On a Priori Energy Estimates for Characteristic Boundary Value Problems

Abstract

Motivated by the study of certain non linear free-boundary value problems for hyperbolic systems of partial differential equations arising in Magneto-Hydrodynamics, in this paper we show that an a priori estimate of the solution to certain boundary value problems, in the conormal Sobolev space $H^1_{ tan}$, can be transformed into an $L^2$ a priori estimate of the same problem.

Appendices

Appendix 1: Proof of Some Technical Lemmata

1.1 Proof of Lemma 10

For a given smooth function $u\in C^\infty _{(0)}(\mathbb {R}^n_+)$, an explicit calculation gives that

$$\begin{aligned} \lambda ^{m,\gamma }_{\chi }(Z)u(x)\!=\! \left\langle \mathcal {F}^{-1}\lambda ^{m,\gamma }(\cdot ),\chi (\cdot )e^{-\frac{(\cdot )_1}{2}}u(x_1e^{-(\cdot )_1},x'\!-\!(\cdot )') \right\rangle ,\quad \forall \,x\!=\!(x_1,x')\in \mathbb {R}^n_+. \end{aligned}$$

We have to prove that, under a suitable choice of $\varepsilon _0$, if $x\notin \mathbb {B}^+$ then $\lambda ^{m,\gamma }_{\chi }(Z)u(x)=0$. This is true if

$$\begin{aligned} y\mapsto v_x(y):=\chi (y)e^{-\frac{y_1}{2}}u(x_1e^{-y_1},x'-y') \end{aligned}$$

is identically zero as long as $x\notin \mathbb {B}^+$.

Since $\mathbb {R}^n_+\setminus \mathbb {B}^+=\{x=(x_1,x'):\,x_1\ge 1,\,\,\forall \,x'\in \mathbb {R}^{n-1}\}\cup \{x=(x_1,x'):\,|x'|\ge 1,\,\,\forall \,x_1\in [0,+\infty [\}$ we need to analyze the following two cases.

1st case: $x_1\ge 1$.

Let $y\in \mathbb {R}^n$ be arbitrarily fixed. If $y\notin \mathrm{supp}\chi$, then $\chi (y)=0$, which implies $v_x(y)=0$. If $y\in \mathrm{supp}\chi$, then we have $-\varepsilon _0\le y_1\le \varepsilon _0$ and $|y'|\le \varepsilon _0$. Hence, we derive that $e^{-\varepsilon _0}\le e^{-y_1}\le e^{\varepsilon _0}$ and, since $x_1\ge 1$, $x_1e^{-y_1}\ge e^{-y_1}\ge e^{-\varepsilon _0}$. Since $u(x_1,x')=0$ when $x_1\ge \delta _0$, if we choose $\varepsilon _0>0$ such that $e^{-\varepsilon _0}>\delta _0$ (that is equivalent to $\varepsilon _0<\log (1/\delta _0)$), then we get that

$$\begin{aligned} \forall \,y\in \mathrm{supp}\chi ,\,\,\forall \,x_1\ge 1\,:\quad u(x_1e^{-y_1},x'-y')=0, \end{aligned}$$

which gives $v_x(y)=0$.

2nd case: $|x'|\ge 1$.

Again, if $y\notin \mathrm{supp}\chi$, then $v_x(y)=0$. If $y\in \mathrm{supp}\chi$ then $|x'-y'|\ge |x'|-|y'|\ge 1-|y'|\ge 1-\varepsilon _0$. To conclude, in this case it is sufficient to choose $\varepsilon _0>0$ such that $1-\varepsilon _0>\delta _0$ in order to have again $v_x(y)=0$.

Finally, the result is proved if we choose $0<\varepsilon _0\le \min \{\log (1/\delta _0),1-\delta _0\}$.

1.2 Proof of Lemma 12

For arbitrary $u\in L^2(\mathbb {R}^n_+)$, we observe that in view of (35), (38)

$$\begin{aligned} (r_m(Z,\gamma )u)^{\sharp }=r_m(D,\gamma )u^\sharp =\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))*u^\sharp \,; \end{aligned}$$

(128)

then, for arbitrary $\beta \in \mathbb {N}^n$:

$$\begin{aligned} \partial ^{\beta }(r_m(Z,\gamma )u)^{\sharp }=(\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))*u^\sharp \,. \end{aligned}$$

Since $H^{p}_{tan,\gamma }(\mathbb {R}^n_+)$ is topologically isomorphic to $H^p_{\gamma }(\mathbb {R}^n)$ for all positive integers $p$, via the $\sharp$ operator, and $u^\sharp \in L^2(\mathbb {R}^n)$, then $r_{m}(Z,\gamma )u\in H^p_{tan,\gamma }(\mathbb {R}^n_+)$ is proven provided that $\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))$ belongs to $L^1(\mathbb {R}^n)$ for all $\beta \in \mathbb {N}^n$ with $|\beta |\le p$.

On the other hand, by the standard properties of the Fourier transform and by (46), we get

$$\begin{aligned}&\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))=\mathcal {F}^{-1}((I-\chi (D))\lambda ^{m,\gamma })=\mathcal {F}^{-1}(\mathcal {F}^{-1}((1-\chi )\widehat{\lambda ^{m,\gamma }}))\nonumber \\&\quad =(2\pi )^{-n}\widetilde{((1-\chi )\widehat{\lambda ^{m,\gamma }})}=(1-\chi )\mathcal {F}^{-1}(\lambda ^{m,\gamma }), \end{aligned}$$

(129)

where we have used the identity $\mathcal {F}^{-1}g= (2\pi )^{-n}\widetilde{\widehat{g}},$ with $\widetilde{g}(x)=g(-x)$, and that $\chi$ is an even function.

Let us firstly focus on $\mathcal {F}^{-1}(\lambda ^{m,\gamma })$. For arbitrary positive integers $N, k$ and $\beta \in \mathbb {N}^n$ one computes

$$\begin{aligned}&{|z|^{2(N+k)}\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)=i^{|\beta |}\sum \limits _{|\alpha |=N+k}\frac{(N+k)!}{\alpha !}z^{2\alpha }\mathcal {F}^{-1}(\xi ^\beta \lambda ^{m,\gamma })(z)}\nonumber \\&\quad {=i^{|\beta |}(-1)^{N+k}\sum \limits _{|\alpha |=N+k}\frac{(N+k)!}{\alpha !}\mathcal {F}^{-1}\left( \partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma })\right) (z)\,.} \end{aligned}$$

(130)

On the other hand, since $\lambda ^{m,\gamma }\in \Gamma ^m$, for $|\alpha |=N+k$ we get

$$\begin{aligned}&|\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi ))|\le C_{\alpha ,\beta }\lambda ^{m+|\beta |-2|\alpha |,\gamma }(\xi )= C_{\alpha ,\beta }\lambda ^{m+|\beta |-2(N+k),\gamma }(\xi )\\&\quad =C_{\alpha ,\beta }\lambda ^{-2k,\gamma }(\xi )\lambda ^{m+|\beta |-2N,\gamma }(\xi )\\&\quad \le C_{\alpha ,\beta }\gamma ^{-2k}\lambda ^{m+|\beta |-2N,\gamma }(\xi ) ,\quad \forall \,\xi \in \mathbb {R}^n,\,\,\forall \,\gamma \ge 1\,. \end{aligned}$$

For fixed $\beta$, we choose the integer $N_{\beta }=N$ such that $2N\ge m+|\beta |+1+n$; then

$$\begin{aligned} \lambda ^{m+|\beta |-2N,\gamma }(\xi )\le \lambda ^{-(1+n),\gamma }(\xi )\le (1+|\xi |^2)^{-\frac{n+1}{2}},\quad \forall \,\xi \in \mathbb {R}^n,\,\,\forall \,\gamma \ge 1 \end{aligned}$$

yields

$$\begin{aligned} |\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi ))|\le C_{\alpha ,\beta }\gamma ^{-2k}(1+|\xi |^2)^{-\frac{n+1}{2}},\quad \forall \,\xi \in \mathbb {R}^n,\,\,\forall \,\gamma \ge 1\,; \end{aligned}$$

hence $\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi ))\in L^1(\mathbb {R}^n)$ and, from Riemann–Lebesgue Theorem, $\mathcal {F}^{-1} (\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi )))\in L^\infty (\mathbb {R}^n)\cap C^0(\mathbb {R}^n)$ and we have

$$\begin{aligned} \begin{array}{ll} {||\mathcal {F}^{-1}(\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi )))||_{L^\infty (\mathbb {R}^n)}\le \int \limits _{\mathbb {R}^n}|\partial ^{2\alpha }_\xi (\xi ^\beta \lambda ^{m,\gamma }(\xi ))|\,d\xi }\\ \qquad {\le C_{\alpha ,\beta }\gamma ^{-2k}\int \limits _{\mathbb {R}^n}(1+|\xi |^2)^{-\frac{n+1}{2}}d\xi \le C_{\alpha ,\beta ,n}\gamma ^{-2k},\quad \forall \,\gamma \ge 1\,.} \end{array} \end{aligned}$$

Therefore, in view of (130),

$$\begin{aligned} |z|^{2(N+k)}\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)\in L^\infty (\mathbb {R}^n)\cap C^0(\mathbb {R}^n) \end{aligned}$$

and

$$\begin{aligned} |z|^{2(N+k)}|\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)|\le C_{k,N,\beta ,n}\gamma ^{-2k},\quad \forall \,z\in \mathbb {R}^n,\gamma \ge 1, \end{aligned}$$

where the constant $C_{k,N,\beta ,n}$ is independent of $\gamma$.

Summarizing, we have proved that:

$$\begin{aligned} \begin{array}{ll} {\forall \,\beta \in \mathbb {N}^n,\forall \,k,N\in \mathbb {N},\,\text {with}\,\, k\ge 1,\,\,\,N\ge \frac{m+|\beta |+1+n}{2},\,\,\,\exists \,C=C_{k,N,\beta ,n}>0:}\\ {1.\quad |z|^{2(N+k)}\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)\in L^\infty (\mathbb {R}^n)\cap C^0(\mathbb {R}^n)}\\ {2.\quad |z|^{2(N+k)}|\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)|\le C_{k,N,\beta ,n}\gamma ^{-2k},\quad \forall \,z\in \mathbb {R}^n,\gamma \ge 1\,.} \end{array} \end{aligned}$$

For arbitrary $\beta \in \mathbb {N}^n$, we consider $\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))$. From (129) we compute, by Leibniz formula,

$$\begin{aligned} \partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))(z)&= -\sum \limits _{\nu <\beta } \left( \begin{array}{ll}\beta \\ \nu \end{array}\right) \partial ^{\beta -\nu }_z\chi (z)\partial ^\nu _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)\nonumber \\&\quad +\, (1-\chi )(z)\partial ^{\beta }_z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)\,. \end{aligned}$$

(131)

Note that $\partial ^{\beta -\nu }\chi$, for all $\nu <\beta$, and $1-\chi$ are identically zero on a neighbourhood of $z=0$. Then, from 1, 2 above we derive that

$$\begin{aligned} \begin{array}{ll} {\forall \,\beta \in \mathbb {N}^n,\forall \,k,N\in \mathbb {N},\,\text {with}\,\, k\ge 1,\,\,\,N\ge \frac{m+|\beta |+1+n}{2},\,\,\,\exists \,C=C_{k,N,\beta ,\chi ,n}>0:}\\ 3.\quad \partial ^{\beta -\nu }_z\chi (z)\partial ^\nu _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z),\, (1-\chi )(z)\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z) \in L^\infty (\mathbb {R}^n)\cap C^0(\mathbb {R}^n),\\ \forall \,\nu <\beta \,;\\ \begin{array}{ll}4.\quad |\partial ^{\beta -\nu }_z\chi (z)\partial ^\nu _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)|\le C_{k,N,\beta ,\chi ,n}\gamma ^{-2k}(1+|z|^2)^{-N},\\ \quad \quad |(1-\chi )(z)\partial ^\beta _z\mathcal {F}^{-1}(\lambda ^{m,\gamma })(z)|\le C_{k,N,\beta ,\chi ,n}\gamma ^{-2k}(1+|z|^2)^{-N},\,\,\forall \,z\in \mathbb {R}^n,\\ \nu <\beta ,\,\,\gamma \ge 1\,.\end{array} \end{array} \end{aligned}$$

Thus, applying 4 for $N\ge \max \left\{ {\frac{n+1}{2}},{\frac{m+|\beta |+n+1}{2}}\right\}$, from (131) we obtain that $\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))\in L^1(\mathbb {R}^n)$ and for all $\gamma \ge 1$:

$$\begin{aligned} ||\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))||_{L^1(\mathbb {R}^n)}&\le C_{N,k,n,\beta ,\chi }\gamma ^{-2k}\int \limits _{\mathbb {R}^n}(1+|z|^2)^{-N}dz\le C_{k,n,\beta ,\chi }\gamma ^{-2k}\nonumber \\&\le C_{k,n,\beta ,\chi }\gamma ^{-k} , \end{aligned}$$

(132)

where the constant $C_{k,n,\beta ,\chi }$ is independent of $\gamma$.

For every positive integer $p$, applying the above result to all multi-indices $\beta \in \mathbb {N}^n$ with $|\beta |\le p$ gives that $\partial ^\beta (r_{m}(Z,\gamma )u)^\sharp =\partial ^\beta \mathcal {F}^{-1}(r_m(\cdot ,\gamma ))*u^\sharp$ belongs to $L^2(\mathbb {R}^n)$ with

$$\begin{aligned} ||\partial ^\beta (r_{m}(Z,\gamma )u)^\sharp ||_{L^2(\mathbb {R}^n)}&\le ||\partial ^{\beta }\mathcal {F}^{-1}(r_m(\cdot ,\gamma ))||_{L^1(\mathbb {R}^n)}||u^{\sharp }||_{L^2(\mathbb {R}^n)}\nonumber \\&\le C_{k,n,\beta ,\chi }\gamma ^{-k}||u||_{L^2(\mathbb {R}^n_+)}\,. \end{aligned}$$

(133)

This gives that $r_m(Z,\gamma )u\in H^p_{tan,\gamma }(\mathbb {R}^n_+)$. Furthermore, for an arbitrary positive integer $h$ we apply (133) for each $\beta \in \mathbb {N}^n$ with $|\beta |\le p$ for $k=p-|\beta |+h$ to get

$$\begin{aligned}&||r_m(Z,\gamma )u||^2_{H^p_{tan,\gamma }(\mathbb {R}^n_+)}\le C_p||(r_m(Z,\gamma )u)^\sharp ||^2_{H^p_\gamma (\mathbb {R}^n)}\nonumber \\&\quad =\sum \limits _{|\beta |\le p}\gamma ^{2(p-|\beta |)}||\partial ^\beta (r_{m}(Z,\gamma )u)^\sharp ||^2_{L^2(\mathbb {R}^n)}\nonumber \\&\quad \le \sum \limits _{|\beta |\le p}\gamma ^{2(p-|\beta |)} C_{h,p,n,\beta ,\chi }\gamma ^{-2(p-|\beta |+h)}||u||^2_{L^2(\mathbb {R}^n_+)}\nonumber \\&\quad \le C_{h,p,n,\chi }\gamma ^{-2h}||u||^2_{L^2(\mathbb {R}^n_+)}, \end{aligned}$$

(134)

for a suitable $\gamma$-independent positive constant $C_{h,p,n,\chi }$. This shows the estimate (49) and completes the proof.

1.3 Proof of Proposition 13

Let $u\in C^{\infty }_{(0)}(\mathbb {R}^n_+)$; to find a symbol $b'_{m}$ satisfying (53), from (46) we firstly compute

$$\begin{aligned}&(\lambda ^{m,\gamma }_{\chi }(Z)u)^{\sharp }(x)=\lambda ^{m,\gamma }_{\chi }(D)(u^{\sharp })(x)=(\mathcal {F}^{-1}(\lambda ^{m,\gamma }_{\chi })*u^{\sharp })(x)\\&\quad =\langle \mathcal {F}^{-1}(\lambda ^{m,\gamma }_{\chi }),u^{\sharp }(x-\cdot )\rangle \\&\quad {=\langle \mathcal {F}^{-1}(\lambda ^{m,\gamma }),\chi (\cdot )e^\frac{{x_1-(\cdot )_1}}{2}u(e^{x_1-(\cdot )_1},x'-(\cdot )')\rangle ,\quad \forall \,(x_1,x')\in \mathbb {R}^n,} \end{aligned}$$

hence, by (31),

$$\begin{aligned} \begin{array}{ll} {\lambda ^{m,\gamma }_{\chi }(Z)u(x)=\langle \mathcal {F}^{-1}(\lambda ^{m,\gamma }),\chi (\cdot )e^\frac{{x_1-(\cdot )_1}}{2}u(e^{x_1-(\cdot )_1},x'-(\cdot )')\rangle ^{\sharp ^{-1}}}\\ \\ {=\frac{1}{\sqrt{x_1}}\left\langle \mathcal {F}^{-1}(\lambda ^{m,\gamma }),\chi (\cdot )e^\frac{{\log x_1-(\cdot )_1}}{2}u(e^{\log x_1-(\cdot )_1},x'-(\cdot )')\right\rangle }\\ \\ \quad {=\left\langle \mathcal {F}^{-1}(\lambda ^{m,\gamma }),\chi (\cdot )e^{\frac{-(\cdot )_1}{2}}u(x_1e^{-(\cdot )_1},x'-(\cdot )')\right\rangle }\\ \\ \quad {=\left\langle \lambda ^{m,\gamma },\mathcal {F}^{-1}\left( \chi (\cdot )e^{\frac{-(\cdot )_1}{2}}u(x_1e^{-(\cdot )_1},x'-(\cdot )')\right) \right\rangle }\\ \\ \quad =(2\pi )^{-n}\int \lambda ^{m,\gamma }(\xi )\left( \int e^{i\xi \cdot y}\chi (y)e^{-\frac{y_1}{2}}u(x_1e^{-y_1},x'-y')dy\right) d\xi ,\quad \forall \,x_1>0,\,\,\\ \qquad \forall \,x'\in \mathbb {R}^{n-1}\,. \end{array} \end{aligned}$$

The regularity of $u$ legitimates all the above calculations. Setting $x_1=0$ in the last expression above, we deduce the corresponding expression for the trace on the boundary of $\lambda ^{m,\gamma }_{\chi }(Z)u$

$$\begin{aligned}&(\lambda ^{m,\gamma }_{\chi }(Z)u)_{|\,x_1=0}(x')\nonumber \\&\quad =(2\pi )^{-n}\int \lambda ^{m,\gamma }(\xi )\left( \int e^{i\xi \cdot y}\chi (y)e^{-\frac{y_1}{2}}(u_{|\,x_1=0})(x'-y')dy\right) d\xi \,.\qquad \end{aligned}$$

(135)

Now we substitute (51) into the $y$-integral appearing in the last expression above; then Fubini’s theorem gives

$$\begin{aligned}&\int e^{i\xi \cdot y}\chi _1(y_1)\widetilde{\chi }(y')e^{-\frac{y_1}{2}}(u_{|\,x_1=0})(x'-y')dy\nonumber \\&\quad {=\int e^{i\xi '\cdot y'}\left( \int e^{i\xi _1y_1}e^{-\frac{y_1}{2}}\chi _1(y_1)dy_1\right) \widetilde{\chi }(y')(u_{|\,x_1=0})(x'-y')dy'}\nonumber \\&\quad {=\int e^{i\xi '\cdot y'}\left( \int e^{-i\xi _1(-y_1)}e^{-\frac{y_1}{2}}\chi _1(y_1)dy_1\right) \widetilde{\chi }(y')(u_{|\,x_1=0})(x'-y')dy'}\nonumber \\&\quad {=\int e^{i\xi '\cdot y'}\left( \int e^{-i\xi _1(-y_1)}e^{-\frac{y_1}{2}}\chi _1(-y_1)dy_1\right) \widetilde{\chi }(y')(u_{|\,x_1=0})(x'-y')dy'}\nonumber \\&\quad {=\left( e^{\frac{(\cdot )_1}{2}}\chi _1\right) ^{\wedge _1}(\xi _1)\int e^{i\xi '\cdot y'}\widetilde{\chi }(y')(u_{|\,x_1=0})(x'-y')dy',} \end{aligned}$$

(136)

where we have used that $\chi _1$ is even and $\wedge _1$ denotes the one-dimensional Fourier transformation with respect to $y_1$. Writing, by the inversion formula, $(u_{|\,x_1=0})(x'-y')=(2\pi )^{-n+1}\int e^{i(x'-y')\cdot \eta '}\widehat{u_{|\,x_1=0}}(\eta ')d\eta '$ and using once more Fubini’s theorem and that $\widetilde{\chi }$ is even, we further obtain

$$\begin{aligned}&\int e^{i\xi '\cdot y'}\widetilde{\chi }(y')(u_{|\,x_1=0})(x'-y')dy'\nonumber \\&\quad =(2\pi )^{-n+1}\int e^{i\xi '\cdot y'}\widetilde{\chi }(y')\left( \int e^{i(x'-y')\cdot \eta '}\widehat{u_{|\,x_1=0}}(\eta ')d\eta '\right) dy'\nonumber \\&\quad {=\int e^{ix'\cdot \eta '}\left( (2\pi )^{-n+1}\int e^{i(\xi '-\eta ')\cdot y'}\widetilde{\chi }(y')dy'\right) \widehat{u_{|\,x_1=0}}(\eta ')d\eta '}\nonumber \\&\quad {=\int e^{ix'\cdot \eta '}\left( (2\pi )^{-n+1}\int e^{-i(\xi '-\eta ')\cdot (-y')}\widetilde{\chi }(-y')dy'\right) \widehat{u_{|\,x_1=0}}(\eta ')d\eta '}\nonumber \\&\quad {=(2\pi )^{-n+1}\int e^{ix'\cdot \eta '}\widehat{\widetilde{\chi }}(\xi '-\eta ')\widehat{u_{|\,x_1=0}}(\eta ')d\eta '\,;} \end{aligned}$$

(137)

here $\wedge$ is used here to denote the $(n-1)$-dimensional Fourier transformation with respect to $x'$. Inserting (136), (137) into (135) then leads to

$$\begin{aligned}&{(\lambda ^{m,\gamma }_{\chi }(Z)u)_{|\,x_1=0}(x')}\nonumber \\&\quad =(2\pi )^{-n}\int \lambda ^{m,\gamma }(\xi )\left( e^{\frac{(\cdot )_1}{2}}\chi _1\right) ^{\wedge _1}(\xi _1){\left( (2\pi )^{-n+1}\int e^{ix'\cdot \eta '}\widehat{\widetilde{\chi }}(\xi '-\eta ')\widehat{u_{|\,x_1=0}}(\eta ')d\eta '\right) d\xi \,.}\nonumber \\ \end{aligned}$$

(138)

Because $\left( e^{\frac{(\cdot )_1}{2}}\chi _1\right) ^{\wedge _1}\in \mathcal {S}(\mathbb {R})$, $\widehat{\widetilde{\chi }}\in \mathcal {S}(\mathbb {R}^{n-1})$ and $\widehat{u_{|\,x_1=0}}\in \mathcal {S}(\mathbb {R}^{n-1})$, the double integral

$$\begin{aligned} \int \int e^{ix'\cdot \eta '} \lambda ^{m,\gamma }(\xi )\left( e^{\frac{(\cdot )_1}{2}}\chi _1\right) ^{\wedge _1}(\xi _1)\widehat{\widetilde{\chi }}(\xi '-\eta ')\widehat{u_{|\,x_1=0}}(\eta ')d\eta 'd\xi \end{aligned}$$

converges absolutely; hence Fubini’s theorem allows to exchange the order of the integrations in (138) and find

$$\begin{aligned} (\lambda ^{m,\gamma }_{\chi }(Z)u)_{|\,x_1=0}(x')=(2\pi )^{-n+1}\int e^{ix'\cdot \eta '} b'_{m}(\eta ',\gamma )\widehat{u_{|\,x_1=0}}(\eta ')d\eta ', \end{aligned}$$

(139)

where $b'_{m}(\eta ',\gamma )$ is defined by (52). This shows the identity (53).

1.4 Proof of Lemma 14

We follow the same lines of the proof of [20, Lemma 4.11]. Setting for short

$$\begin{aligned} \phi (x):=e^{x_1/2}\chi _1(x_1)\widetilde{\chi }(x'), \end{aligned}$$

(140)

the symbol (52) can be re-written as

$$\begin{aligned} b'_{m}(\xi ',\gamma )=(2\pi )^{-n}\int \lambda ^{m,\gamma }(\eta _1,\eta '+\xi ')\widehat{\phi }(\eta )\,d\eta \,. \end{aligned}$$

(141)

Substituting in (141) the function $\eta \mapsto \lambda ^{m,\gamma }(\eta _1,\eta '+\xi ')$ by its Taylor expansion about $\eta =0$

$$\begin{aligned}&\lambda ^{m,\gamma }(\eta _1,\eta '+\xi ')=\sum \limits _{|\alpha |<N}\frac{(\partial ^{\alpha }\lambda ^{m,\gamma })(0,\xi ')}{\alpha !}\eta ^\alpha \nonumber \\&\quad +\, N\sum \limits _{|\alpha |=N}\frac{\eta ^\alpha }{\alpha !} \int \limits _0^1(\partial ^\alpha \lambda ^{m,\gamma })(t\eta _1,\xi '+t\eta ')(1-t)^{N-1}dt \end{aligned}$$

(142)

for $N=2$, we get

$$\begin{aligned}&{b'_{m}(\xi ',\gamma )} =(2\pi )^{-n}\int \left[ \lambda ^{m,\gamma }(\xi ')+\sum \limits _{j=1}^n\eta _j(\partial _j\lambda ^{m,\gamma })(0,\xi ')\right. \nonumber \\&\quad \qquad \qquad \qquad \left. +\, 2\sum \limits _{|\alpha |=2}\frac{\eta ^\alpha }{\alpha !} \int \limits _0^1(\partial ^\alpha \lambda ^{m,\gamma })(t\eta _1,\xi '+t\eta ')(1-t)dt\right] \widehat{\phi }(\eta )\,d\eta \nonumber \\&\quad {=(2\pi )^{-n}\lambda ^{m,\gamma }(\xi ')\int \widehat{\phi }(\eta )\,d\eta -i(2\pi )^{-n}\sum \limits _{j=1}^n(\partial _j\lambda ^{m,\gamma })(0,\xi ')\int \widehat{\partial _j\phi }(\eta )d\eta }\nonumber \\&\quad \qquad {-\,2(2\pi )^{-n}\sum \limits _{|\alpha |=2}\frac{1}{\alpha !}\int \left( \int \limits _0^1\partial ^{\alpha }\lambda ^{m,\gamma }(t\eta _1,t\eta '+\xi ')(1-t)\,dt\right) \widehat{\partial ^{\alpha }\phi }(\eta )\,d\eta \,.}\nonumber \\ \end{aligned}$$

(143)

From Plancherel’s identity and (140) (cf. also (45), (51)) we compute

$$\begin{aligned} \begin{array}{ll} {(2\pi )^{-n}\int \widehat{\phi }(\eta )\,d\eta =\phi (0)=1,}\\ {(2\pi )^{-n}\int \widehat{\partial _1\phi }(\eta )\,d\eta =\partial _1\phi (0)=-\frac{1}{2},}\\ {(2\pi )^{-n}\int \widehat{\partial _j\phi }(\eta )\,d\eta =\partial _j\phi (0)=0,\quad j\ge 2\,.} \end{array} \end{aligned}$$

(144)

On the other hand, from (14) one trivially computes that $(\partial _1\lambda ^{m,\gamma })(0,\xi ')=0$ for all $\xi '\in \mathbb {R}^{n-1}$. Inserting the last relation and (144) into (143) then gives (54), where we set

$$\begin{aligned} \beta _{m,\delta }(\xi ',\gamma )&:= -2(2\pi )^{-n}\sum \limits _{|\alpha |=2}\frac{1}{\alpha !}\int \left( \int \limits _0^1\partial ^\alpha \lambda ^{m,\gamma }(t\eta _1,t\eta '+\xi ')(1-t)\,dt\right) \nonumber \\&\times \, \widehat{\partial ^\alpha \phi }(\eta )\,d\eta \,. \end{aligned}$$

(145)

To prove that $\beta _m$ belongs to $\Gamma ^{m-2}$, differentiation under the integral sign of (145) gives, for an arbitrary $\nu '\in \mathbb {N}^{n-1}$,

$$\begin{aligned}&\partial ^{\nu '}_{\xi '}\beta _{m,\delta }(\xi ',\gamma )\nonumber \\&\quad =-2(2\pi )^{-n}\sum \limits _{|\alpha |=2}^n\frac{1}{\alpha !}\int \left[ \partial ^{\nu '}_{\xi '} \left( \int \limits _0^1(\partial ^\alpha \lambda ^{m,\gamma })(t\eta _1,t\eta '+\xi ')(1-t)\,dt\right) \right] \times \,\widehat{\partial ^\alpha \phi }(\eta ) d\eta \nonumber \\&\quad =-2(2\pi )^{-n} {\sum \limits _{|\alpha |=2}^n\frac{1}{\alpha !}\int \left[ \!\int \limits _0^1(\partial ^{\alpha +(0,\nu ')}\lambda ^{m,\gamma })(t\eta _1,t\eta '\!+\!\xi ')(1-t)\,dt\right] \widehat{\partial ^\alpha \phi }(\eta )\,d\eta \,;}\quad \nonumber \\ \end{aligned}$$

(146)

hence from $\lambda ^{m,\gamma }\in \Gamma ^m$ we obtain

$$\begin{aligned} |\partial ^{\nu '}_{\xi '}\beta _{m,\delta }(\xi ',\gamma )| \le C_{m,\nu '}\sum \limits _{|\alpha |=2}\int \left( \int \limits _0^1\lambda ^{m-2-|\nu '|,\gamma }(t\eta _1,t\eta '+\xi ')\,dt\right) |\widehat{\partial ^\alpha \phi }(\eta )|\,d\eta ,\nonumber \\ \end{aligned}$$

(147)

for a suitable $\gamma -$independent positive constant $C_{m,\nu '}$.

Recall that, for all $s\in \mathbb {R}$, $\gamma \ge 1$ and $\xi ,\eta \in \mathbb {R}^n$

$$\begin{aligned} \lambda ^{s,\gamma }(\xi )\le 2^{|s|}\lambda ^{s,\gamma }(\xi -\eta )\lambda ^{|s|}(\eta ), \end{aligned}$$

(148)

see [6], [27, Lemma 1.18]. Then, we apply (148) (for $s=m-2-|\nu '|$) to estimate $\lambda ^{m-2-|\nu '|,\gamma }(t\eta _1,t\eta '+\xi ')$ within the right-hand side of (147) by

$$\begin{aligned} \begin{array}{ll} {\lambda ^{m-2-|\nu '|,\gamma }(t\eta _1,t\eta '+\xi ')\le 2^{|m-2-|\nu '||}\lambda ^{m-2-|\nu '|,\gamma }(\xi ')\lambda ^{|m-2-|\nu '||}(t \eta )}\\ \qquad \qquad \qquad \qquad \le 2^{|m-2-|\nu '||}\lambda ^{m-2-|\nu '|,\gamma }(\xi ')\lambda ^{|m-2-|\nu '||}(\eta ),\\ \quad \forall \,\xi '\in \mathbb {R}^{n-1},\,\eta \in \mathbb {R}^n,t\in [0,1], \end{array} \end{aligned}$$

and combine with (147) to finally get

$$\begin{aligned} |\partial ^{\nu '}_{\xi '}\beta _{m}(\xi ',\gamma )|&\le C'_{m,\nu '}\lambda ^{m-2-|\nu '|,\gamma }(\xi ')\sum \limits _{|\alpha |=2}\int \lambda ^{|m-2-|\nu '||}(\eta )|\widehat{\partial ^\alpha \phi }(\eta )|d\eta \nonumber \\&\le C''_{m,\nu '}\lambda ^{m-2-|\nu '|,\gamma }(\xi '), \end{aligned}$$

(149)

for $C'_{m,\nu '}, C''_{m,\nu '}$ suitable positive constants independent of $\gamma$ (notice in particular that the integrals in the sum involved in the right-hand side of the first inequality in (149) are absolutely convergent, because $\widehat{\partial ^{\alpha }{\phi }}\in \mathcal {S}(\mathbb {R}^n)$ for all $|\alpha |=2$).

1.5 Proof of Corollary 15

For all $\psi \in C^\infty _0(\mathbb {R}^{n-1})$ under the above assumptions, let $\Psi \in C^\infty _{(0)}(\mathbb {R}^n_+)$ be chosen in such a way that

$$\begin{aligned} \mathrm{supp}\,\Psi \subseteq \mathbb {B}^+_{\delta _0},\quad \Psi _{|\,x_1=0}=\psi \,. \end{aligned}$$

(150)

Such a function $\Psi$ could be for instance obtained as

$$\begin{aligned} \Psi (x_1,x'):=\eta (x_1)\psi (x'),\quad \forall \,x_1\ge 0,\,\,x'\in \mathbb {R}^{n-1}, \end{aligned}$$

with $\eta =\eta (x_1)\in C^\infty _{(0)}([0,+\infty [)$ such that

$$\begin{aligned} \eta (x_1)=1,\quad 0\le x_1<\frac{\delta _0}{2},\quad \eta (x_1)=0,\quad x_1>\delta _0\,. \end{aligned}$$

Then, in view of Proposition 13 one has

$$\begin{aligned} b'_{m}(D',\gamma )\psi =b'_{m}(D',\gamma )(\Psi _{|\,x_1=0})=(\lambda ^{m,\gamma }_{\chi }(Z)\Psi )_{|\,x_1=0}\,. \end{aligned}$$

Then, from (150) and Lemma 10,

$$\begin{aligned} \mathrm{supp}\,b'_{m}(D',\gamma )\psi \subset \mathbb {B}^+\cap \{x_1=0\}=\mathcal {B}(0;1)\,. \end{aligned}$$

1.6 Proof of Lemma 17

Recall that we have defined for each $k=1,\dots ,n$

$$\begin{aligned} q_{k,m}(x,\xi ,\gamma ):=(2\pi )^{-n}\int \limits _{\mathbb {R}^n}\widehat{b}_k(x,\eta )\partial _k\lambda ^{m,\gamma }(\xi -\eta )\,d\eta , \end{aligned}$$

(151)

where the functions $b_k=b_k(x,y)$ (cf. (72)) are given in $C^{\infty }(\mathbb {R}^n\times \mathbb {R}^n)$, have bounded derivatives in $\mathbb {R}^n\times \mathbb {R}^n$, and satisfy for all $x\in \mathbb {R}^n$

$$\begin{aligned} \mathrm{supp}\,b_k(x,\cdot )\subseteq \{|y|\le 2\varepsilon _0\}\,. \end{aligned}$$

Recall also that $\widehat{b}_k(x,\zeta )$ denotes the partial Fourier transform of $b_k(x,y)$ with respect to $y$.

The following lemma is concerned with the behavior at infinity of $\widehat{b}_k(x,\zeta )$.

Lemma 19

Let the function $b_k=b_k(x,y)\in C^{\infty }(\mathbb {R}^n\times \mathbb {R}^n)$ obey all of the preceding assumptions. Then, for every positive integer $N$ and all multi-indices $\alpha \in \mathbb {N}^n$ there exists a positive constant $C_{N,\alpha }$ such that

$$\begin{aligned} (1+|\zeta |^2)^N|\partial ^{\alpha }_x\widehat{b}_k(x,\zeta )|\le C_{N,\alpha },\quad \forall \,x,\,\zeta \in \mathbb {R}^n\,. \end{aligned}$$

(152)

Proof

Since for each $x\in \mathbb {R}^n$, the function $b_k(x,\cdot )$ has compact support (independent of $x$), integrating by parts we get for an arbitrary integer $N>0$

$$\begin{aligned}&{(1+|\zeta |^2)^N\widehat{b}_k(x,\zeta )=\sum \limits _{|\alpha |\le N}\frac{N!}{\alpha !(N-|\alpha |)!}\int \limits _{\{|y|\le 2\varepsilon _0\}} \zeta ^{2\alpha } e^{-i\zeta \cdot y}b_k(x,y)\,dy}\nonumber \\&\quad {=\sum \limits _{|\alpha |\le N}\frac{N!}{\alpha !(N-|\alpha |)!}(-1)^{|\alpha |}\int \limits _{\{|y|\le 2\varepsilon _0\}} \partial ^{2\alpha }_{y}(e^{-i\zeta \cdot y})b_k(x,y)\,dy}\nonumber \\&\quad {=\sum \limits _{|\alpha |\le N}\frac{N!}{\alpha !(N-|\alpha |)!}(-1)^{|\alpha |}\int \limits _{\{|y|\le 2\varepsilon _0\}} e^{-i\zeta \cdot y}\partial ^{2\alpha }_y b_k(x,y)\,dy,} \end{aligned}$$

(153)

from which (152) trivially follows, using that $y-$derivatives of $b_k(x,y)$ are bounded in $\mathbb {R}^n\times \mathbb {R}^n$ by a positive constant independent of $x$. $\square$

We are going now to analyze the behavior at infinity of the derivatives of $q_{k,m}(x,\xi ,\gamma )$ defined as in (151). For all multi-indices $\alpha ,\beta \in \mathbb {N}^n$, differentiation under the integral sign in (151) gives

$$\begin{aligned} \begin{array}{ll} \partial ^{\alpha }_{\xi }\partial ^{\beta }_xq_{k,m}(x,\xi ,\gamma )=(2\pi )^{-n}\int \partial ^{\beta }_x\widehat{b}_k(x,\eta )\partial ^{\alpha +e^k}\lambda ^{m,\gamma }(\xi -\eta )\,d\eta , \end{array} \end{aligned}$$

(154)

where $e^k:=(0,\dots ,\underbrace{1}_{k},\dots ,0)$. Then using that $\lambda ^{m,\gamma }$ is a symbol of order $m$ together with (152) and combining with (148), for $s=m-1-|\alpha |$, we obtain

$$\begin{aligned}&{|\partial ^{\alpha }_{\xi }\partial ^{\beta }_x q_{k,m}(x,\xi ,\gamma )|\le C_{N,\beta }C_{m,\alpha }\int \lambda ^{-2N}(\eta )\lambda ^{m-1-|\alpha |,\gamma }(\xi -\eta )\,d\eta }\nonumber \\&\quad {\le C_{N,m,\alpha ,\beta }\lambda ^{m-1-|\alpha |,\gamma }(\xi )\int \lambda ^{|m-1-|\alpha ||-2N}(\eta )\,d\eta ,} \end{aligned}$$

(155)

where the integral in the last line is finite, provided that the integer $N$ is taken to be sufficiently large. This provides the estimate (80), with constant $C_{N,m,\alpha ,\beta }\int \lambda ^{|m-1-|\alpha ||-2N}(\eta )\,d\eta$ independent of $\gamma$.

Appendix 2: Some Examples from MHD

1.1 Current-Vortex Sheets

Consider the equations of ideal compressible MHD:

$$\begin{aligned} \left\{ \begin{array}{l} \partial _t\rho +\mathrm{div}\, (\rho {v} )=0,\\ \partial _t(\rho {v} ) +\mathrm{div}\,(\rho {v}\otimes {v} -{H}\otimes {H} ) + {\nabla }q=0, \\ \partial _t{H} -{\nabla }\times ({v} {\times }{H})=0,\\ \partial _t\bigl ( \rho e +\frac{1}{2}|{H}|^2\bigr )+ \mathrm{div}\, \bigl ((\rho e +p){v} +{H}{\times }({v}{\times }{H})\bigr )=0, \end{array}\right. \end{aligned}$$

(156)

where $\rho$ denotes density, $v\in \mathbb {R}^3$ plasma velocity, $H \in \mathbb {R}^3$ magnetic field, $p=p(\rho ,S )$ pressure, $q =p+\frac{1}{2}|{H} |^2$ total pressure, $S$ entropy, $e=E+\frac{1}{2}|{v}|^2$ total energy, and $E=E(\rho ,S )$ internal energy. With a state equation of gas, $\rho =\rho (p ,S)$, and the first principle of thermodynamics, (156) is a closed system. The system is symmetric hyperbolic provided $\rho >0, \frac{\partial \rho }{\partial {p}} >0.$ System (156) is supplemented by the divergence constraint

$$\begin{aligned} \mathrm{div}\, {H} =0 \end{aligned}$$

(157)

on the initial data.

Current-vortex sheets are weak solutions of (156) that are smooth on either side of a smooth hypersurface $\Gamma (t)=\{x_1=\psi (t,x')\}$ in $[0,T]\times \Omega$, where $\Omega \subset {\mathbb R}^3,\, x'=(x_2,x_3)$ and that satisfy suitable jump conditions at each point of the front $\Gamma (t)$.

Let us denote $\Omega ^\pm (t)=\{x_1\gtrless \psi (t,x')\}$, where $\Omega =\Omega ^+(t)\cup \Omega ^-(t)\cup \Gamma (t)$; given any function $g$ we denote $g^\pm =g$ in $\Omega ^\pm (t)$ and $[g]=g^+_{|\Gamma }-g^-_{|\Gamma }$ the jump across $\Gamma (t)$.

One looks for smooth solutions $(v^\pm ,H^\pm ,p^\pm ,S^\pm )$ of (156) in $\Omega ^\pm (t)$ such that $\Gamma (t)$ is a tangential discontinuity, namely the plasma does not flow through the discontinuity front and the magnetic field is tangent to $\Gamma (t)$, see e.g. [16], so that the boundary conditions take the form

$$\begin{aligned} \partial _{t}\psi =v^\pm \cdot N \, ,\quad H^\pm \cdot N=0 \, ,\quad [q]=0 \quad \mathrm{on }\;\Gamma (t) \, , \end{aligned}$$

(158)

with $N:=(1,-\partial _{x_2}\psi ,-\partial _{x_3}\psi )$. Because of the possible jump in the tangential velocity and magnetic fields, there is a concentration of vorticity and current along the discontinuity $\Gamma (t)$. Notice that the function $\psi$ describing the discontinuity front is part of the unknown of the problem, i.e. this is a free-boundary problem. The well-posedness of the nonlinear problem (156)–(158) is shown in [8, 34] under the assumption of the structural stability condition $|H^+\times H^-|>0$ on $\Gamma (t)$.

After a change of independent variables that “flattens”the boundary, a linearization around a suitable basic state and some reductions, Trakhinin [33, 34] (see also [8]) gets a linearized problem for $u=(v^\pm ,H^\pm ,p^\pm ,S^\pm )$ of the form (1) with $\mathcal {L}_\gamma$ as in (2), $b_\gamma$ as in (3a), $\mathcal {M}_\gamma$ as in (3b) but with $M_2=M_3=0$, that is the boundary operator has order zero in $u$. Moreover, because of the special reductions, the boundary data are zero, i.e. $g=0$ in (1b), and $F$ in (1a) is such that the solution satisfies some additional constraints.

It is proved that the solution of the linearized problem satisfies an a priori estimate similar to (11) (with $g=0$). Instead, the linearized problem with general data $F$ and $g\not =0$ admits an a priori estimate with a loss of two derivatives, see [34] for details.

Analogous results for incompressible current-vortex sheets are obtained in [4] and [23].

1.2 Plasma-Vacuum 1

Using the previous notations, let $\Omega ^+(t)$ and $\Omega ^-(t)$ be space-time domains occupied by the plasma and the vacuum respectively. That is, in the domain $\Omega ^+(t)$ we consider system (156), (157) governing the motion of an ideal plasma and in the domain $\Omega ^-(t)$ we consider the so-called pre-Maxwell dynamics

$$\begin{aligned} \nabla \times \mathcal {H} =0,\qquad \mathrm{div}\, \mathcal {H}=0, \end{aligned}$$

(159)

describing the vacuum magnetic field $\mathcal {H}\in \mathbb {R}^3$, see [13].

The plasma variable $(v,H,p,S)$ is connected with the vacuum magnetic field $\mathcal {H}$ through the relations [13]

$$\begin{aligned} \partial _{t}\psi =v \cdot N,\quad \!H\cdot N=0, \!\quad \mathcal {H}\cdot N=0 ,\!\quad [q]=0,\quad \text{ on }\ \Gamma (t), \end{aligned}$$

(160)

where the jump of the total pressure across the interface is $[q]= q|_{\Gamma }-\frac{1}{2}|\mathcal {H}|^2_{|\Gamma }$. The well-posedness of the nonlinear problem (156), (157), (159), (160) is shown in [31, 32] under the assumption of the structural stability condition $|H \times \mathcal {H}|>0$ on $\Gamma (t)$.

As in the case of current-vortex sheets, after a change of independent variables that “flattens”the boundary, a linearization around a suitable basic state and some reductions, the authors obtain a linearized problem for $u=(v,H,p,S,\mathcal {H})$ of the form (1) with $\mathcal {L}_\gamma$ as in (2), $b_\gamma$ as in (3a), $\mathcal {M}_\gamma$ as in (3b) with $M_2=M_3=0$, that is the boundary operator has order zero in $u$. Moreover, because of the special reductions, the boundary data are zero, i.e. $g=0$ in (1b), and $F$ in (1a) is such that the solution satisfies some additional constraints.

In [31] it is proved that the solution of the linearized problem satisfies an a priori estimate similar to (11) (with $g=0$). The vacuum magnetic field $\mathcal {H}$ is estimated in the standard Sobolev space $H^1$ with full regularity. Instead, the linearized problem with general data $F$ and $g\not =0$ admits an a priori estimate similar to (10), with loss of one derivative in $F$ and $g$, see [32].

For similar results in the case of the incompressible plasma - vacuum problem, see [25].

1.3 Plasma-Vacuum 2

In the domain $\Omega ^+(t)$ we consider system (156), (157) governing the motion of an ideal plasma and in the domain $\Omega ^-(t)$ we consider the Maxwell equations

$$\begin{aligned} {\left\{ \begin{array}{ll} \partial _{t}\mathcal {H} + \nabla \times \mathcal {E} = 0 , \\ \partial _{t}\mathcal {E} - \nabla \times \mathcal {H} = 0 , \\ \mathrm{div}\, \mathcal {H} \,= \, \mathrm{div}\, \mathcal {E} = 0 , \end{array}\right. } \end{aligned}$$

(161)

describing the vacuum magnetic and electric fields $\mathcal {H},\mathcal {E}\in \mathbb {R}^3$, see [13].

The plasma variable $(v,H,p,S)$ is connected with the vacuum variable $(\mathcal {H},\mathcal {E})$ through the relations [13]

$$\begin{aligned} \partial _{t}\psi =v \cdot N,\ H\cdot N=0, \ \!\mathcal {H}\cdot N=0 \, ,\ [q]=0,\quad N \!\times \!\mathcal {E} = (N \cdot v) \mathcal {H},\quad \text{ on }\ \Gamma (t), \nonumber \\ \end{aligned}$$

(162)

where the jump of the total pressure across the interface is $[q]= q|_{\Gamma }-\frac{1}{2}|\mathcal {H}|^2_{|\Gamma }+\frac{1}{2}|\mathcal {E}|^2_{|\Gamma }$.

The stability of the linearized problem obtained from (156), (157), (161), (162) is shown in [5] under suitable stability conditions on $\Gamma (t)$. The authors obtain a linearized problem for $u=(v,H,p,S,\mathcal {H},\mathcal {E})$ of the form (1) with $\mathcal {L}_\gamma$ as in (2), $b_\gamma$ as in (3a), $\mathcal {M}_\gamma$ as in (3b) with $M_2=M_3=0$, that is the boundary operator has order zero in $u$. Moreover, because of the special reductions, the boundary data are zero, i.e. $g=0$ in (1b), and $F$ in (1a) is such that the solution satisfies some additional constraints. It is proved that the solution of the linearized problem satisfies an a priori estimate similar to (11) (with $g=0$). The vacuum variable $(\mathcal {H},\mathcal {E})$ is estimated in the standard Sobolev space $H^1$ with full regularity.

1.4 Contact Discontinuities

We consider the equations of ideal compressible MHD (156) for two-dimensional planar flows with respect to the unknown vector $U=(p, v, H, S)$, with $v(t,x)=(v_1,v_2)\in \mathbb R^2$, $H(t,x)=(H_1,H_2)\in \mathbb R^2$, $x=(x_1,x_2)$. For simplicity, let us assume that the plasma obeys the state equation of a polytropic gas

$$\begin{aligned} \rho (p, S)=Ap^{1/\gamma }e^{-S/\gamma },\quad A>0,\,\,\,\gamma >1\,. \end{aligned}$$

(163)

Following the notations already introduced for current vertex sheets, contact discontinuities are weak solutions of (156), that are smooth on either side of a smooth hypersurface $\Gamma (t)=\{x_1=\psi (t,x_2)\}$ in $[0,T]\times {\mathbb R}^2$, satisfying at each point of the front $\Gamma (t)$ suitable jump conditions. More precisely, one looks for smooth solutions $U^\pm$ of (156) in $\Omega ^\pm (t):=\{x_1\gtrless \psi (t,x_2)\}$, satisfying on $\Gamma (t)$ the following conditions

$$\begin{aligned} v^+_N - \partial _t \psi =0,\quad \left[ v\right] =0,\quad \left[ H\right] =0,\quad H_N^\pm \ne 0,\quad \left[ p\right] =0, \end{aligned}$$

(164)

where $N:=(1, -\partial _2\psi )$ is the space normal to the front $\Gamma (t)$, $H_N=H_1-\partial _2\psi H_2$.

After a change of independent variables that “flattens” the boundary, in [24] the authors perform a linearization of the free-boundary problem (156), (164) for contact discontinuities, around a suitable sufficiently smooth basic state $(\hat{p}, \hat{v}, \hat{H}, \hat{S}, \hat{\varphi })$, obeying the “stability” condition

$$\begin{aligned} \left[ \partial _1\hat{p}\right] \ge c_0>0,\quad \text{ on }\,\,\{x_1=\hat{\varphi }(t,x_2)\}\,. \end{aligned}$$

(165)

Under the preceding assumptions, the linearized problem can be recast in the form of (1) with $\mathcal {L}_\gamma$ as in (2), $b_\gamma =0$ and $\mathcal {M}_\gamma$ of order one in $U$ as in (3b). Moreover, because of the special reductions, the boundary data are zero, i.e. $g=0$ in (1b), whereas the only nonzero components of $F$ in (1a) are the ones corresponding to the equation for $v$.

In [24] it is proved that the solution of the above linearized problem satisfies an a priori estimate in the Sobolev space $H^1_{tan}$ similar to (11).