Structural Stability of Supersonic Solutions to the Euler–Poisson System

Abstract

The structural stability for supersonic solutions of the Euler–Poisson system for hydrodynamical model in semiconductor devices and plasmas in two dimensional domain is established, under the perturbation of the flow velocity and the strength of electric field in the horizontal direction at the entrance of a channel. First, the Euler–Poisson system in the supersonic region is reformulated into a second order hyperbolic–elliptic coupled system together with several transport equations. One of the key ingredients of the analysis is to obtain the well-posedness of the boundary value problem for the associated linearized hyperbolic–elliptic coupled system, which is achieved via a delicate choice of multiplier to gain energy estimate. The nonlinear structural stability of supersonic solution in the general situation is established by combining the iteration method with the estimate for hyperbolic–elliptic system and the transport equations together.

Appendices

Appendix A. Proof of Lemma 2.8

In this appendix, we prove Lemma 2.8.

Proof of Lemma 2.8

The proof is divided into 3 steps. For a fixed $n\in \mathbb {N}$, let $(\hat{\mathfrak {f}}_1^{(n)}, \hat{\mathfrak {f}}_2^{(n)})$ be given by (2.68). To simplify notations, $(\hat{\mathfrak {f}}_1^{(n)}, \hat{\mathfrak {f}}_2^{(n)})$ is abbreviated as $(\hat{\mathfrak {f}}_1, \hat{\mathfrak {f}}_2)$ hereafter. Note that $(\hat{\mathfrak {f}}_1, \hat{\mathfrak {f}}_2)$ are smooth in $\overline{\Omega _L}$. For the rest of proof, we also fix $m\in \mathbb {N}$, and set

$$\begin{aligned} \hat{\mathfrak {f}}_{l,m}:=\sum _{j=0}^m\langle \hat{\mathfrak {f}}_l, \eta _j \rangle \eta _j\quad \text{ in } \Omega _L \text {for } l=1,2, \end{aligned}$$

(A.1)

where $\{\eta _j\}_{j=0}^\infty$ is the orthonormal basis given in (2.73). And, let $(V_m, W_m)$ given in the form (2.74) be the solution to (2.75)–(2.76).

Step 1. $H^2$ estimate for $W_m$. It follows from the definition of $\mathcal {L}_2$ given by (2.12), (2.13) and (2.75) that

$$\begin{aligned} \langle \mathcal {L}_2(V_m, W_m)-\hat{\mathfrak {f}}_{2,m}, \eta _k \rangle =0 \quad \text{ for } \text{ all } k\in \mathbb {Z}_+, 0<x_1<L. \end{aligned}$$

(A.2)

By (2.76)–(2.77) and (A.2), $W_m$ becomes a classical solution to the elliptic boundary value problem:

$$\begin{aligned} \begin{aligned} \Delta W_m-\bar{h}_1W_m=&\hat{\mathfrak {f}}_{2,m}+\bar{h}_2\partial _{x_1}V_m \quad \text{ in } \Omega _L,\\ \partial _{x_1}W_m=&0\,\,\text{ on } \Gamma _0,\quad \partial _{x_2}W_m=0\,\,\text{ on } \Lambda _L,\quad W_m=0\,\,\text{ on } \Gamma _L. \end{aligned} \end{aligned}$$

(A.3)

Applying [12, Theorems 8.8 and 8.12] and the method of reflection to (A.3) yields

$$\begin{aligned} \Vert W_m\Vert _{H^2\left( \Omega _L \right) }\leqq C(\Vert \hat{\mathfrak {f}}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert W_m\Vert _{L^2\left( \Omega _L \right) }+\Vert V_m\Vert _{H^1\left( \Omega _L \right) }) \end{aligned}$$

for a constant $C>0$ depending only on $(\gamma , S_0, J_0, \rho _0, E_0, \epsilon _0, L)$. Combining this estimate with (2.80) gives

$$\begin{aligned} \Vert W_m\Vert _{H^2\left( \Omega _L \right) }\leqq C\Bigl (\Vert (\hat{\mathfrak {f}}_1\Vert _{L^2\left( \Omega _L \right) }+\Vert \hat{\mathfrak {f}}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^0(\overline{\Gamma _0})}\Bigr ) \end{aligned}$$

(A.4)

for a constant $C>0$ depending only on $(\gamma , S_0, J_0, \rho _0, E_0, \epsilon _0, L)$.

Step 2. $H^2$ estimate for $V_m$. The proof for the $H^2$-estimate for $V_m$ is divided into three parts.

Part 1. Energy estimate. The first equation in (2.75) can be written as

$$\begin{aligned} \langle \mathcal {L}_1^{(n)}(V_m, W_m),\eta _k\rangle =\langle \hat{\mathfrak {f}}_{1,m}, \eta _k \rangle \quad \text{ for } 0<x_1<L, \text{ and } k=0,1,\ldots , m. \end{aligned}$$

(A.5)

We define a linear hyperbolic differential operator $\mathcal {L}_\mathrm{{hyp}}$ by

$$\begin{aligned} \mathcal {L}_\mathrm{{hyp}}(V) := V_{x_1x_1}+2\tilde{a}_{12}^{(n)}V_{x_1x_2} -\tilde{a}_{22}^{(n)}V_{x_2x_2}+\bar{a}_1({\mathrm{x}})V_{x_1}, \end{aligned}$$

and use this definition to rewrite (A.5) as

$$\begin{aligned} \langle \mathcal {L}_\mathrm{{hyp}}(V_m), \eta _k \rangle =\langle \hat{\mathfrak {f}}_{1,m}-\bar{b}_1 \partial _{x_1}W_m-\bar{b}_2W_m, \eta _k \rangle \quad \text{ for } 0<x_1<L, \text{ and } k=0,1,\ldots , m. \end{aligned}$$

(A.6)

Denote

$$\begin{aligned} q_m:=\partial _{x_1}V_m\quad \text{ in } \Omega _L. \end{aligned}$$

One can differentiate (A.6) with respect to $x_1$, then multiply the resultant equation by $\vartheta ''_k$ for each $k=0,1,\ldots ,m$, and add up the results over $k=0$ to m, finally integrate the summation with respect to $x_1$ on the interval [0, t] for t varying in the interval [0, L] to get

$$\begin{aligned} \begin{aligned} \int _{\Omega _t} \mathcal {L}_\mathrm{{hyp}}(q_m) \partial _{x_1}q_m \,\mathrm{d}\mathrm{x}&=\int _{\Omega _t}\left( \partial _{x_1}f_{1,m}-\partial _{x_1}(\bar{b}_1\partial _{x_1}W_m+\bar{b}_2 W_m) -\partial _{x_1}\bar{a}_1q_m\right) \partial _{x_1}q_m\,\mathrm{d}\mathrm{x}\\&\quad +\,\int _{\Omega _t} (-\partial _{x_1}\tilde{a}_{12}^{(n)}\partial _{x_2}q_m + \partial _{x_1}\tilde{a}_{22}^{(n)}\partial _{x_2x_2}V_m) \partial _{x_1}q_m\,\mathrm{d}\mathrm{x}\end{aligned} \end{aligned}$$

(A.7)

for $\Omega _t:=\{\mathrm{x}=(x_1,x_2){:}\,0<x_1<t,\,\, -1<x_2<1\}$. Using (2.78) gives

$$\begin{aligned} \begin{aligned} {\text{ LHS } \text{ of } \text{ A.7) }}=&\frac{1}{2} \left( \int _{\Gamma _t}-\int _{\Gamma _0}\right) \left[ \left( \partial _{x_1}q_m\right) ^2+\tilde{a}_{22}^{(n)}\left( \partial _{x_2}q_m\right) ^2\right] \,\mathrm{d}x_2\\&\quad +\,\int _{\Omega _t}\left( \bar{a}_1-\partial _{x_2}\tilde{a}_{12}^{(n)}\right) \left( \partial _{x_1}q_m\right) ^2 -\frac{\partial _{x_1}\tilde{a}_{12}^{(n)}}{2}\left( \partial _{x_2}q_m\right) ^2 +\partial _{x_2}\tilde{a}_{22}^{(n)}\partial _{x_1}q_m\partial _{x_2}q_m \,\mathrm{d}\mathrm{x}\end{aligned} \end{aligned}$$

for $\Gamma _t=\{(t,x_2)\in \mathbb {R}^2{:}\,-1<x_2<1\}$. By Lemmas 2.1 and 2.3, (2.78), Morrey’s inequality and Cauchy-Schwarz inequality, there exist positive constants $\lambda$, $\mu$, and C depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0)$ to satisfy

$$\begin{aligned} \int _{\Omega _t} \mathcal {L}_\mathrm{{hyp}}(q_m) \partial _{x_1}q_m\,\mathrm{d}\mathrm{x}\geqq \lambda \int _{\Gamma _t} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2 -\mu \int _{\Gamma _0} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2 -C\int _{\Omega _t} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}\mathrm{x}. \end{aligned}$$

Substituting this inequality into the left-hand side of (A.7) and applying (A.4) and Cauchy–Schwarz inequality yield

$$\begin{aligned} \begin{aligned} \int _{\Gamma _t} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2\leqq&\frac{\mu }{\lambda }\int _{\Gamma _0} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2 +C\Big (\int _{\Omega _t}|\nabla _{\mathrm{x}}q_m|^2+(\partial _{x_2}^2V_m)^2\,\mathrm{d}\mathrm{x}\\&\quad +\left( \Vert \hat{\mathfrak {f}}_1\Vert _{H^1\left( \Omega _L \right) } +\Vert \hat{\mathfrak {f}}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^0\left( \overline{\Gamma _0}\right) }\right) ^2\Big ) \end{aligned} \end{aligned}$$

(A.8)

for some constant $C>0$ depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0)$. Next, one can estimate $\int _{\Gamma _0} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2$ and $\int _{\Omega _t}(\partial _{x_2}^2V_m)^2\,\mathrm{d}\mathrm{x}$, separately.

Part 2. Estimate of $\int _{\Gamma _0} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2$. Differentiating the boundary condition $\partial _{x_1}V_m=\sum _{j=0}^m \langle g_1, \eta _j\rangle \eta _j$ on $\Gamma _0$ with respect to $x_2$ gives

$$\begin{aligned} \int _{\Gamma _0} (\partial _{x_2}q_m)^2\,\mathrm{d}x_2 =\int _{\Gamma _0}\Bigl (\sum _{j=0}^m\langle g_1, \eta _j\rangle \eta _j'(x_2)\Bigr )^2\,\mathrm{d}x_2. \end{aligned}$$

(A.9)

By (2.73), for $0\leqq j,k\leqq m$, one has

$$\begin{aligned} \int _{\Gamma _0}\langle g_1, \eta _j\rangle \langle g_1, \eta _k\rangle \eta _j'\eta _k' \mathrm{d}x_2={\left\{ \begin{array}{ll} \langle g_1, \eta _j\rangle ^2(j\pi )^2 \quad &{}\text{ for } j=k,\\ 0\quad &{}\text{ otherwise }. \end{array}\right. } \end{aligned}$$

(A.10)

For any $j\in \mathbb {N}$, since $-\eta _j''=(j\pi )^2\eta _j$, integration by parts with using $\eta _j'(\pm 1)=0$ yields

$$\begin{aligned} \langle g_1, \eta _j\rangle =\frac{1}{(j\pi )^2}\langle g_1, -\eta _j''\rangle =\frac{1}{j\pi }\langle g_1', \frac{\eta _j'}{j\pi }\rangle . \end{aligned}$$

(A.11)

Note that the set $\{\frac{\eta _j'}{j\pi }\}_{j=1}^\infty$ forms an orthonormal basis in $L^2([-1,1])$. Therefore, it is concluded from (A.9)–(A.11) that

$$\begin{aligned} \int _{\Gamma _0} \left( \partial _{x_2}q_m\right) ^2\,\mathrm{d}x_2\leqq \int _{\Gamma _0}|g_1'|^2\,\mathrm{d}x_2\leqq 2\Vert g_1\Vert ^2_{C^1\left( \overline{\Gamma _0}\right) }. \end{aligned}$$

(A.12)

For each $k=0, 1,\ldots , m$, multiplying (A.6) by $\vartheta _k''$, summing up over $k=0$ to m, and integrating the result over $\Gamma _0$ with respect to $x_2$ give

$$\begin{aligned} \int _{\Gamma _0}\left( \partial _{x_1}q_m\right) ^2 +2\tilde{a}_{12}^{(n)}\partial _{x_2}q_m\partial _{x_1}q_m+\bar{a}_1q_m\partial _{x_1}q_m\,\mathrm{d}x_2 =\int _{\Gamma _0} \left( \hat{f}_{1,m}-\bar{b}_2 W_m\right) \partial _{x_1}q_m\,\mathrm{d}x_2, \end{aligned}$$

(A.13)

where $V_m=\partial _{x_1}W_m=0$ on $\Gamma _0$ because of (2.76). It follows from (2.76), (2.80), (A.4), (A.10), Cauchy-Schwarz inequality and trace inequality, and (A.13) that

$$\begin{aligned} \int _{\Gamma _0}\left( \partial _{x_1}q_m\right) ^2\,\mathrm{d}x_2 \leqq C\left( \Vert \hat{f}_1\Vert _{H^1\left( \Omega _L \right) } +\Vert \hat{f}_2\Vert _{L^2\left( \Omega _L \right) }+\Vert g_1\Vert _{C^1\left( \overline{\Gamma _0}\right) }\right) ^2, \end{aligned}$$

where the constant $C>0$ depends only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$. Combining this integral estimate with (A.12) yields

$$\begin{aligned} \int _{\Gamma _0}|\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}x_2 \leqq C\Bigl (\Vert \hat{f}_1^{(n)}\Vert _{H^1\left( \Omega _L \right) } +\Vert \hat{f}_2^{(n)}\Vert _{L^2\left( \Omega _L \right) }+\Vert g_1\Vert _{C^1(\overline{\Gamma _0})}\Bigr )^2, \end{aligned}$$

(A.14)

where the constant $C>0$ depends only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$.

Part 3. Estimate of $\int _{\Omega _t}(\partial _{x_2}^2V_m)^2\,\mathrm{d}\mathrm{x}$. First, (A.6) can be rewritten as

$$\begin{aligned} \langle \tilde{a}_{22}^{(n)}\partial _{x_2}^2V_m, \eta _k\rangle =\langle \partial _{x_1}q_m+2\tilde{a}_{12}^{(n)}\partial _{x_2}q_m+\bar{a}_1q_m +\bar{b}_1\partial _{x_1}W_m+\bar{b}_2 W_m-\hat{\mathfrak {f}}_{1,m},\eta _k\rangle \end{aligned}$$

(A.15)

for $x_1\in (0,L)$, $k=0,1,\ldots , m$. Since $\eta _k''=-(k\pi )^2\eta _k$ for each $k\in \mathbb {Z}^+$, it follows from (A.15) that

$$\begin{aligned}&\int _0^t\sum _{k=0}^m\vartheta _k\langle \tilde{a}_{22}^{(n)}\partial _{x_2}^2V_m, \eta _k''\rangle \mathrm{d}x_1 =\int _0^t\sum _{k=0}^m \vartheta _k\langle \partial _{x_1}q_m\\&\quad +\,2\tilde{a}_{12}^{(n)}\partial _{x_2}q_m+\bar{a}_1q_m +\bar{b}_1\partial _{x_1}W_m+\bar{b}_2 W_m-\hat{\mathfrak {f}}_{1,m},\eta _k''\rangle \mathrm{d}x_1, \end{aligned}$$

which is the same as

$$\begin{aligned} \begin{aligned}&\int _{\Omega _t}\tilde{a}_{22}^{(n)}(\partial _{x_2}^2V_m)^2\,\mathrm{d}\mathrm{x}\\&\quad =\,\,\int _{\Omega _t}\left( \partial _{x_1}q_m+2\tilde{a}_{12}^{(n)}\partial _{x_2}q_m+\bar{a}_1q_m +\bar{b}_1\partial _{x_1}W_m+\bar{b}_2 W_m-\hat{\mathfrak {f}}_{1,m}\right) \partial _{x_2}^2V_m\,\mathrm{d}\mathrm{x}. \end{aligned} \end{aligned}$$

(A.16)

Combining Lemmas 2.1 and 2.3, (2.78), Morrey’s inequality and Cauchy–Schwarz inequality, (A.4), and (A.16) yields that

$$\begin{aligned} \int _{\Omega _t} \left( \partial _{x_2}^2V_m\right) ^2\,\mathrm{d}\mathrm{x}\leqq C \left( \int _{\Omega _t} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}\mathrm{x}+(\Vert \hat{\mathfrak {f}}_1\Vert _{L^2\left( \Omega _L \right) }+\Vert \hat{\mathfrak {f}}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^0(\overline{\Gamma _0})})^2\right) \end{aligned}$$

(A.17)

for some constant $C>0$ depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0,L)$.

For notational convenience, define

$$\begin{aligned} \mathscr {Z}_m(t):=\int _{\Omega _t} |\nabla _{\mathrm{x}}q_m|^2\,\mathrm{d}\mathrm{x}\quad \text {for}\,\, t\in [0, L] \end{aligned}$$

and

$$\begin{aligned} \mathscr {E}\left( \hat{f}_1, \hat{f}_2, g_1\right) := \left( \Vert \hat{f}_1\Vert _{H^1\left( \Omega _L \right) }+\Vert \hat{f}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^1\left( \overline{\Gamma _0}\right) }\right) ^2. \end{aligned}$$

(A.18)

It follows from (A.8), (A.14), and (A.17) that $\mathscr {Z}_m$ satisfies a differential inequality

$$\begin{aligned} \mathscr {Z}_m'(t)\leqq \alpha \mathscr {Z}_m(t)+\beta \mathscr {E}\left( \hat{f}_1, \hat{f}_2, g_1\right) \quad \text{ for } 0<t<L, \end{aligned}$$

(A.19)

where the constants $\alpha$ and $\beta$ depend only on $(\gamma , J_0, S_0, \rho _0, \epsilon _0, L)$. Applying Gronwall’s inequality to (A.19) gives

$$\begin{aligned} \mathscr {Z}_m(L)\leqq C \mathscr {E}\left( \hat{f}_1, \hat{f}_2, g_1\right) . \end{aligned}$$

(A.20)

Finally, the estimate (A.20), together with (A.16), yields

$$\begin{aligned} \Vert V_m\Vert _{H^2\left( \Omega _L \right) }\leqq C\left( \Vert \hat{f}_1\Vert _{H^1\left( \Omega _L \right) }+\Vert \hat{f}_2\Vert _{L^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^1\left( \overline{\Gamma _0}\right) }\right) . \end{aligned}$$

(A.21)

In (A.20)–(A.21), the constants C may vary, but they depend only on $(\gamma , J_0, S_0, \rho _0, \epsilon _0, L)$.

Step 3. Estimate for higher order weak derivatives of $(V_m, W_m)$. In order to complete a priori $H^4$ estimates of $(V_m, W_m)$ in $\Omega _L$, one can apply the bootstrap argument. All the details can be given by employing the ideas in Steps 1 and 2, but they are much more lengthy and technical. So, in this step, we only describe main differences in establishing the estimate for higher order weak derivatives of $(V_m, W_m)$ in $\Omega _L$.

Note that the extension of $\hat{\mathfrak {f}}_{2,m}+\bar{h}_2\partial _{x_1}V_m$ given by even reflection about $\Lambda _L$ is $H^1$ across $\Lambda _L$ without any additional compatibility condition. Therefore, back to (A.3), applying Lemma 2.4, (A.21) and the method of reflection yield

$$\begin{aligned} \Vert W_m\Vert _{H^3\left( \Omega _L \right) }\leqq C\left( \Vert \hat{\mathfrak {f}}_2\Vert _{H^1\left( \Omega _L \right) }+\sqrt{\mathscr {E}(\hat{\mathfrak f}_1, \hat{\mathfrak f}_2, g_1)}\right) , \end{aligned}$$

(A.22)

where the constant $C>0$ depends only on $(\gamma , S_0, J_0, \rho _0, E_0, \epsilon _0, L)$ and $\mathscr {E}(\hat{\mathfrak f}_1, \hat{\mathfrak f}_2, g_1)$ is given in (A.18).

For a priori $H^3$ estimate of $V_m$, we adapt the argument in Step 2. The main difference is that the compatibility condition

$$\begin{aligned} \frac{\mathrm{d}g_1}{\mathrm{d}x_2}(\pm 1)=0 \end{aligned}$$

(A.23)

derived from (1.47) is used. For example, it follows from

$$\begin{aligned} \int _{\Gamma _0} (\partial _{x_2}^3V_m )^2\,\mathrm{d}x_2=\int _{\Gamma _0} \left( \sum _{j=0}^m \langle g_1, \eta _j\rangle \eta _j''\right) ^2\,\mathrm{d}x_2= \int _{\Gamma _0} \left( \sum _{j=0}^m \langle g_1, \eta _j''\rangle \eta _j \right) ^2\,\mathrm{d}x_2 \end{aligned}$$

that there is a differential inequality similar to (A.19). This, together with (A.23), gives

$$\begin{aligned} \langle g_1, \eta _j'' \rangle =\langle g_1'', \eta _j \rangle , \end{aligned}$$

from which one has

$$\begin{aligned} \int _{\Gamma _0} (\partial _{x_2}^3V_m )^2\,\mathrm{d}x_2 \leqq 2\left( \Vert g_1\Vert _{C^2(\overline{\Gamma _0})}\right) ^2. \end{aligned}$$

Furthermore, it can be directly checked from (2.8), (2.20), (2.67) and (2.68) that

$$\begin{aligned} \partial _{x_2}\hat{\mathfrak {f}}_{1} =0\quad \text{ on } \Lambda _L. \end{aligned}$$

Using this yields

$$\begin{aligned} \Vert \hat{\mathfrak {f}}_{1,m}\Vert _{H^2\left( \Omega _L \right) } \leqq \Vert \hat{\mathfrak {f}}_1\Vert _{H^2\left( \Omega _L \right) }\quad \text{ for } \text{ all } m\in \mathbb {Z}^+. \end{aligned}$$

With the aid of (A.21)– (A.23), it follows from lengthy but straightforward computations that

$$\begin{aligned} \Vert V_m\Vert _{H^3\left( \Omega _L \right) }\leqq C\left( \Vert \hat{f}_1\Vert _{H^2\left( \Omega _L \right) } +\Vert \hat{f}_2\Vert _{H^1\left( \Omega _L \right) } +\Vert g_1\Vert _{C^2\left( \overline{\Gamma _0}\right) }\right) , \end{aligned}$$

(A.24)

where the constant $C>0$ depending only on $(\gamma , S_0, J_0, \rho _0, E_0, \epsilon _0, L)$.

Similar to $H^3$ estimate of $W_m$ given in (A.22), a priori $H^4$-estimate of $W_m$ can be obtained by applying (A.24), [12, Theorems 8.8 and 8.12] and the method of reflection to (A.3) because the compatibility condition $\partial _{x_2}(\hat{\mathfrak {f}}_{2,m}+\bar{h}_2\partial _{x_1}V_m)=0$ holds on $\Lambda _{L}$. Furthermore, it can be directly checked from (2.20), (2.14), (2.67) and (2.68) that one has

$$\begin{aligned} \partial _{x_2}\hat{\mathfrak {f}}_{2}=0\quad \text{ on } \Lambda _L. \end{aligned}$$

Thus it holds that

$$\begin{aligned} \Vert \hat{\mathfrak {f}}_{2,m}\Vert _{H^2\left( \Omega _L \right) } \leqq \Vert \hat{\mathfrak {f}}_2\Vert _{H^2\left( \Omega _L \right) }\quad \text{ for } \text{ all } m\in \mathbb {Z}^+. \end{aligned}$$

Therefore, one has

$$\begin{aligned} \Vert W_m\Vert _{H^4\left( \Omega _L \right) }\leqq C\left( \Vert \hat{\mathfrak {f}}_1\Vert _{H^2\left( \Omega _L \right) } +\hat{\mathfrak {f}}_2\Vert _{H^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^2(\overline{\Gamma _0})}\right) . \end{aligned}$$

(A.25)

Finally, adapting the argument in Step 2 and using the estimate (A.25) yield

$$\begin{aligned} \Vert V_m\Vert _{H^4\left( \Omega _L \right) }\leqq C\left( \Vert \hat{\mathfrak {f}}_1\Vert _{H^3\left( \Omega _L \right) }+ \Vert \hat{\mathfrak {f}}_2\Vert _{H^2\left( \Omega _L \right) } +\Vert g_1\Vert _{C^3(\overline{\Gamma _0})} \right) . \end{aligned}$$

(A.26)

This finishes the proof of Lemma 2.8. $\square$

Appendix B. Proof of Lemma 3.5

Proof of Lemma 3.5

For a fixed $\tilde{Y}\in \mathcal {J}^\mathrm{ent}_{\delta _\mathrm{e}, L}$, denote ${{\varvec{{\mathcal {M}}}}}(\mathrm{x})=(\mathcal {M}_1, \mathcal {M}_2)(\mathrm{x})$ by

$$\begin{aligned} {{\varvec{{\mathcal {M}}}}}(\mathrm{x}):=\mathbf{M}\left( \mathrm{x}, \tilde{Y}, \Psi , \nabla \psi , \nabla \phi \right) \end{aligned}$$

for $\mathbf{M}(\mathrm{x}, \tilde{Y}, \Psi , \nabla \psi , \nabla \phi )$ defined by Definition 3.1(iii). Define a function $w:\overline{\Omega _L}\rightarrow \mathbb {R}$ by

$$\begin{aligned} w(x_1,x_2):=\int _{-1}^{x_2} \mathcal {M}_1(x_1, y)\,dy. \end{aligned}$$

By adjusting the proof of [4, Lemma 3.3] with using (1.55), (3.40), (3.41) and Lemma 3.3(a), the following properties hold:

• (i) $\displaystyle {\nabla ^{\perp } w={\varvec{\mathcal {M}}}}$ in $\overline{\Omega _L}$;

• (ii) $\displaystyle {0=w(0,-1)\leqq w(\mathrm{x})\leqq w(0,1)}$ in $\overline{\Omega _L}$;

• (iii) The function $w_0:=w(0,\cdot ):[-1,1]\rightarrow [0, w(0,1)]$ is strictly increasing, and its inverse $w_0^{-1}{:}\,[0, w(0,1)] \rightarrow [-1,1]$ is well defined;

• (iv) Define a Lagrangian coordinate mapping $\mathscr {L}_{\tilde{Y}}:\overline{\Omega _L}\rightarrow [-1,1]$ by

  $$\begin{aligned} \mathscr {L}_{\tilde{Y}}(\mathrm{x}):=w_0^{-1}\circ w(\mathrm{x}). \end{aligned}$$

  (B.1)

  From the definition of $\mathscr {L}_{\tilde{Y}}$, (3.44) stated in Lemma 3.5(a) can be directly checked. And, the function Y given by

  $$\begin{aligned} Y=\left( S_\mathrm{en}-S_0\right) \circ \mathscr {L}_{\tilde{Y}} \end{aligned}$$

  solves the boundary value problem (3.43).

In order to complete the proof of Lemma 3.5, it suffices to show that there exists a constant $C_{**}>0$ depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$ to satisfy

$$\begin{aligned} \Vert \mathscr {L}_{\tilde{Y}}\Vert _{H^4\left( \Omega _L \right) }\leqq C_{**}\quad \text{ for } \text{ all } \tilde{Y}\in \mathcal {J}^\mathrm{ent}_{\delta _\mathrm{e},L}. \end{aligned}$$

(B.2)

Once (B.2) is verified, then the rest of Lemma 3.5 can be easily proved by direct computations. In particular, the continuity of $\mathscr {L}_{\tilde{Y}}$ with respect to $\tilde{Y}$ in $H^3(\Omega _L)$ stated in Lemma 3.5(a) follows from the smooth dependence of $\mathbf{M}$ on $(\mathrm{x}, \tilde{Y},\Psi , \nabla \psi , \nabla \phi )$, and the continuous dependence of the fixed point $(\psi , \Psi , \phi )\in \mathcal {J}^\mathrm{pot}_{\delta _\mathrm{p},L}\times \mathcal {J}^\mathrm{vort}_{\delta _\mathrm{v},L}$ of the iteration mapping $\mathfrak {F}_1^{\tilde{Y}}$ on $\tilde{Y}\in \mathcal {J}^\mathrm{ent}_{\delta _\mathrm{e},L}$.

The rest of the proof devotes to verify (B.2). Due to the smooth dependence of $\mathbf{M}$ on $(\mathrm{x}, \tilde{Y}, \Psi , \nabla \psi , \nabla \phi )$, there exists a constant $C>0$ depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$ to satisfy

$$\begin{aligned} \Vert \mathbf{M}\left( \cdot , \tilde{Y}, \Psi , \nabla \psi , \nabla \phi \right) \Vert _{H^3\left( \Omega _L \right) }\leqq C\quad \text{ for } \text{ all } \tilde{Y}\in \mathcal {J}^\mathrm{ent}_{\delta _\mathrm{e},L}. \end{aligned}$$

(B.3)

It follows from (1.31), (3.13), (3.15), (3.17), and Definition 3.1(iii) that one has

$$\begin{aligned} \mathbf{M}(\mathrm{x}, \tilde{Y},\Psi , \nabla \psi , \nabla \phi )\cdot \mathbf{e}_1= \left( \frac{\gamma -1}{\gamma S_\mathrm{en}}\left( \Phi _0+\Psi -\frac{1}{2}(u_\mathrm{en}^2+v_\mathrm{en}^2)\right) \right) ^{\frac{1}{\gamma -1}}u_\mathrm{en}\quad \text{ on } \Gamma _0. \end{aligned}$$

(B.4)

Note that the fixed point $(\psi , \Psi , \phi )\in \mathcal {J}^\mathrm{pot}_{\delta _\mathrm{p},L}\times \mathcal {J}^\mathrm{vort}_{\delta _\mathrm{v},L}$ of the iteration mapping $\mathfrak {F}_1^{\tilde{Y}}$ solves the nonlinear boundary value problem (3.11) with boundary conditions (3.13)–(3.15) with Y being replaced by $\tilde{Y}$. Thus $\Psi$ can be considered as a solution to the linear boundary value problem

$$\begin{aligned} \begin{aligned} \Delta \Psi&=F_2\quad \text{ in } \Omega _L, \\ \Psi _{x_1}&=E_\mathrm{en}-E_0\,\,\text{ on } \Gamma _0,\quad \Psi _{x_2}=0\,\,\text{ on } \Lambda _L, \end{aligned} \end{aligned}$$

(B.5)

where

$$\begin{aligned} F_2:=\bar{h}_1\Psi +\bar{h}_2\psi _{x_1}+f_2\left( \mathrm{x}, \tilde{Y}, \Psi , \nabla \psi , \nabla \phi \right) \end{aligned}$$

with $(\bar{h}_1, \bar{h}_2)$ given by (2.13), and $f_2$ given by Definition 3.1(v). Using (3.32) and (3.35) yields

$$\begin{aligned} \Vert F_2\Vert _{H^3\left( \Omega _L \right) }\leqq C\left( \delta _\mathrm{e}+\sigma _\mathrm{v}\right) \end{aligned}$$

(B.6)

for $\sigma _\mathrm{v}$ given by (3.28). From (3.13), (3.14) and Definition 3.1, it can be directly checked that

$$\begin{aligned} \partial _{x_2}F_2=0\quad \text{ on } \Lambda _L. \end{aligned}$$

(B.7)

Applying the Morrey’s inequality to $F_2$ to obtain from (B.6) and (B.7) that

$$\begin{aligned} \Vert F_2\Vert _{C^{1,\frac{1}{2}}\left( \overline{\Omega _L}\right) }\leqq C\left( \delta _\mathrm{e}+\sigma _\mathrm{v}\right) . \end{aligned}$$

In addition, one has $\partial _{x_2}(E_\mathrm{en}-E_0)(\pm 1)=0$ due to (1.55). Then, by the standard Schauder estimates and the method of reflection, one has

$$\begin{aligned} \Vert \Psi \Vert _{C^{3, \frac{1}{2}}\left( \overline{\Omega _L\cap \left\{ x_1<\frac{L}{2}\right\} }\right) }\leqq C\left( \delta _\mathrm{e}+\sigma _\mathrm{v}\right) . \end{aligned}$$

(B.8)

The estimate constants C appeared so far vary, but they all depend only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$. It follows from (B.4) and (B.8) that $\mathbf{M}(\mathrm{x}, \tilde{Y},\Psi , \nabla \psi , \nabla \phi )\cdot \mathbf{e}_1\in C^3(\overline{\Gamma _0})$. Hence straightforward computations together with (3.42), (3.44), (B.3), (B.4), (B.8), the chain rule and the Sobolev inequality show that there exists a constant $C_{**}>0$ depending only on $(\gamma , J_0, S_0, \rho _0, E_0, \epsilon _0, L)$ to satisfy the estimate (B.2). This finishes the proof of Lemma 3.5. $\square$