|\varphi-\varphi_{2}\|_{C^{1,\alpha}(\overline{\mathcal{D}})}\leq
C(\frac{\pi}{2}-\theta_{\rm w}),$

which is small when $\frac{\pi}{2}-\theta_{w}\ll 1$, and

$\|\varphi-\varphi_{2}\|_{C^{1,1}(\overline{\mathcal{D}}\cap{\mathcal{N}}_{\varepsilon}(\Gamma_{\rm
sonic}))}<\infty.$

However,
$\|\varphi-\varphi_{2}\|_{C^{1,1}(\overline{\mathcal{D}}\cap{\mathcal{N}}_{\varepsilon}(\Gamma_{\rm
sonic}))}$ is not small even if $\frac{\pi}{2}-\theta_{\rm w}$ is small; the
reasons for that will be discussed below.

Given a function $\hat{\varphi}$ from the iteration set, we define domain
$\Omega(\hat{\varphi}):=\\{\hat{\varphi}<\varphi_{1}\\}$ so that the iteration
free boundary is $\Gamma_{\rm
shock}(\hat{\varphi})=\partial\Omega(\hat{\varphi})\cap{\mathcal{D}}$. This is
similar to (2.41), and the corresponding non-degeneracy similar to (2.40) in
the present case is:

$\partial_{\xi_{1}}(\varphi_{1}-\varphi_{2}-\phi)\geq\frac{u_{1}}{2}\qquad\,\,\mbox{in
${\mathcal{D}}$ \, if $\|\phi\|_{C^{1}(\overline{{\mathcal{D}}})}$ and
$\frac{\pi}{2}-\theta_{\rm w}$ are small}.$

Then we define the iteration equation by using form (4.12) of equation (4.8),
by making an elliptic truncation (which is somewhat different from Step 1 in
§2.2) and substituting $\hat{\varphi}$ in some terms of the coefficients of
(4.12). The iteration boundary condition on $\Gamma_{\rm
shock}(\hat{\varphi})$ is an oblique derivative condition obtained by
combining two conditions in Problem 4.3(ii) and making some truncations. On
$\Gamma_{\rm sonic}$, we prescribe $\varphi=\varphi_{2}$, i.e., one of two
conditions in Problem 4.3(iii). On $\Gamma_{\rm wedge}$ and $\Gamma_{\rm
sym}(\hat{\varphi})$, we prescribe the conditions given in Problem 4.3(iv).
The iteration map: $\hat{\varphi}\to\varphi$ is defined by solving the
iteration problem to obtain $\varphi$ and then extending $\varphi$ from
$\Omega(\hat{\varphi})$ to ${\mathcal{D}}$.

The fundamental differences between the iteration procedure in the shock
reflection-diffraction problem and the previous procedures on transonic shocks
in the steady case in §2–§3 (such as [29, 30, 32, 111] and follow-up papers)
include:

  1. (i)

The procedures on steady transonic shocks in §2–§3 are for the perturbation
case. In particular, the ellipticity of the iteration equation and the removal
of the elliptic cutoff are achieved by making the iteration set sufficiently
close to the background solution in $C^{1}$ or a stronger norm. For the
regular reflection problem, this cannot be done because of the elliptic
degeneracy near the sonic arc.

  2. (ii)

Only one condition on $\Gamma_{\rm sonic}$ can be prescribed; however, both
$\varphi=\varphi_{2}$ and $D\varphi=D\varphi_{2}$ on $\Gamma_{\rm sonic}$ are
needed to be matched to obtain a global entropy solution. This is resolved by
exploiting the elliptic degeneracy on $\Gamma_{\rm sonic}$.

2\. In order to see the elliptic degeneracy on $\Gamma_{\rm sonic}$ more
explicitly, we fix the wedge angle $\theta_{\rm w}$ and the corresponding
pseudo-potential $\varphi_{2}=\varphi_{2}^{(\theta_{\rm w})}$ of the weak
state (2), and rewrite equation (4.10) in terms of the function:

$\psi=\varphi-\varphi_{2}$

in the following coordinates flattening $\Gamma_{\rm sonic}$:

$x=c_{2}-r,\qquad y=\theta-\theta_{\rm w},$ (4.23)

where $(r,\theta)$ are the polar coordinates centered at
$O_{2}=\mathbf{u}_{2}$ of the sonic circle of state (2). Then

$\Omega_{\varepsilon}:=\Omega\cap{\mathcal{N}}_{\varepsilon}(\Gamma_{\rm
sonic})\subset\\{x>0\\}\,\,$ for small $\varepsilon>0$, $\Gamma_{\rm
sonic}\subset\\{x=0\\}$.

In what follows, we always assume that $\varphi\in
C^{1,1}(\overline{\Omega}_{\varepsilon})$ as in Theorem 4.1 for the supersonic
case. Then, by the conditions in Problem 4.3(iii) and the definition of
$\psi$,

$\displaystyle\psi=0\qquad\quad\,\mbox{on $\Gamma_{\rm sonic}$},$ (4.24)
$\displaystyle D\psi=0\qquad\,\,\mbox{on $\Gamma_{\rm sonic}$}.$ (4.25)

Moreover, we a priori assume that solution $\varphi$ satisfies (4.22) in
$\Omega$ to derive the required estimates of the solution; with these
estimates, we then construct such a solution and verify that it satisfies
(4.22). The heuristic motivation of (4.22) is the following: From Figs.
4.2–4.2, it appears that $\Gamma_{\rm shock}$ (and hence $\Omega$) is located
below line $\mathcal{S}_{1}$, i.e., in the half-plane
$\\{\varphi_{1}>\varphi_{2}\\}$. Thus, $\varphi=\varphi_{1}>\varphi_{2}$ on
$\Gamma_{\rm shock}$, and $\varphi_{1}>\varphi_{2}=\varphi$ on $\Gamma_{\rm
sonic}$. Also, the potential functions $\phi_{1}$ and $\phi_{2}$ of states (1)
and (2) are linear functions, thus they satisfy equation (4.12) with
coefficients determined by $\varphi$, considered as a linear equation for
$\phi$. Taking into account the inequalities on $\Gamma_{\rm shock}$ and
$\Gamma_{\rm sonic}$ noted above, and the oblique boundary conditions on
$\Gamma_{\rm wedge}$ and $\Gamma_{\rm sym}$, we obtain (4.22) by the maximum
principle. Then, from (4.22), we have

$\psi>0\qquad\mbox{in $\Omega$}.$ (4.26)

Even though the previous argument is heuristic, the fact that it comes from
the structure of the problem allows us to include the condition that $\psi\geq
0$ in the definition of the iteration set and close the iteration argument for
constructing the solutions within this set.

Equation (4.10) in $\Omega\cap{\mathcal{N}}_{\varepsilon}(\Gamma_{\rm sonic})$
for $\psi$ in the $(x,y)$–coordinates (4.23) is

$\big{(}2x-(\gamma+1)\psi_{x}+O_{1}\big{)}\psi_{xx}+O_{2}\psi_{xy}+({1\over
c_{2}}+O_{3})\psi_{yy}-(1+O_{4})\psi_{x}+O_{5}\psi_{y}=0,$ (4.27)

where

$\begin{split}O_{1}(D\psi,\psi,x)&=-\frac{x^{2}}{c_{2}}+{\gamma+1\over
2c_{2}}(2x-\psi_{x})\psi_{x}-{\gamma-1\over c_{2}}\Big{(}\psi+{1\over
2(c_{2}-x)^{2}}\psi_{y}^{2}\Big{)},\\\
O_{2}(D\psi,\psi,x)&=-\frac{2(\psi_{x}+c_{2}-x)\psi_{y}}{c_{2}(c_{2}-x)^{2}},\\\
O_{3}(D\psi,\psi,x)&={1\over
c_{2}(c_{2}-x)^{2}}\Big{(}x(2c_{2}-x)-(\gamma-1)\big{(}\psi+(c_{2}-x)\psi_{x}+{1\over
2}\psi_{x}^{2}\big{)}-\frac{\gamma+1}{2(c_{2}-x)^{2}}\psi_{y}^{2}\Big{)},\\\
O_{4}(D\psi,\psi,x)&=\frac{1}{c_{2}-x}\Big{(}x-{\gamma-1\over
c_{2}}\big{(}\psi+(c_{2}-x)\psi_{x}+{1\over
2}\psi_{x}^{2}+\frac{(\gamma+1)\psi_{y}^{2}}{2(\gamma-1)(c_{2}-x)^{2}}\big{)}\Big{)},\\\
O_{5}(D\psi,\psi,x)&=-\frac{2(\psi_{x}+c_{2}-x)\psi_{y}}{c_{2}(c_{2}-x)^{3}}.\end{split}$
(4.28)

Since $\psi\in C^{1,1}(\overline{\Omega}_{\varepsilon})$, it follows from
(4.24)–(4.25) that $|\psi(x,y)|\leq Cx^{2}$ and

$|D\psi(x,y)|\leq Cx\qquad\mbox{in $\Omega_{\varepsilon}$},$ (4.29)

so that

$|O_{1}(D\psi,\psi,x)|\leq N|x|^{2},\quad|O_{k}(D\psi,\psi,x)|\leq
N|x|\qquad\,\,\,\mbox{for $k=2,\dots,5$}.$ (4.30)

Using (4.30), we can show that $O_{k}(D\psi,\psi,x)$ are small perturbations
of the leading terms of equation (4.27) in
$\Omega_{\varepsilon}=\Omega\cap{\mathcal{N}}_{\varepsilon}(\Gamma_{\rm
sonic})$. Also, if (4.29) holds, equation (4.27) is strictly elliptic in
$\overline{\Omega}_{\varepsilon}\setminus\overline{\Gamma_{\rm sonic}}$ if

$\psi_{x}(x,y)\leq\frac{2\mu}{\gamma+1}x$ (4.31)

for $\mu\in(0,1)$, when $\varepsilon=\varepsilon(\mu,N)$ is small. For
$\theta_{\rm w}$ close to $\frac{\pi}{2}$, it can be shown that any solution
of Problem 4.3 (with some natural regularity properties) satisfies that, for
any small $\delta>0$,

$|\psi_{x}(x,y)|\leq\frac{1+\delta}{\gamma+1}x\qquad\mbox{in
$\Omega_{\varepsilon}$ for small $\varepsilon=\varepsilon(\delta)$,}$ (4.32)

which verifies (4.31) with any $\mu\in(\frac{1}{2},1)$ (e.g., with
$\mu=\frac{2}{3}$) if $\delta$ is correspondingly small.

3\. The iteration equation near $\Gamma_{\rm sonic}$ is defined based on the
above facts. The iteration set ${\mathcal{K}}_{M}$ used in [33] is such that
every $\hat{\psi}=\hat{\varphi}-\varphi_{2}\in{\mathcal{K}}_{M}$ satisfies
(4.24) and (4.29) for some $C,\varepsilon>0$. Then the iteration equation for
$\psi$ is

$\big{(}2x-(\gamma+1)x\eta(\frac{\psi_{x}}{x})+O_{1}^{(\hat{\psi})}\big{)}\psi_{xx}+O_{2}^{(\hat{\psi})}\psi_{xy}+({1\over
c_{2}}+O_{3}^{(\hat{\psi})})\psi_{yy}-(1+O_{4}^{(\hat{\psi})})\psi_{x}+O_{5}^{(\hat{\psi})}\psi_{y}=0,$
(4.33)

where the cutoff function $\eta\in C^{\infty}({\mathbb{R}})$ satisfies
$|\eta|\leq\frac{5}{3(\gamma+1)}$, $\eta^{\prime}\geq 0$, and $\eta(s)=s$ if
$|s|\leq\frac{4}{3(\gamma+1)}$, and some other technical conditions. The terms
$O_{k}^{(\hat{\psi})}$, $k=1,\dots,5$, are obtained from $O_{k}$ by
substituting $\hat{\psi}$ into certain terms in (4.28) and performing the
cutoff in the remaining terms, so that estimates (4.30) hold. Then (4.33) is
strictly elliptic in
$\overline{\Omega}_{\varepsilon}\setminus\overline{\Gamma_{\rm sonic}}$ for
small $\varepsilon$, and its ellipticity degenerates on $\Gamma_{\rm sonic}$.
Since the solution of Problem 4.3 satisfies equation (4.27) and inequality
(4.32) with $\delta=\frac{1}{3}$ in $\Omega_{\varepsilon}$ for small
$\varepsilon$, then it satisfies equation (4.33) in $\Omega_{\varepsilon}$
with $\hat{\psi}=\psi$. Indeed, we have the estimate:
$|\psi_{x}|\leq\frac{4}{3(\gamma+1)}x$, so that
$x\eta(\frac{\psi_{x}}{x})=\psi_{x}$; and the cutoffs in the terms of
$O_{k}^{(\hat{\psi})}$ are removed similarly.

We also note that the degenerate ellipticity structure of equation (4.33) is
the following: Writing (4.33) in the form

$\displaystyle\sum_{i,j=1}^{2}A_{ij}(D\psi,\psi,x)D_{ij}\psi+\sum_{i=1}^{2}A_{i}(D\psi,\psi,x)D_{i}\psi=0$
(4.34)

with $A_{12}=A_{21}$, we see that, for any
${\bm{\xi}}=(\xi_{1},\xi_{2})\in{\mathbb{R}}^{2}$,

$\lambda|{\bm{\xi}}|^{2}\leq A_{11}({\bf
p},z,x)\frac{\xi_{1}^{2}}{x}+2A_{12}({\bf
p},z,x)\frac{\xi_{1}\xi_{2}}{x^{1/2}}+A_{22}({\bf
p},z,x)\xi_{2}^{2}\leq\frac{1}{\lambda}|{\bm{\xi}}|^{2}$ (4.35)

for all $({\bf p},z)\in{\mathbb{R}}^{2}\times{\mathbb{R}}$ and
$\mathbf{x}\in(0,\varepsilon)$.

We consider solutions of (4.33) in $\Omega_{\varepsilon}$ satisfying (4.24)
and (4.26). Since condition (4.25) can not be prescribed in the iteration
problem as discussed above, we have to obtain (4.25) from the estimates of the
solutions by exploiting the elliptic degeneracy. The estimates of the positive
solutions of (4.33) with (4.24) in $\Omega_{\varepsilon}$ are based on the
fact that, for any $\delta>0$, the function:

$w_{\delta}(x,y)=\frac{1+\delta}{2(\gamma+1)}x^{2}$

is a supersolution of (4.33) in $\Omega_{\varepsilon}$ if
$\varepsilon=\varepsilon(\delta)$ is small; that is,
${\mathcal{N}}(w_{\delta})<0$ in $\Omega_{\varepsilon}$, where
${\mathcal{N}}(\cdot)$ denotes the operator determined by the left-hand side
of (4.33). Using this, the boundary conditions on $\Gamma_{\rm shock}$ and
$\Gamma_{\rm wedge}$, and (4.26), we obtain by the comparison principle that

$0\leq\psi\leq Cx^{2}\qquad\,\,\mbox{in $\Omega_{\varepsilon}$},$ (4.36)

where $\varepsilon$ and $C$ are uniform with respect to the wedge angles near
$\frac{\pi}{2}$. Note that $-w_{\delta}$ is not a subsolution of (4.27) so
that it cannot be used to bound $\psi$ from below. Thus, property (4.26),
which is derived from the global structure of the solution, is crucially used
in this argument. Then, in (4.36), the upper bound is from the local estimates
near $\Gamma_{\rm sonic}$, while the lower bound is from the global structure
of the problem.

In particular, (4.36) implies that $D\psi=0$ on $\Gamma_{\rm sonic}$, which
resolves the issue described in (ii) above. Furthermore, from (4.36), using
the non-isotropic parabolic rescaling corresponding to the elliptic degeneracy
(4.35) of equation (4.33) near $x=0$, we obtain the estimates in the
appropriately weighted and scaled Hölder norm in $\Omega_{\varepsilon}$, which
also imply the uniform $C^{1,1}$ estimates:

$|D^{2}\psi|\leq C\qquad\mbox{in $\Omega_{\varepsilon}$}.$ (4.37)

More precisely, we denote this norm by
$\|\psi\|_{2,\alpha,{\Omega_{\varepsilon}}}^{\rm(par)}$ and define it as
follows: Denote $\mathbf{z}=(x,y)$ and
$\tilde{\mathbf{z}}=(\tilde{x},\tilde{y})$ with
$x,\tilde{x}\in(0,2\varepsilon)$ and

$\delta^{\rm(par)}_{\alpha}(\mathbf{z},\tilde{\mathbf{z}}):=\left(|x-\tilde{x}|^{2}+\max(x,\tilde{x})|y-\tilde{y}|^{2}\right)^{\alpha/2}.$

Then, for $\psi\in C^{2}({\Omega_{\varepsilon}})\cap
C^{1,1}(\overline{\Omega_{\varepsilon}})$ written in the $(x,y)$–coordinates,
we define

$\begin{split}&\qquad\,\,\|\psi\|^{\rm(par)}_{2,0,{\Omega_{\varepsilon}}}:=\sum_{0\leq
k+l\leq
2}\sup_{\mathbf{z}\in{\Omega_{\varepsilon}}}\left(x^{k+l/2-2}|\partial_{x}^{k}\partial_{y}^{l}\psi(\mathbf{z})|\right),\\\
&\qquad\,\,[\psi]^{\rm(par)}_{2,\alpha,{\Omega_{\varepsilon}}}:=\sum_{k+l=2}\sup_{\mathbf{z},\tilde{\mathbf{z}}\in{\Omega_{\varepsilon}},\mathbf{z}\neq\tilde{\mathbf{z}}}\bigg{(}\min(x^{k+l/2-2},\tilde{x}^{k+l/2-2})\frac{|\partial_{x}^{k}\partial_{y}^{l}\psi(\mathbf{z})-\partial_{x}^{k}\partial_{y}^{l}\psi(\tilde{\mathbf{z}})|}{\delta^{\rm(par)}_{\alpha}(\mathbf{z},\tilde{\mathbf{z}})}\bigg{)},\\\
&\qquad\,\,\|\psi\|^{\rm(par)}_{2,\alpha,{\Omega_{\varepsilon}}}:=\|\psi\|^{\rm(par)}_{2,0,{\Omega_{\varepsilon}}}+[\psi]^{\rm(par)}_{2,\alpha,{\Omega_{\varepsilon}}}.\end{split}$
(4.38)

Now we obtain the required estimates in the norm in (4.38), under the
assumption that (4.36) holds in $\Omega_{2\varepsilon}$. For every
$\mathbf{z}_{0}=(x_{0},y_{0})\in\overline{\Omega}_{\varepsilon}\setminus\Gamma_{\rm
sonic}$ (so that $x_{0}\in(0,\varepsilon]$), we define

$R_{\mathbf{z}_{0}}=\Big{\\{}(x,y)\;\;:\;\;|x-x_{0}|<\frac{x_{0}}{10},\;|y-y_{0}|<\frac{\sqrt{x_{0}}}{10}\Big{\\}}\cap\Omega.$
(4.39)

Note that $\mbox{dist}(R_{\mathbf{z}_{0}},\Gamma_{\rm
sonic})=\frac{9}{10}x_{0}>0$. We rescale the rectangle in (4.39) to the unit
square $Q_{1}=(-1,1)^{2}$:

$Q_{1}^{(\mathbf{z}_{0})}:=\Big{\\{}(S,T)\in
Q_{1}\;:\;(x_{0}+\frac{x_{0}}{10}S,\;y_{0}+\frac{\sqrt{x_{0}}}{10}T)\in\Omega\Big{\\}},$
(4.40)

and define the scaled version of $\psi$ in the $(S,T)$–coordinates in
$Q_{1}^{(\mathbf{z}_{0})}$:

$\psi^{(\mathbf{z}_{0})}(S,T):=\frac{1}{x_{0}^{2}}\psi(x_{0}+\frac{x_{0}}{10}S,\;y_{0}+\frac{\sqrt{x_{0}}}{10}T)\qquad\mbox{for
$(S,T)\in Q_{1}^{(z_{0})}$}.$ (4.41)

Note that this rescaling is non-isotropic with respect to the two variables
$x$ and $y$. By (4.36), we have

$\|\psi^{(\mathbf{z}_{0})}\|_{L^{\infty}(\overline{Q_{1}^{(\mathbf{z}_{0})}})}\leq
C\qquad\text{for any
$\mathbf{z}_{0}=(x_{0},y_{0})\in\overline{\Omega}_{\varepsilon}\setminus\overline{\Gamma_{\rm
sonic}}$}.$ (4.42)

Rewriting equation (4.33) in terms of $\psi^{(\mathbf{z}_{0})}$ in the
$(S,T)$–coordinates and noting the degenerate ellipticity structure (4.35), we
find that $\psi^{(\mathbf{z}_{0})}$ satisfies a uniformly elliptic equation in
$Q_{1}^{(\mathbf{z}_{0})}$ with the ellipticity constants and certain Hölder
norms of the coefficients independent of $\mathbf{z}_{0}$. We also rescale the
boundary conditions on $\Gamma_{\rm shock}\cap\partial\Omega_{\varepsilon}$
and $\Gamma_{\rm wedge}\cap\partial\Omega_{\varepsilon}$ in a similar way,
when $\mathbf{z}_{0}$ is on the corresponding part of the boundary. Then we
apply the local elliptic $C^{2,\alpha}$–estimates for
$\psi^{(\mathbf{z}_{0})}$ in $Q_{1}^{(\mathbf{z}_{0})}$ in the following
cases:

  1. (i)

Interior rectangles $R_{\mathbf{z}_{0}}$, i.e., all $\mathbf{z}_{0}$ such that
$Q_{1}^{(\mathbf{z}_{0})}=Q_{1}$ holds,

  2. (ii)

Rectangles $R_{\mathbf{z}_{0}}$ centered on the shock:
$\mathbf{z}_{0}\in\Gamma_{\rm shock}\cap\partial\Omega_{\varepsilon}$,

  3. (iii)

Rectangles $R_{\mathbf{z}_{0}}$ centered on the wedge:
$\mathbf{z}_{0}\in\Gamma_{\rm wedge}\cap\partial\Omega_{\varepsilon}$,

where, in the last two cases, we use the local estimates for the corresponding
boundary value problems. Using (4.42), we obtain

$\|\psi^{(\mathbf{z}_{0})}\|_{C^{2,\alpha}(\overline{Q_{1/2}^{(\mathbf{z}_{0})}})}\leq
C\qquad\,\,\mbox{with $C$ independent of $\mathbf{z}_{0}$},$

where
$Q_{1/2}^{(\mathbf{z}_{0})}=Q_{1}^{(\mathbf{z}_{0})}\cap(-\frac{1}{2},\;\frac{1}{2})^{2}$.
Rewriting in terms of $\psi$ in the $(x,y)$–coordinates and combining the
estimates for all $\mathbf{z}_{0}$ as above, we obtain the estimate:
$\|\psi\|_{2,\alpha,{\Omega_{\varepsilon}}}^{\rm(par)}\leq C$ in norm (4.38),
which also implies the $C^{1,1}$–estimates (4.37).

######  Remark 4.5.

Note that
$\psi^{(\mathbf{z}_{0})}_{SS}(S,T)=\frac{1}{100}\psi_{xx}(x_{0}+\frac{x_{0}}{10}S,\;y_{0}+\frac{\sqrt{x_{0}}}{10}T)$.
It follows that $\|D^{2}\psi\|_{L^{\infty}}$ cannot be made small by choosing
the parameters, e.g., choosing $\varepsilon$ small or $\theta_{\rm w}$ close
to $\frac{\pi}{2}$.

######  Remark 4.6.

The above argument, starting from (4.39), is also used for the a priori
estimates of the positive solutions of (4.27)–(4.28) with condition (4.24),
satisfying (4.29) and the ellipticity condition (4.31) with some
$\mu\in(0,1)$. Note that (4.24), (4.29), and $\psi\geq 0$ imply (4.36), which
is used in the argument.

######  Remark 4.7.

Remark 4.6 applies only to the positive solutions of (4.27) with condition
(4.24). For the negative solutions of (4.27) with condition (4.24), the
equation is uniformly elliptic up to $\\{x=0\\}$ and, similar to Hopf’s lemma,
the negative solutions have linear growth: $|\psi(x,y)|\geq\frac{1}{C}x$, in a
contrast with (4.36). This feature is used in the proof of certain geometric
properties of the free boundary for the wedge angles away from
$\frac{\pi}{2}$, where we note that $\varphi-\varphi_{1}<0$ by (4.22).

4\. In order to remove the ellipticity cutoff in (4.33), i.e., to show that
the fixed point solution of (4.33) (i.e., with $\psi=\hat{\psi}$) actually
satisfies (4.27), we need to show that $|\psi_{x}|\leq\frac{4}{3(\gamma+1)}x$,
as we have discussed right after (4.33). Combining (4.37) with $D\psi=0$ on
$\Gamma_{\rm sonic}$, we obtain that $|D\psi(x,y)|\leq Cx$ in
$\Omega_{\varepsilon}$, which does not remove the ellipticity cutoff, unless
we show the explicit bound $C\leq\frac{4}{3(\gamma+1)}$. However, this bound
does not follow from the estimates discussed above (cf. Remark 4.5).

Note that the only explicit solution we know is the normal reflection for
$\theta_{\rm w}=\frac{\pi}{2}$, for which
$\varphi=\varphi_{2}^{(\frac{\pi}{2})}$, i.e., $\psi=0$ in $\Omega$. Also, the
analysis in Bae-Chen-Feldman [5] has shown that the solutions of Problem 4.3
for the supersonic regular shock reflection-diffraction configuration satisfy
that, for small $\varepsilon$,

$\psi_{x}\thicksim\frac{x}{\gamma+1}\qquad\mbox{in
$\Omega_{\varepsilon}\cap\\{(x,y)\;:\;\mbox{dist}\,((x,y),\Gamma_{\rm
shock})>\sqrt{x}\\}$,}$

but

$D\psi=o(x)\qquad\mbox{in
$\Omega_{\varepsilon}\cap\\{(x,y)\;:\;\mbox{dist}\,((x,y),\Gamma_{\rm
shock})<x^{2}\\}$}.$

This shows that the convergence of solutions $\varphi^{(\theta_{\rm w})}$ of
Problem 4.3 to $\varphi^{(\frac{\pi}{2})}$ as $\theta_{\rm
w}\to{\frac{\pi}{2}}^{-}$ does not hold in $C^{2}$ up to the sonic arc
$\Gamma_{\rm sonic}$ (but holds in $C^{1,\alpha}$) after mapping
$\Omega^{(\theta_{\rm w})}$ to a fixed domain for all $\theta_{\rm w}$.
Moreover, the difference between the behaviors of $D\psi$ near $\Gamma_{\rm
shock}$ and away from $\Gamma_{\rm shock}$ within $\Omega_{\varepsilon}$ shows
that there is no clear background solution such that the appropriate iteration
set would lie in its small neighborhood in the norm sufficiently strong to
remove the ellipticity cutoff in (4.33) by the smallness of the norm. Then, in
order to remove the ellipticity cutoff for the fixed point of the iteration,
we derive an equation for $\psi_{x}$ in $\Omega_{\varepsilon}$ and boundary
conditions on $\Gamma_{\rm shock}\cap\\{x<\varepsilon\\}$ and $\Gamma_{\rm
wedge}\cap\\{x<\varepsilon\\}$, and prove that

$\psi_{x}\leq\frac{4}{3(\gamma+1)}x$

from this boundary value problem, if the wedge angle $\theta_{\rm w}$ is
sufficiently close to $\frac{\pi}{2}$. The estimate from below:

$\psi_{x}\geq-\frac{4}{3(\gamma+1)}x$

is proved from the global setting of Problem 4.3 under the same condition on
$\theta_{\rm w}$. This use of the local and global structure is similar to
that in the proof of (4.36).

Note that, in this argument for the wedge angles near $\frac{\pi}{2}$, the
non-perturbative nature of the problem is seen only in the estimates of the
solution near $\Gamma_{\rm sonic}$, specifically in the fact that $D^{2}\psi$
on $\Gamma_{\rm sonic}$ does not tend to zero as $\theta_{\rm
w}\to\frac{\pi}{2}$. The free boundary $\Gamma_{\rm shock}$ in this case is
near $\mathcal{S}_{1}(\theta_{\rm w})$, and also close to the reflected shock
of the normal reflection as in Fig. 4.3, which is the vertical line
$\mathcal{S}_{1}(\frac{\pi}{2})$. Also, $\|\varphi-\varphi_{2}^{(\theta_{\rm
w})}\|_{C^{1}(\Omega)}\leq C(\frac{\pi}{2}-\theta_{\rm w})$, which is small.
Thus, away from $\Gamma_{\rm sonic}$, the argument is perturbative for the
wedge angles near $\frac{\pi}{2}$. In the case of general wedge angles in
Theorem 4.1, the free boundary $\Gamma_{\rm shock}$ is no longer close to a
line, its structure is not known a priori, thus the study of geometric
properties of the free boundary is a part of the argument.

Case II. General wedge angles up to the detachment angle. For the general case
and the proof of Theorem 4.1, we follow the approach introduced in Chen-
Feldman [35]. Similar to the case of wedge angles near $\frac{\pi}{2}$ where
we have restricted our consideration to the class of solutions satisfying
$\psi\geq 0$ in $\Omega$ and established the existence of such solutions, for
the general case, we define a class of admissible solutions, make the
necessary a priori estimates of such solutions, and then employ these
estimates to prove the existence of solutions in this class. Our motivation
for the definition of admissible solutions is from the following properties of
supersonic regular reflection solutions $\varphi$ for the wedge angles close
to $\frac{\pi}{2}$; or more generally, for the supersonic regular reflection
solutions $\varphi$ satisfying that $\|\varphi-\varphi_{2}^{(\theta_{\rm
w})}\|_{C^{1}(\Omega)}$ is small: If (4.8) is strictly elliptic for $\varphi$
in $\overline{\Omega}\setminus\overline{\Gamma}_{\rm sonic}$, then $\varphi$
satisfies (4.22) and the monotonicity properties:

$\partial_{\xi_{2}}(\varphi_{1}-\varphi)\leq 0,\quad
D(\varphi_{1}-\varphi)\cdot{\bf e}_{\mathcal{S}_{1}}\leq
0\qquad\,\,\,\,\mbox{in $\Omega\,\,$ for ${\bf
e}_{\mathcal{S}_{1}}=\frac{{P_{0}P_{1}}}{|P_{0}P_{1}|}$}.$ (4.43)

We now present the outline of the proof of Theorem 4.1 in the following four
steps:

1\. Motivated by the discussion above, for the general case, we define the
admissible solutions as the solutions of Problem 4.3 (thus the solutions with
weak regular reflection-diffraction configuration of either supersonic or
subsonic type) satisfying the following properties:

######  Definition 4.8.

Let $\theta_{\rm w}\in(\theta_{\rm w}^{\rm d},\frac{\pi}{2})$. A function
$\varphi\in C^{0,1}(\overline{\Lambda})$ is an admissible solution of the
regular reflection problem if $\varphi$ is a solution of Problem 4.3 extended
to $\Lambda$ by (4.21) $($where $P_{0}P_{1}P_{4}$ is a point in the subsonic
and sonic cases$)$ and satisfies the following properties:

  1. (i)

The structure of solutions:

     * •

If $|D\varphi_{2}(P_{0})|>c_{2}$, then $\varphi$ is of the supersonic regular
shock reflection-diffraction configuration shown on Fig. 4.2 and satisfies
that the curved part of reflected-diffracted shock $\Gamma_{\text{\rm shock}}$
is $C^{2}$ in its relative interior; curves $\Gamma_{\text{\rm shock}}$,
$\Gamma_{\rm sonic}$, $\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$ do not have
common points except their endpoints; $\varphi\in C^{0,1}(\Lambda)\cap
C^{1}(\Lambda\setminus({\mathcal{S}_{0}}\cup\overline{P_{0}P_{1}P_{2}}))$ and
$\varphi\in C^{1}(\overline{\Omega})\cap
C^{3}(\overline{\Omega}\setminus(\overline{\Gamma}_{\text{\rm
sonic}}\cup\\{P_{2},P_{3}\\}))$.

     * •

If $|D\varphi_{2}(P_{0})|\leq c_{2}$, then $\varphi$ is of the subsonic
regular shock reflection-diffraction configuration shown on Fig. 4.2 and
satisfies that the reflected-diffracted shock $\Gamma_{\text{\rm shock}}$ is
$C^{2}$ in its relative interior; curves $\Gamma_{\text{\rm shock}}$,
$\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$ do not have common points except
their endpoints; $\varphi\in C^{0,1}(\Lambda)\cap
C^{1}(\Lambda\setminus({\mathcal{S}_{0}}\cup\overline{\Gamma}_{\text{\rm
shock}}))$ and $\varphi\in C^{1}(\overline{\Omega})\cap
C^{3}(\overline{\Omega}\setminus\\{P_{0},P_{3}\\})$.

Moreover, in both the supersonic and subsonic cases, the extended curve
$\Gamma_{\text{\rm shock}}^{\rm ext}:=\Gamma_{\text{\rm
shock}}\cup\\{P_{0}\\}\cup\Gamma_{\text{\rm shock}}^{-}$ is $C^{1}$ in its
relative interior, where $\Gamma_{\text{\rm shock}}^{-}$ is the reflection of
$\Gamma_{\text{\rm shock}}$ with respect to the $\xi_{1}$–axis.

  2. (ii)

Equation (4.8) is strictly elliptic in
$\overline{\Omega}\setminus\,\overline{\Gamma_{\text{\rm sonic}}}$, i.e.,
$|D\varphi|<c(|D\varphi|^{2},\varphi)$ in
$\overline{\Omega}\setminus\,\overline{\Gamma_{\text{\rm sonic}}}$.

  3. (iii)

$\partial_{{\bm{\nu}}}\varphi_{1}>\partial_{{\bm{\nu}}}\varphi>0$ on
$\Gamma_{\text{\rm shock}}$, where ${\bm{\nu}}$ is the normal to
$\Gamma_{\text{\rm shock}}$, pointing to the interior of $\Omega$.

  4. (iv)

Inequalities hold:

$\varphi_{2}\leq\varphi\leq\varphi_{1}\qquad\mbox{in $\Omega$}.$ (4.44)

  5. (v)

(4.43) is satisfied, where vector ${\bf e}_{\mathcal{S}_{1}}$ is defined as
the unit vector parallel to $\mathcal{S}_{1}$ and pointing into $\Lambda$ at
$P_{0}$ for the general case.

Note that (4.43) implies that

$D(\varphi_{1}-\varphi)\cdot{\bf e}\leq 0\qquad\,\,\mbox{in
$\overline{\Omega}\,\,$ for all ${\bf e}\in\overline{Cone({\bf
e}_{\xi_{2}},{\bf e}_{\mathcal{S}_{1}})}$},$ (4.45)

where $Cone({\bf e}_{\xi_{2}},{\bf e}_{\mathcal{S}_{1}})=\\{a\,{\bf
e}_{\xi_{2}}+b\,{\bf e}_{\mathcal{S}_{1}}\;:\;a,b>0\\}$ with ${\bf
e}_{\xi_{2}}=(0,1)$. Notice that ${\bf e}_{\xi_{2}}$ and ${\bf
e}_{\mathcal{S}_{1}}$ are not parallel if $\theta_{\rm w}\neq\frac{\pi}{2}$.

2\. To prove the existence of admissible solutions for each wedge angle in
Theorem 4.1, we derive uniform a priori estimates for admissible solutions
with any wedge angle $\theta_{\rm w}\in[\theta_{\rm w}^{\rm
d}+\sigma,\frac{\pi}{2}]$ for each small $\sigma>0$, show the compactness of
this subset of admissible solutions in the appropriate norm, and then apply
the degree theory to establish the existence of admissible solutions for each
$\theta_{\rm w}\in[\theta_{\rm w}^{\rm d}+\sigma,\frac{\pi}{2}]$, starting
from the unique normal reflection solution for $\theta_{\rm w}=\frac{\pi}{2}$.
To derive the a priori estimates for admissible solutions, we first obtain the
required estimates related to the geometry of shock $\Gamma_{\rm shock}$ and
domain $\Omega$, as well as the basic estimates of solution $\varphi$. We
prove:

  1. (a)

The inequality in (4.45) is strict for any ${\bf e}\in Cone({\bf
e}_{\xi_{2}},{\bf e}_{\mathcal{S}_{1}})$. Combined with the first inequality
in (4.44) and the fact that $\varphi=\varphi_{1}$ on $\Gamma_{\rm shock}$,
this implies that $\Gamma_{\rm shock}$ is a Lipschitz graph with a uniform
Lipschitz estimate for all admissible solutions.

  2. (b)

The uniform bounds on $\text{diam}(\Omega),\|\varphi\|_{C^{0,1}(\Omega)}$, and
the directional monotonicity of $\varphi-\varphi_{2}$ near the sonic arc for a
cone of directions.

  3. (c)

The uniform positive lower bound for the distance between the shock and the
wedge, and the uniform separation of the shock and the symmetry line (that is,
$\Gamma_{\rm shock}$ is away from a uniform conical neighborhood of
$\Gamma_{\rm sym}$ with vertex at their common endpoint $P_{2}$);

  4. (d)

The uniform positive lower bound for the distance between the shock and the
sonic circle ${B_{c_{1}}((u_{1},0))}$ of state (1), by using the properties
described in Remark 4.7. This allows us to estimate the ellipticity of (4.8)
for $\varphi$ in $\Omega$ (depending on the distance to the sonic arc
$P_{1}P_{4}$ for the supersonic regular shock reflection-diffraction
configuration and to ${P_{0}}$ for the subsonic regular shock reflection-
diffraction configuration).

  5. (e)

Estimate (4.29) holds in the supersonic case, by using the monotonicity of
$\psi=\varphi-\varphi_{2}$ near the sonic arc in a cone of directions shown in
(b) and the conditions on $\Gamma_{\rm sonic}$ in Problem 4.3.

The results of (a)–(c) are obtained via the maximum principle, by considering
equation (4.12) as a linear elliptic equation for $\phi$ and using the
boundary conditions on $\Gamma_{\rm shock}$, $\Gamma_{\rm sonic}$,
$\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$ in Problem 4.3 and (4.44)–(4.45).
The results of (c), combined with (a), show the structure of $\Omega$ which
allows us to perform the uniform local elliptic estimates in various parts of
$\Omega$: the interior, near a point $P$ in a relative interior of
$\Gamma_{\rm shock}$, $\Gamma_{\rm wedge}$, and $\Gamma_{\rm sym}$, and
locally near corners $P_{2}$ and $P_{3}$.

Based on estimates (a)–(d), we show the uniform regularity estimates for the
solution and the free boundary in the weighted and scaled $C^{2,\alpha}$ norms
away from the sonic arc in the supersonic case and away from $P_{0}$ in the
subsonic case, i.e., in $\Omega\setminus\Omega_{\varepsilon}$, for any small
$\varepsilon>0$, for some $\alpha\in(0,1)$. The equation is uniformly elliptic
in this region, with the ellipticity constant depending on $\varepsilon$.
Thus, the estimates depend on $\varepsilon$.

3\. Below we discuss the estimates near $\Gamma_{\rm sonic}$ (resp. near
$P_{0}$ in the subsonic/sonic case), i.e., the estimates in
$\Omega_{2\varepsilon}$ for some $\varepsilon$, independent of $\theta_{\rm
w}\in[\theta_{\rm w}^{\rm d}+\sigma,\frac{\pi}{2}]$, which allows us to
complete the uniform a priori estimates for admissible solutions with wedge
angles $\theta_{\rm w}\in[\theta_{\rm w}^{\rm d}+\sigma,\frac{\pi}{2}]$. We
obtain the estimates near $\Gamma_{\rm sonic}$ (or ${P_{0}}$ for the subsonic
reflection), i.e., in $\Omega_{2\varepsilon}$, in scaled and weighted
$C^{2,\alpha}$ for $\varphi$ and the free boundary $\Gamma_{\rm
shock}\cap\partial\Omega_{2\varepsilon}$, by considering separately four cases
depending on $\frac{|D\varphi_{2}|}{c_{2}}$ at ${P_{0}}$:

  * (i)

Supersonic: $\frac{|D\varphi_{2}|}{c_{2}}\geq 1+\delta$,

  * (ii)

Supersonic (almost sonic): $1<\frac{|D\varphi_{2}|}{c_{2}}<1+\delta$,

  * (iii)

Subsonic (almost sonic, including sonic):
$1-\delta\leq\frac{|D\varphi_{2}|}{c_{2}}\leq 1$,

  * (iv)

Subsonic: $\frac{|D\varphi_{2}|}{c_{2}}\leq 1-\delta$,

for small $\delta>0$ chosen so that the estimates can be obtained. The choice
of $\delta$ determines $\varepsilon$.

For cases (i)–(ii), equation (4.8) is degenerate elliptic in $\Omega$ near
${P_{1}}{P_{4}}$ on Fig. 4.2. For case (iii), except the sonic case
$\frac{|D\varphi_{2}(P_{0})|}{c_{2}}=1$, the equation is uniformly elliptic in
$\overline{\Omega}$, but the ellipticity constant is small and tends to zero
near ${P_{0}}$ on Fig. 4.2 as $\frac{|D\varphi^{(\theta_{\rm
w})}_{2}(P_{0})|}{c_{2}}\to 1^{-}$, i.e., as the subsonic angles $\theta_{\rm
w}$ tend to the sonic angle. Thus, for cases (i)–(iii), we exploit the local
elliptic degeneracy, which allows us to find a comparison function in each
case, to show the appropriately fast decay of $\varphi-\varphi_{2}$ near
${P_{1}}{P_{4}}$ for cases (i)–(ii) and near ${P_{0}}$ for case (iii);
furthermore, combining with appropriate local non-isotropic rescaling to
obtain the uniform ellipticity, we obtain the a priori estimates in the
weighted and scaled $C^{2,\alpha}$–norms. In cases (i)–(ii), the norms are
(4.38). For case (iii), we use the different norms to obtain the estimates
that imply the standard $C^{2,\alpha}$–estimates. To obtain these estimates,
for case (i), we use the argument developed in Chen-Feldman [33] and described
above (see Remark 4.6), where the ellipticity estimate (4.31) follows from the
estimates described in (d) above and (4.29) obtained in (e). These estimates
hold in $\Omega_{\varepsilon}$ with
$\varepsilon\lesssim(\mbox{length}(\Gamma_{\rm sonic}))^{2}$ because the
rectangles $R_{(x_{0},y_{0})}$ defined by (4.39) do not fit into $\Omega$ for
larger $x_{0}$, which means, for example, that
$R_{(x_{0},y_{0})}\cap\Gamma_{\rm wedge}\neq\emptyset$ for
$(x_{0},y_{0})\in\Gamma_{\rm shock}\cap\partial\Omega_{\varepsilon}$ with
$x_{0}\geq C(\mbox{length}(\Gamma_{\rm sonic}))^{2}$ if $C$ is fixed and
$\mbox{length}(\Gamma_{\rm sonic})$ is small, because the length of the
$y$-side of $R_{(x_{0},y_{0})}$ is $\frac{\sqrt{x_{0}}}{10}$, and $\Gamma_{\rm
shock}$ and $\Gamma_{\rm wedge}$ are smooth curves that intersect $\Gamma_{\rm
sonic}$ transversally. However, $\mbox{length}(\Gamma_{\rm sonic})$ tends to
zero, as $\displaystyle\frac{|D\varphi^{(\theta_{\rm
w})}_{2}(P_{0})|}{c_{2}}\to 1^{+}$, i.e., when the supersonic wedge angle
tends to the sonic angle. Thus, a different argument, involving an appropriate
scaling, is employed for case (ii) in order to keep $\varepsilon$ uniform for
all $\theta_{\rm w}\in[\theta_{\rm w}^{\rm d}+\delta,\frac{\pi}{2}]$. Another
version of that argument (with a different scaling) is applied for case (iii).
For both cases (ii)–(iii), we need to use smaller rectangles than those for
case (i), but this requires stronger growth estimates than (4.36) to obtain a
bound in $C^{1,1}$ from the corresponding weighted and scaled estimates. We
obtain such growth estimates by using the conditions of cases (ii)–(iii) for
sufficiently small $\delta$. For case (iv), the equation is uniformly elliptic
in $\Omega$ for the admissible solution, where the ellipticity constant is not
small, and the estimates are more technically challenging than those for cases
(i)–(iii). This can be seen as follows: For all cases (i)–(iv), the free
boundary has a lower a priori regularity in the sense that only the Lipschitz
estimate of $\Gamma_{\rm shock}$ is obtained in (a) above; however, for case
(iv), the uniform ellipticity combined with oblique boundary conditions does
not allow a comparison function that leads to the fast decay of
$|\varphi-\varphi_{2}|$ near ${P_{0}}$. Thus, we prove the
$C^{\alpha}$–estimates of $D(\varphi-\varphi_{2})$ near $P_{0}$, by deriving
the equations and boundary conditions for two directional derivatives of
$\varphi-\varphi_{2}$ near $P_{0}$, and performing the hodograph transform to
flatten the free boundary.

4\. In order to prove the existence of solutions, we perform an iteration,
which is an extension of the iteration process used in Chen-Feldman [33].
First, given an admissible solution $\varphi$ for the wedge angle $\theta_{\rm
w}$, we map its elliptic domain $\Omega(\varphi,\theta_{\rm w})$ to a unit
square $Q=(0,1)^{2}$ so that, for the supersonic case, the boundary parts
$\Gamma_{\rm shock}$, $\Gamma_{\rm sonic}$, $\Gamma_{\rm wedge}$, and
$\Gamma_{\rm sym}$ are mapped to the respective sides of $Q$, and the other
properties of this map are satisfied. For the subsonic case, the map is
discontinuous at $P_{0}=\overline{\Gamma_{\rm sonic}}$ (mapping the triangular
domain to a square). Moreover, we define a function $u$ on $Q$ by expressing
$u:=\varphi-\tilde{\varphi}_{2}^{(\theta_{\rm w})}$ in the coordinates on $Q$,
where $\tilde{\varphi}_{2}^{(\theta_{\rm w})}$ is a function determined by
$\theta_{\rm w}$ and equals to $\varphi_{2}$ near $\overline{\Gamma_{\rm
sonic}}$; we skip the complete technical definition here. For appropriate
functions $u$ on $Q$ and the wedge angle $\theta_{\rm w}$, this map can be
inverted, i.e., the elliptic domain $\Omega(u,\theta_{\rm w})$ and the
iteration free boundary $\Gamma_{\rm shock}(u,\theta_{\rm w})$ can be
determined, and a function $\varphi^{(u,\theta_{\rm w})}$ on
$\Omega(u,\theta_{\rm w})$ is defined by expressing $u$ in the coordinates on
$\Omega(u,\theta_{\rm w})$ and adding $\tilde{\varphi}_{2}^{(\theta_{\rm w})}$
so that, if $u$ is obtained from the admissible solution $\varphi$ with the
elliptic domain $\Omega$ as described above, then $\Omega(u,\theta_{\rm
w})=\Omega$ and $\varphi^{(u,\theta_{\rm w})}=\varphi$ in $\Omega$. Moreover,
the map: $\Omega(u,\theta_{\rm w})\to Q$ and its inverse satisfy certain
continuity properties with respect to $(u,\theta_{\rm w})$. The iteration is
performed in terms of the functions defined on $Q$. The iteration set consists
of pairs $(u,\theta_{\rm w})$, where $u$ is in a weighted and scaled
$C^{2,\alpha}$ space on $Q$, denoted as $C^{2,\alpha}_{**}$ (its definition is
technical, so we skip it here), and satisfy

  1. (i)

$\|u\|_{C^{2,\alpha}_{**}}\leq M(\theta_{\rm w})$, where $M(\theta_{\rm w})$
is defined explicitly, based on the a priori estimates discussed above;

  2. (ii)

$\Omega(u,\theta_{\rm w})$, $\Gamma_{\rm shock}(u,\theta_{\rm w})$, and
$\varphi^{(u,\theta_{\rm w})}$ on $\Omega(u,\theta_{\rm w})$ satisfy some
geometric and analytical properties.

The iteration map: $(\hat{u},\theta_{\rm w})\to(u,\theta_{\rm w})$ is defined
by solving the iteration problem on $\Omega(u,\theta_{\rm w})$ and then
mapping its solution $\varphi$ to a function $u$ on $Q$. This mapping includes
additional steps, compared to the one described above. Specifically, we modify
the iteration free boundary by using the solution $\varphi$ of the iteration
problem so that, in the mapping: $(\varphi,\theta_{\rm w})\to u$, the
resulting function $u$ on $Q$ keeps the regularity obtained from solving the
iteration problem. This yields the compactness of the iteration map. We prove
that, for a fixed point $(u,\theta_{\rm w})$ of the iteration map,
$\varphi^{(u,\theta_{\rm w})}$ on $\Omega(u,\theta_{\rm w})$ is an admissible
solution. We use the degree theory to establish the existence of admissible
solutions as fixed points of the iteration map for each $\theta_{\rm
w}\in[\theta_{\rm w}^{\rm d}+\delta,\frac{\pi}{2}]$, starting from the unique
normal reflection solution for $\theta_{\rm w}=\frac{\pi}{2}$. The compactness
of the iteration map described above is necessary for that. The a priori
estimates of admissible solutions discussed above are used in the degree
theory argument in order to define the iteration set such that a fixed point
of the iteration map (i.e., admissible solution) cannot occur on the boundary
of the iteration set, since that would contradict the a priori estimates. With
all of these arguments, we complete the proof of Theorem 4.1. This provides a
solution to the von Neumann’s conjectures.

More details can be found in Chen-Feldman [35]; also see [33].

###  4.2. The Prandtl-Meyer Problem for Unsteady Supersonic Flow onto Solid
Wedges

As we discussed in §2–§3, steady shocks appear when a steady supersonic flow
hits a straight wedge; see Figure 3.1. Since both weak and strong steady shock
solutions are stable in the steady regime, the static stability analysis alone
is not able to single out one of them in this sense, unless an additional
condition is posed on the speed of the downstream flow at infinity. Then the
dynamic stability analysis becomes more significant to understand the non-
uniqueness issue of the steady oblique shock solutions. However, the problem
for the dynamic stability of the steady shock solutions for supersonic flow
past solid wedges involves several additional difficulties. The recent efforts
have been focused on the construction of the global Prandtl-Meyer reflection
configurations in the self-similar coordinates for potential flow.

As we discussed earlier, if a supersonic flow with a constant density
$\rho_{0}>0$ and a velocity ${\bf u}_{0}=(u_{10},0)$,
$u_{10}>c_{0}:=c(\rho_{0})$, impinges toward wedge $W$ in (3.11), and if
$\theta_{\rm w}$ is less than the detachment angle $\theta_{\rm w}^{\rm d}$,
then the well-known _shock polar analysis_ shows that there are two different
steady weak solutions: _the steady weak shock solution_ $\bar{\Phi}$ and _the
steady strong shock solution_ , both of which satisfy the entropy condition
and the slip boundary condition (see Fig. 3.1).

Then the dynamic stability of the weak transonic shock solution for potential
flow can be formulated as the following problem:

######  Problem 4.4 (Initial-Boundary Value Problem).

Given $\gamma>1$, fix $(\rho_{0},u_{10})$ with $u_{10}>c_{0}$. For a fixed
$\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm d})$, let $W$ be given by (3.11).
Seek a global entropy solution $\Phi\in W^{1,\infty}_{\rm
loc}(\mathbb{R}_{+}\times(\mathbb{R}^{2}\setminus W))$ of Eq. (4.5) with
$\rho$ determined by (4.4) and $B=\frac{u_{10}^{2}}{2}+h(\rho_{0})$ so that
$\Phi$ satisfies the initial condition at $t=0$:

$(\rho,\Phi)|_{t=0}=(\rho_{0},u_{10}x_{1})\qquad\,\,\,\text{for}\;\;\mathbf{x}\in\mathbb{R}^{2}\setminus
W,$ (4.46)

and the slip boundary condition along the wedge boundary $\partial W$:

$\nabla_{\bf x}\Phi\cdot{\bm{\nu}}_{\rm w}|_{\partial W}=0,$ (4.47)

where ${\bm{\nu}}_{\rm w}$ is the exterior unit normal to $\partial W$.

In particular, we seek a solution $\Phi\in W^{1,\infty}_{\rm
loc}(\mathbb{R}_{+}\times(\mathbb{R}^{2}\setminus W))$ that converges to the
steady weak oblique shock solution $\bar{\Phi}$ corresponding to the fixed
parameters $(\rho_{0},u_{10},\gamma,\theta_{\rm w})$ with
$\bar{\rho}=h^{-1}(B-\frac{1}{2}|\nabla\bar{\Phi}|^{2})$, when $t\to\infty$,
in the following sense: For any $R>0$, $\Phi$ satisfies

$\lim_{t\to\infty}\|(\nabla_{\bf x}\Phi(t,\cdot)-\nabla_{\bf
x}\bar{\Phi},\rho(t,\cdot)-\bar{\rho})\|_{L^{1}(B_{R}({\bf 0})\setminus W)}=0$
(4.48)

for $\rho(t,{\bf x})$ given by (4.4).

Since the initial data functions in (4.46) do not satisfy the boundary
condition (4.47), a boundary layer is generated along the wedge boundary
starting at $t=0$, which forms the Prandtl-Meyer reflection configurations;
see Bae-Chen-Feldman [6] and the references cited therein.

Notice that the initial-boundary value problem, Problem 4.4, is invariant
under scaling (4.6). Thus, we study the existence of self-similar solutions
determined by equation (4.8) with (4.9) through (4.7).

As the upstream flow has the constant velocity $(u_{10},0)$, noting the choice
of $B$ in Problem 4.4, the corresponding pseudo-potential $\varphi_{0}$ has
the expression of

$\varphi_{0}=-\frac{1}{2}|{\bm{\xi}}|^{2}+u_{10}\xi_{1}$ (4.49)

in self-similar coordinates ${\bm{\xi}}=\frac{{\bf x}}{t}$, as shown directly
from (4.14). Notice also the symmetry of the domain and the upstream flow in
Problem 4.4 with respect to the $x_{1}$–axis. Problem 4.4 can then be
reformulated as the following boundary value problem in the domain:

$\Lambda:=\mathbb{R}^{2}_{+}\setminus\\{{\bm{\xi}}\,:\,\xi_{2}\leq\xi_{1}\tan\theta_{\rm
w},\,\xi_{1}\geq 0\\}$

in the self-similar coordinates ${\bm{\xi}}$, which corresponds to domain
$\\{(t,{\bf x})\,:\,{\bf x}\in\mathbb{R}^{2}_{+}\setminus W,\,t>0\\}$ in the
$(t,{\bf x})$–coordinates, where
$\mathbb{R}^{2}_{+}=\\{{\bm{\xi}}\,:\,\xi_{2}>0\\}$.

######  Problem 4.5 (Boundary Value Problem).

Seek a solution $\varphi$ of equation (4.8) in the self-similar domain
$\Lambda$ with the slip boundary condition:

$D\varphi\cdot{\bm{\nu}}|_{\partial\Lambda}=0$ (4.50)

and the asymptotic boundary condition:

$\varphi-\varphi_{0}\longrightarrow 0$ (4.51)

along each ray $R_{\theta}:=\\{\xi_{1}=\xi_{2}\cot\theta,\xi_{2}>0\\}$ with
$\theta\in(\theta_{\rm w},\pi)$ as $\xi_{2}\to\infty$ in the sense that

$\lim_{r\to\infty}\|\varphi-\varphi_{0}\|_{C(R_{\theta}\setminus
B_{r}(0))}=0.$ (4.52)

In particular, we seek a global entropy solution of Problem 4.5 with two types
of Prandtl-Meyer reflection configurations whose occurrence is determined by
the wedge angle $\theta_{\rm w}$ for the two different cases: One contains a
straight weak oblique shock ${\mathcal{S}}_{0}$ attached to the wedge vertex
$O$ and connected to a normal shock ${\mathcal{S}}_{1}$ through a curved shock
$\Gamma_{\rm shock}$ when $\theta_{\rm w}<\theta_{\rm w}^{\rm s}$, as shown in
Fig. 4.4; the other contains a curved shock $\Gamma_{\rm shock}$ attached to
the wedge vertex and connected to a normal shock ${\mathcal{S}}_{1}$ when
$\theta_{\rm w}^{\rm s}\leq\theta_{\rm w}<\theta_{\rm w}^{\rm d}$, as shown in
Fig. 4.5, in which the curved shock $\Gamma_{\rm shock}$ is tangential to the
straight weak oblique shock ${\mathcal{S}_{0}}$ at the wedge vertex.

Figure 4.4. Self-similar solutions for $\theta_{\rm w}\in(0,\theta_{\rm
w}^{\rm s})$ in the self-similar coordinates ${\bm{\xi}}$ (cf. [6]) Figure
4.5. Self-similar solutions for $\theta_{\rm w}\in[\theta_{\rm w}^{\rm
s},\theta_{\rm w}^{\rm d})$ in the self-similar coordinates ${\bm{\xi}}$ (cf.
[6])

To seek a global entropy solution of Problem 4.5 with the structure of Fig.
4.4 or Fig. 4.5, one needs to compute the pseudo-potential function $\varphi$
below ${\mathcal{S}_{0}}$.

Given $M_{0}>1$, $\rho_{1}$ and $\mathbf{u}_{1}$ are determined by using the
shock polar as in Fig. 3.1 for steady potential flow (note that the shock
polar is now different from the one for the full Euler system but has the same
shape as in Fig. 3.1). Similar to those in §3.1, in the potential flow case,
for any wedge angle $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm s})$, line
$u_{2}=u_{1}\tan\theta_{\rm w}$ and the shock polar intersect at a point
$\mathbf{u}_{1}$ with $|\mathbf{u}_{1}|>c_{1}$ and $u_{11}<u_{10}$; while, for
any wedge angle $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm
d})$, they intersect at a point $\mathbf{u}_{1}$ with $u_{11}>u_{\rm 1d}$ and
$|\mathbf{u}_{1}|<c_{1}$, where $u_{\rm 1d}$ is the $u_{1}$–component of the
unique detachment state $\mathbf{u}_{\rm d}$ when $\theta_{\rm w}=\theta_{\rm
w}^{\rm d}$. The intersection state $\mathbf{u}_{1}$ is the velocity for
steady potential flow behind an oblique shock $\mathcal{S}_{0}$ attached to
the wedge vertex with angle $\theta_{\rm w}$. The strength of shock
$\mathcal{S}_{0}$ is relatively weak compared to the other shock given by the
other intersection point on the shock polar, hence we call $\mathcal{S}_{0}$ a
_weak oblique shock_ , and the corresponding state $\mathbf{u}_{1}$ is a _weak
state_.

We also note that states $\mathbf{u}_{1}$ depend smoothly on $u_{10}$ and
$\theta_{\rm w}$, and such states are supersonic when $\theta_{\rm
w}\in(0,\theta_{\rm w}^{\rm s})$ and subsonic when $\theta_{\rm
w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm d})$.

Once $\mathbf{u}_{1}$ is determined, by (4.17) and (4.49), the pseudo-
potential $\varphi_{1}$ below the weak oblique shock $\mathcal{S}_{0}$ is

$\varphi_{1}=-\frac{1}{2}|{\bm{\xi}}|^{2}+\mathbf{u}_{1}\cdot{\bm{\xi}}.$
(4.53)

Similarly, by (4.16)–(4.17) and (4.49)–(4.50), the pseudo-potential
$\varphi_{2}$ below the normal shock $\mathcal{S}_{1}$ is of the form:

$\varphi_{2}=-\frac{1}{2}|{\bm{\xi}}|^{2}+\mathbf{u}_{2}\cdot{\bm{\xi}}+k_{2}$
(4.54)

for constant state $\mathbf{u}_{2}$ and constant $k_{2}$; see (4.14). Then it
follows from (4.9) and (4.53)–(4.54) that the corresponding densities
$\rho_{1}$ and $\rho_{2}$ are constants, respectively. In particular, we have

$\rho_{k}^{\gamma-1}=\rho_{0}^{\gamma-1}+\frac{\gamma-1}{2}\big{(}u_{10}^{2}-|\mathbf{u}_{k}|^{2}\big{)}\qquad\,\,\mbox{for
$k=1,2$}.$ (4.55)

Denote $\Gamma_{\rm wedge}:=\partial W\cap\partial\Lambda$. Next, we define
the sonic arcs $\Gamma_{\rm sonic}^{1}=P_{1}P_{4}$ on Fig. 4.4 and
$\Gamma_{\rm sonic}^{2}=P_{2}P_{3}$ on Figs. 4.4–4.5. The sonic circle
$\partial B_{c_{1}}(\mathbf{u}_{1})$ of the uniform state $\varphi_{1}$
intersects line ${\mathcal{S}_{0}}$, where
$c_{1}=\rho_{1}^{\frac{\gamma-1}{2}}$ by (4.11). For the supersonic case
$\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm s})$, there are two arcs of this
sonic circle between ${\mathcal{S}_{0}}$ and $\Gamma_{\rm wedge}$ in
$\Lambda$. We denote by $\Gamma_{\rm sonic}^{1}$ the lower arc (i.e., located
to the left from another arc) in the orientation on Fig. 4.4. Note that
$\Gamma_{\rm sonic}^{1}$ tends to point $O$ as $\theta_{\rm
w}\nearrow\theta_{\rm w}^{\rm s}$ and is outside of $\Lambda$ for the subsonic
case $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm d})$.
Similarly, the sonic circle $\partial B_{c_{2}}(\mathbf{u}_{2})$ of the
uniform state $\varphi_{2}$ intersects line ${\mathcal{S}_{1}}$, where
$c_{2}=\rho_{2}^{\frac{\gamma-1}{2}}$. There are two arcs of this circle
between ${\mathcal{S}_{1}}$ and the line containing $\Gamma_{\rm wedge}$. For
all $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm d})$, the upper arc (i.e.,
located to the right of the other arc) in the orientation on Figs. 4.4–4.5 is
within $\Lambda$, which is denoted as $\Gamma_{\rm sonic}^{2}$.

Then Problem 4.5 can be reformulated into the following free boundary problem:

######  Problem 4.6 (Free Boundary Problem).

For $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm d})$, find a free boundary
(curved shock) $\Gamma_{\rm shock}$ and a function $\varphi$ defined in domain
$\Omega$, as shown in Figs. 4.4–4.5, such that $\varphi$ satisfies

  * (i)

Equation (4.8) in $\Omega$,

  * (ii)

$\varphi=\varphi_{0}$ and $\rho D\varphi\cdot{\bm{\nu}}_{\rm
s}=\rho_{0}D\varphi_{0}\cdot{\bm{\nu}}_{\rm s}$ on $\Gamma_{\rm shock}$,

  * (iii)

$\varphi=\hat{\varphi}$ and $D\varphi=D\hat{\varphi}$ on $\Gamma_{\rm
sonic}^{1}\cup\Gamma_{\rm sonic}^{2}\,\,$ when $\theta_{\rm
w}\in(0,\theta_{\rm w}^{\rm s})$ and on $\Gamma_{\rm sonic}^{2}\cup\\{O\\}$
when $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm d})$ for
$\hat{\varphi}:=\max(\varphi_{1},\varphi_{2})$,

  * (iv)

$D\varphi\cdot{\bm{\nu}}_{\rm w}=0$ on $\Gamma_{\rm wedge}$,

where ${\bm{\nu}}_{\rm s}$ and ${\bm{\nu}}_{\rm w}$ are the unit normals to
$\Gamma_{\rm shock}$ and $\Gamma_{\rm wedge}$ pointing to the interior of
$\Omega$, respectively.

######  Remark 4.9.

It can be shown that $\varphi_{1}>\varphi_{2}$ on $\Gamma_{\rm sonic}^{1}$ and
the opposite inequality holds on $\Gamma_{\rm sonic}^{2}$. This justifies the
requirements in Problem 4.6(iii) above.

######  Remark 4.10.

Similar to Problem 4.3, the conditions in Problem 4.6(ii)–(iii) are the
Rankine-Hugoniot conditions (4.16)–(4.17) on $\Gamma_{\rm shock}$ and
$\Gamma_{\rm sonic}^{1}\cup\Gamma_{\rm sonic}^{2}$ or $\Gamma_{\rm
sonic}^{2}\cup\\{O\\}$, respectively; see the discussions right after Problem
4.3.

Let $\varphi$ be a solution of Problem 4.6 such that $\Gamma_{\rm shock}$ is a
$C^{1}$–curve up to its endpoints and $\varphi\in C^{1}(\overline{\Omega})$.
To obtain a solution of Problem 4.5 from $\varphi$, we consider two cases:

For the supersonic case $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm s})$, we
divide region $\Lambda$ into four separate regions; see Fig. 4.4. We denote by
${\mathcal{S}_{0,{\rm seg}}}$ the line segment $OP_{1}\subset\mathcal{S}_{0}$,
and by ${\mathcal{S}_{1,{\rm seg}}}$ the portion (half-line) of
$\mathcal{S}_{1}$ with left endpoint $P_{2}$ so that ${\mathcal{S}_{1,{\rm
seg}}}\subset\Lambda$. Let $\Omega_{\mathcal{S}}$ be the unbounded domain
below curve $\overline{{\mathcal{S}_{0,{\rm seg}}}\cup\Gamma_{\rm
shock}\cup{\mathcal{S}_{1,{\rm seg}}}}$ and above $\Gamma_{\rm wedge}$ (see
Fig. 4.4). In $\Omega_{\mathcal{S}}$, let $\Omega_{1}$ be the bounded domain
enclosed by $\mathcal{S}_{0},\Gamma^{1}_{\rm sonic}$, and $\Gamma_{\rm
wedge}$. Set
$\Omega_{2}:=\Omega_{\mathcal{S}}\setminus\overline{\Omega_{1}\cup\Omega}$.
Define a function $\varphi_{*}$ in $\Lambda$ by

$\varphi_{*}=\begin{cases}\varphi_{0}&\qquad\text{in}\,\Lambda\setminus\Omega_{\mathcal{S}},\\\
\varphi_{1}&\qquad\text{in}\,\Omega_{1},\\\
\varphi&\qquad\text{in}\,\Gamma^{1}_{\rm sonic}\cup\Omega\cup\Gamma^{2}_{\rm
sonic},\\\ \varphi_{2}&\qquad\text{in}\,\Omega_{2}.\end{cases}$ (4.56)

By Problem 4.6(ii)–(iii), $\varphi_{*}$ is continuous in
$\Lambda\setminus\Omega_{\mathcal{S}}$ and $C^{1}$ in
$\overline{\Omega_{\mathcal{S}}}$. In particular, $\varphi_{*}$ is $C^{1}$
across $\Gamma^{1}_{\rm sonic}\cup\Gamma^{2}_{\rm sonic}$. Moreover, using
Problem 4.6(i)–(iii), we obtain that $\varphi_{*}$ is a global entropy
solution of equation (4.8) in $\Lambda$.

For the subsonic case $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm
w}^{\rm d})$, region $\Omega_{1}\cup\Gamma^{1}_{\rm sonic}$ in $\varphi_{*}$
reduces to one point $\\{O\\}$; see Fig. 4.5. The corresponding function
$\varphi_{*}$ is a global entropy solution of equation (4.8) in $\Lambda$.

The first unsteady analysis of the steady supersonic weak shock solution as
the long-time behavior of an unsteady flow is due to Elling-Liu [56], in which
they succeeded in establishing a stability theorem for an important class of
physical parameters determined by certain assumptions for the wedge angle
$\theta_{\rm w}$ less than the sonic angle $\theta^{\rm s}_{\rm
w}\in(0,\theta^{\rm d}_{\rm w})$ for potential flow.

Recently, in Bae-Chen-Feldman [6], we have removed the assumptions [56] and
established the stability theorem for the steady (supersonic or transonic)
weak shock solutions as the long-time asymptotics of the global Prandtl-Meyer
reflection configurations for unsteady potential flow for all the admissible
physical parameters even up to the detachment angle $\theta^{\rm d}_{\rm w}$
(beyond the sonic angle $\theta^{\rm s}_{\rm w}<\theta^{\rm d}_{\rm w}$).

To achieve this, we solve the free boundary problem (Problem 4.6), involving
transonic shocks, for all the wedge angles $\theta_{\rm w}\in(0,\theta_{\rm
w}^{\rm d})$ by employing the techniques developed in Chen-Feldman [35],
described in §4.1 above. Similar to Definition 4.8, we define admissible
solutions in the present case:

######  Definition 4.11.

Let $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm d})$. A function $\varphi\in
C^{0,1}(\overline{\Lambda})$ is an admissible solution of Problem 4.6 if
$\varphi$ is a solution of Problem 4.6 extended to $\Lambda$ by (4.56) and
satisfies the following properties:

  1. (i)

The structure of solutions is as follows:

     * •

If $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm s})$, then $\varphi$ has the
configuration shown on Fig. 4.4 such that $\Gamma_{\text{\rm shock}}$ is
$C^{2}$ in its relative interior, $\varphi\in C^{0,1}(\Lambda)\cap
C^{1}(\Lambda\setminus(\overline{{\mathcal{S}_{0,{\rm
seg}}}}\cup\overline{\Gamma}_{\text{\rm
shock}}\cup\overline{{\mathcal{S}_{1,{\rm seg}}}}))$, and $\varphi\in
C^{1}(\overline{\Omega})\cap
C^{2}(\overline{\Omega}\setminus(\overline{\Gamma_{\rm
sonic}^{1}}\cup\overline{\Gamma_{\rm sonic}^{2}}))\cap C^{3}(\Omega)$.

     * •

If $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm d})$, then
$\varphi$ has the configuration shown on Fig. 4.5 such that $\Gamma_{\text{\rm
shock}}$ is $C^{2}$ in its relative interior, $\varphi\in C^{0,1}(\Lambda)\cap
C^{1}(\Lambda\setminus(\Gamma_{\text{\rm
shock}}\cup\overline{{\mathcal{S}_{1,{\rm seg}}}}))$, and $\varphi\in
C^{1}(\overline{\Omega})\cap
C^{2}(\overline{\Omega}\setminus(\\{O\\}\cup\overline{\Gamma_{\rm
sonic}^{2}}))\cap C^{3}(\Omega)$.

  2. (ii)

Equation (4.8) is strictly elliptic in
$\overline{\Omega}\setminus(\overline{\Gamma_{\rm
sonic}^{1}}\cup\overline{\Gamma_{\rm sonic}^{2}})$, i.e.,
$|D\varphi|<c(|D\varphi|^{2},\varphi)$ in
$\overline{\Omega}\setminus(\overline{\Gamma_{\rm
sonic}^{1}}\cup\overline{\Gamma_{\rm sonic}^{2}})$.

  3. (iii)

$\partial_{{\bm{\nu}}}\varphi_{0}>\partial_{{\bm{\nu}}}\varphi>0$ on
$\Gamma_{\text{\rm shock}}$, where ${\bm{\nu}}$ is the normal to
$\Gamma_{\text{\rm shock}}$, pointing to the interior of $\Omega$.

  4. (iv)

The inequalities hold:

$\max\\{\varphi_{1},\varphi_{2}\\}\leq\varphi\leq\varphi_{0}\qquad\mbox{in
$\Omega$},$ (4.57)

  5. (v)

The monotonicity properties hold:

$D(\varphi_{0}-\varphi)\cdot\mathbf{e}_{\mathcal{S}_{1}}\geq 0,\quad
D(\varphi_{0}-\varphi)\cdot\mathbf{e}_{\mathcal{S}_{0}}\leq
0\qquad\,\,\,\,\mbox{in $\Omega$},$ (4.58)

where $\mathbf{e}_{\mathcal{S}_{0}}$ and $\mathbf{e}_{\mathcal{S}_{1}}$ are
the unit vectors along lines $\mathcal{S}_{0}$ and $\mathcal{S}_{1}$ pointing
to the positive $\xi_{1}$–direction, respectively.

Similar to (4.45), the monotonicity properties in (4.58) imply that

$D(\varphi_{1}-\varphi)\cdot{\bf e}\leq 0\qquad\,\,\mbox{in
$\overline{\Omega}\,\,$ for all ${\bf
e}\in\overline{Cone(-\mathbf{e}_{\mathcal{S}_{1}},\mathbf{e}_{\mathcal{S}_{0}})}$},$
(4.59)

where
$Cone(-\mathbf{e}_{\mathcal{S}_{1}},\mathbf{e}_{\mathcal{S}_{0}})=\\{-a\,\mathbf{e}_{\mathcal{S}_{1}}+b\,\mathbf{e}_{\mathcal{S}_{0}}\;:\;a,b>0\\}$.
We note that $\mathbf{e}_{\mathcal{S}_{0}}$ and $\mathbf{e}_{\mathcal{S}_{1}}$
are not parallel if $\theta_{\rm w}\neq 0$. Then we establish the following
theorem.

######  Theorem 4.2.

Let $\gamma>1$ and $u_{10}>c_{0}$. For any $\theta_{\rm w}\in(0,\theta_{\rm
w}^{\rm d})$, there exists a global entropy solution $\varphi$ of Problem 4.6
such that the following regularity properties are satisfied for some
$\alpha\in(0,1)$:

  1. (i)

If $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm s})$, the reflected shock
$\overline{{\mathcal{S}_{0,{\rm seg}}}}\cup\Gamma_{\rm
shock}\cup\overline{{\mathcal{S}_{1,{\rm seg}}}}$ is $C^{2,\alpha}$–smooth,
and $\varphi\in C^{1,\alpha}(\overline{\Omega})\cap
C^{\infty}(\overline{\Omega}\setminus(\overline{\Gamma_{\rm
sonic}^{1}}\cup\overline{\Gamma_{\rm sonic}^{2}}))$;

  2. (ii)

If $\theta_{\rm w}\in[\theta_{\rm w}^{\rm s},\theta_{\rm w}^{\rm d})$, the
reflected shock $\overline{\Gamma_{\rm
shock}}\cup\overline{{\mathcal{S}_{1,{\rm seg}}}}$ is $C^{1,\alpha}$ near $O$
and $C^{2,\alpha}$ away from $O$, and $\varphi\in
C^{1,\alpha}(\overline{\Omega})\cap
C^{\infty}(\overline{\Omega}\setminus(\\{O\\}\cup\overline{\Gamma_{\rm
sonic}^{2}}))$.

Moreover, in both cases, $\varphi$ is $C^{1,1}$ across the sonic arcs, and
$\Gamma_{\rm shock}$ is $C^{\infty}$ in its relative interior.

Furthermore, $\varphi$ is an admissible solution in the sense of Definition
4.11, so $\varphi$ satisfies further properties listed in Definition 4.11.

We follow the argument described in §4.1 so that, for any small $\delta>0$, we
obtain the required uniform estimates of admissible solutions with wedge
angles $\theta_{\rm w}\in[0,\theta_{\rm w}^{\rm d}-\delta]$. Using these
estimates, we apply the Leray-Schauder degree theory to obtain the existence
for each $\theta_{\rm w}\in[0,\theta_{\rm w}^{\rm d}-\delta]$ in the class of
admissible solutions, starting from the unique normal solution for
$\theta_{\rm w}=0$. Since $\delta>0$ is arbitrary, the existence of a global
entropy solution for any $\theta_{\rm w}\in(0,\theta_{\rm w}^{\rm d})$ can be
established. More details can be found in Bae-Chen-Feldman [6]; see also Chen-
Feldman [35].

The existence results in Bae-Chen-Feldman [6] indicate that the steady weak
supersonic/transonic shock solutions are the asymptotic limits of the dynamic
self-similar solutions, the Prandtl-Meyer reflection configurations, in the
sense of (4.52) in Problem 4.5 for all $\theta_{\rm w}\in(0,\theta_{\rm
w}^{\rm d})$ and all $\gamma>1$.

On the other hand, it is shown in Elling [55] and Bae-Chen-Feldman [6] that,
for each $\gamma>1$, there is no self-similar strong Prandtl-Meyer reflection
configuration for the unsteady potential flow in the class of admissible
solutions. This means that the situation for the dynamic stability of the
strong steady oblique shocks is more sensitive.

##  5\. Convexity of Self-Similar Transonic Shocks and Free Boundaries

We now discuss some recent developments in the analysis of geometric
properties of transonic shocks as free boundaries in the 2-D self-similar
coordinates for compressible fluid flows. In Chen-Feldman-Xiang [36], we have
developed a general framework for the analysis of the convexity of transonic
shocks as free boundaries. For both applications discussed above, the von
Neumann problem for shock reflection-diffraction in §4.1 and the Prandtl-Meyer
problem for unsteady supersonic flow onto solid wedges in §4.2, the admissible
solutions satisfy the conditions of this abstract framework, as shown in [36].
For simplicity, we present below the results on the convexity properties of
transonic shocks for these two problems (without discussion on the abstract
framework).

For the regular shock reflection-diffraction configurations, we recall that,
for admissible solutions in the sense of Definition 4.8, the inequality in
(4.45) is shown to be strict for any ${\bf e}\in Cone({\bf e}_{\xi_{2}},{\bf
e}_{\mathcal{S}_{1}})$. From this, it is proved that, for admissible
solutions, the shock is a graph in the coordinate system $(S,T)$ with respect
to basis $\\{\mathbf{e},\mathbf{e}^{\perp}\\}$ for any unit vector
$\mathbf{e}\in Cone({\bf e}_{\xi_{2}},{\bf e}_{\mathcal{S}_{1}})$, where
$\mathbf{e}^{\perp}$ is the unit vector orthogonal to $\mathbf{e}$ and
oriented so that $T_{P_{1}}>T_{P_{2}}$, and we have used notation
$(S_{P},T_{P})$ for the coordinates of point $P$. That is, there exists
$f_{\mathbf{e}}\in C^{\infty}((T_{P_{2}},T_{P_{1}}))\cap
C^{1}([T_{P_{2}},T_{P_{1}}])$ such that

$\Gamma_{\text{\rm
shock}}=\\{(S,T):\;S=f_{\mathbf{e}}(T),\;T_{P_{2}}<T<T_{P_{1}}\\},\quad\;\Omega\cap\\{T_{P_{2}}<T<T_{P_{1}}\\}\subset\\{S<f_{\mathbf{e}}(T)\\},$
(5.1)

where we have used the notational convention (4.20) in the subsonic/sonic
case.

Since the convexity or concavity of a shock as a graph depends on the
orientation of the coordinate system and $\Omega$ will be shown to be a convex
domain (corresponding to the concavity of $f_{\mathbf{e}}$ in (5.1)), we do
not distinguish them and instead use the term convexity for either case below.
Then we have

######  Theorem 5.1 (Convexity of transonic shocks for the regular shock
reflection-diffraction configurations).

If a solution of the von Neumann problem for shock reflection-diffraction is
admissible in the sense of Definition 4.8, then its domain $\Omega$ is convex,
and the shock curve $\Gamma_{\text{\rm shock}}$ is a strictly convex graph in
the following sense: For any $\mathbf{e}\in Cone({\bf e}_{\xi_{2}},{\bf
e}_{\mathcal{S}_{1}})$, the function $f_{\mathbf{e}}$ in (5.1) satisfies

$f^{\prime\prime}_{\mathbf{e}}<0\qquad\mbox{ on $(T_{P_{2}},T_{P_{1}})$}.$

That is, $\Gamma_{\text{\rm shock}}$ is uniformly convex on any closed subset
of its relative interior.

Moreover, for the solution of Problem 4.3 extended to $\Lambda$ by (4.21),
with pseudo-potential $\varphi\in C^{0,1}(\Lambda)$ satisfying Definition
4.8(i)–(iv), the shock is strictly convex if and only if Definition 4.8(v)
holds.

For the Prandtl-Meyer problem for unsteady supersonic flow onto solid wedges,
the results are similar. We first note that, based on (4.59) (which is strict
for ${\bf
e}\in{Cone(-\mathbf{e}_{\mathcal{S}_{1}},\mathbf{e}_{\mathcal{S}_{0}})}$) and
the maximum principle, it is proved that, for admissible solutions in the
sense of Definition 4.11, the shock is a graph in the coordinate system
$(S,T)$ with respect to basis $\\{\mathbf{e},\mathbf{e}^{\perp}\\}$ for any
unit vector $\mathbf{e}\in
Cone(-\mathbf{e}_{\mathcal{S}_{1}},\mathbf{e}_{\mathcal{S}_{0}})$, i.e., (5.1)
holds, with $f_{\mathbf{e}}\in C^{\infty}((T_{P_{2}},T_{P_{1}}))\cap
C^{1}([T_{P_{2}},T_{P_{1}}])$, where we have used the notational convention
$P_{1}=P_{0}$ for the subsonic/sonic case $\theta_{\rm w}\in[\theta_{\rm
w}^{\rm s},\theta_{\rm w}^{\rm d})$.

######  Theorem 5.2 (Convexity of transonic shocks for the Prandtl-Meyer
reflection configurations).

If a solution of the Prandtl-Meyer problem is admissible in the sense of
Definition 4.11, then its domain $\Omega$ is convex, and the shock curve
$\Gamma_{\text{\rm shock}}$ is a strictly convex graph in the following sense:
For any $\mathbf{e}\in
Cone(-\mathbf{e}_{\mathcal{S}_{1}},\mathbf{e}_{\mathcal{S}_{0}})$, the
function $f_{\mathbf{e}}$ in (5.1) satisfies

$f^{\prime\prime}_{\mathbf{e}}<0\qquad\mbox{ on $(T_{P_{2}},T_{P_{1}})$}.$

That is, $\Gamma_{\text{\rm shock}}$ is uniformly convex on any closed subset
of its relative interior.

Moreover, for the solution of Problem 4.6 extended to $\Lambda$ by (4.56)
$($with the appropriate modification for the subsonic/sonic case$)$ with
pseudo-potential $\varphi\in C^{0,1}(\Lambda)$ satisfying Definition
4.11(i)–(iv), the shock is strictly convex if and only if Definition 4.11(v)
holds.

Theorems 5.1–5.2 indicate that the curvature of $\Gamma_{\text{\rm shock}}$:

$\kappa=-\frac{f^{\prime\prime}_{\mathbf{e}}(T)}{\big{(}1+(f^{\prime}_{\mathbf{e}}(T))^{2}\big{)}^{3/2}}$

has a positive lower bound on any closed subset of $(T_{P_{2}},T_{P_{1}})$.

Now we discuss the techniques developed in [36] by giving the main steps in
the proofs of Theorems 5.1–5.2. While the argument in [36] is carried out in a
more general setting, we focus here on the specific cases of the regular shock
reflection-diffraction and Prandtl-Meyer reflection configurations; see [36]
for the results in the more general setting and the detailed proofs.

For the von Neumann problem, define

$\phi:=\varphi-\varphi_{1}.$

For the Prandtl-Meyer problem, define

$\phi:=\varphi-\varphi_{0}.$

Then, in both cases, $\phi=0$ on $\Gamma_{\rm shock}$. From this, using
Definition 4.8(iii) for the regular reflection-diffraction case and Definition
4.11(iii) for the Prandtl-Meyer reflection case, it follows that, in both
problems, $\phi<0$ in $\Omega$ near $\Gamma_{\text{\rm shock}}$. Since
$\Gamma_{\text{\rm shock}}$ is the zero level set of $\phi$, the conclusion of
Theorems 5.1–5.2 on the strict convexity of $\Gamma_{\text{\rm shock}}$ is
equivalent to the following: $\phi_{{\bm{\tau}}{\bm{\tau}}}>0$ along
$\Gamma_{\text{\rm shock}}^{0}$, where $\Gamma_{\text{\rm shock}}^{0}$ is the
relative interior of $\Gamma_{\text{\rm shock}}$. Also, denote by $Con$ the
cone from (4.45) for the von Neuamnn problem and the cone from (4.59) for the
Prandtl-Meyer problem.

First, we establish the relation between the strict convexity/concavity of a
portion of the shock and the possibility for $\partial_{\mathbf{e}}\phi$, with
$\mathbf{e}\in Con$, to attain its local minimum or maximum with respect to
$\overline{\Omega}$ on that portion of the shock. More precisely, on a portion
of “wrong” convexity on which $f^{\prime\prime}_{\mathbf{e}}\geq 0$
(equivalently, $\phi_{{\bm{\tau}}{\bm{\tau}}}\leq 0$), $\phi_{\mathbf{e}}$
cannot attain its local minimum relative to $\overline{\Omega}$. Then,
assuming that a portion of the free boundary has a “wrong” convexity
$f^{\prime\prime}_{\mathbf{e}}>0$, we show that $\phi_{\mathbf{e}}$ for
$\mathbf{e}\in Con$ attains its local minimum relative to $\Gamma_{\rm shock}$
on the closure of that portion. As we discussed above, it cannot be a local
minimum with respect to $\overline{\Omega}$. Starting from that, through a
nonlocal argument, with the use of the maximum principle for equation (4.12),
considered as a linear elliptic PDE for $\phi$, in $\Omega$, and the boundary
conditions on various parts of $\partial\Omega$, we reach a contradiction,
which implies that the shock is convex, possibly non-strictly, i.e.,
$f^{\prime\prime}_{\mathbf{e}}\leq 0$ on $(T_{P_{2}},T_{P_{1}})$, or
equivalently, $\phi_{{\bm{\tau}}{\bm{\tau}}}\geq 0$ on $\Gamma_{\rm shock}$.
Extending the previous argument with use of the real analyticity of
$\Gamma_{\rm shock}^{0}$, we improve the result to the locally uniform
convexity as in Theorems 5.1–5.2.

Furthermore, with the convexity of reflected-diffracted transonic shocks, the
uniqueness and stability of global regular shock reflection-diffraction
configurations have also been established in the class of admissible
solutions; see Chen-Feldman-Xiang [37] for the details.

The nonlinear method, ideas, techniques, and approaches that we have presented
above for solving M-D transonic shocks and free boundary problems should be
useful to analyze other longstanding and newly emerging problems. Examples of
such problems include the unsolved M-D steady transonic shock problems for the
full Euler equations (including steady detached shock problems), the unsolved
M-D self-similar transonic shock problems (such as the 2-D Riemann problems
and the conic body problems) for potential flow, as well as the longstanding
open transonic shock problems for both the isentropic and the full Euler
equations; also see Chen-Feldman [35]. Certainly, further new ideas,
techniques, and methods are still required to be developed in order to solve
these mathematically challenging and fundamentally important problems.

Acknowledgements. The research of Gui-Qiang G. Chen was supported in part by
the UK Engineering and Physical Sciences Research Council Awards EP/L015811/1,
EP/V008854, and EP/V051121/1, and the Royal Society–Wolfson Research Merit
Award WM090014. The research of Mikhail Feldman was supported in part by the
National Science Foundation under DMS-1764278 and DMS-2054689.

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