Anisotropic Regularity of the Free-Boundary Problem in Compressible Ideal Magnetohydrodynamics

Abstract

We consider that 3D free-boundary compressible ideal magnetohydrodynamic (MHD) system under the Rayleigh-Taylor sign condition. This describes the motion of a free-surface perfect conducting fluid in an electro-magnetic field. A local existence and uniqueness result was recently proved by Trakhinin and Wang (Arch Ration Mech Anal 239(2):1131–1176, 2021) by using Nash–Moser iteration. However, that result loses regularity going from data to solution. In this paper, we show that the Nash–Moser iteration scheme in Trakhinin and Wang (2021) can be improved such that the local-in-time smooth solution exists and is unique when the initial data is smooth and satisfies the compatibility condition up to infinite order. Second, we prove the a priori estimates without loss of regularity for the free-boundary compressible MHD system in Lagrangian coordinates in anisotropic Sobolev space, with more regularity tangential to the boundary than in the normal direction. This is based on modified Alinhac good unknowns, which take into account the covariance under the change of coordinates to avoid the derivative loss; full utilization of the cancellation structures of MHD system, to turn normal derivatives into tangential ones; and delicate analysis in anisotropic Sobolev spaces. As a result, we can prove the uniqueness and the continuous dependence on initial data provided the local existence, and a continuation criterion for smooth solution. Finally, we extend the local well-posedness theorem to the case of initial data only satisfying compatibility conditions up to finite order, assuming these can be approximated by data satisfying infinitely many compatibility conditions.

Construction of Smooth Data Satisfying Compatibility Conditions Up to Infinite Order

In the appendix, we prove the existence of smooth data satisfying compatibility conditions up to infinite order. Assume $m\geqq 8$ is an integer and we are given an initial data $(w_0,b_0,P_0)$ satisfying the compatibility conditions up to $(m-1)$-th order in $H^m(\Omega )$, where $P_0$ is the total pressure, while the fluid pressure is denoted by $p_0=P_0-\frac{1}{2}|b_0|^2$. For simplicity of notations, we assume $R'(q)/R|_{R=1}=1$.

The initial constraints and compatibility conditions for $(w_0,b_0,P_0)$ are

• (Compatibility conditions) $P_{(j)}:=\partial _t^j P|_{t=0}=0$ on $\Gamma$, $0\leqq j\leqq m-1$.

• (Initial constraints) $\text {div }b_0=0$ in $\Omega$, $b_0^3|_{\Gamma }=0$, and the Rayleigh-Taylor sign condition $-\frac{\partial P_0}{\partial N}\geqq c_0>0$ on $\Gamma$.

1.1 Compatibility Conditions in Terms of Initial Data

First we express the compatibility conditions in terms of $w_0,b_0,P_0$. The zero-th order compatibility condition is $P_0|_{\Gamma }=0$. To express the first-order compatibility condition, we use $P=p+\frac{1}{2}|b_0|^2$, where p is the fluid pressure, the continuity equation $\partial _t p+\text {div }v=0$, and $\partial _t b=b\cdot \nabla v-b\text {div }v$ to get (we omit the coefficient A as it equals to Id at $t=0$. The appearance of A does not affect the essence of the proof.)

$$\begin{aligned} \text {div }w_0=-|b_0|^2\text {div }w_0+({\bar{b}}_0\cdot \overline{\nabla })w_0\cdot b_0\text { on }\Gamma , \end{aligned}$$

(A.1)

and we define the right side of (A.1) to be the functional ${\mathcal {M}}_{-1}(w_0,b_0)$. Next we take divergence in the momentum equation to get a wave equation of P

$$\begin{aligned} \partial _t^2 P-\Delta P=\partial _t^2(\frac{1}{2}|b|^2)+\nabla _iw^j\nabla _jw^i-\nabla _ib^j\nabla _jb^i,~~P_0|_{\Gamma }=0 \end{aligned}$$

(A.2)

and again use $\partial _t b=b\cdot \nabla v-b\text {div }v$ and $\partial _t v\sim (b_0\cdot \partial )b-\nabla P$ to get

$$\begin{aligned} \partial _t^2 P-\Delta P={\mathcal {M}}_0(w_0,b_0,P_0)+{\mathcal {N}}_0(w_0,b_0)~~\text { on }\{t=0\}, \end{aligned}$$

where ${\mathcal {N}}_0(w_0,b_0):=\nabla _iw_0^j\nabla _jw_0^i-\nabla _ib_0^j\nabla _jb_0^i$ and ${\mathcal {M}}_0(w_0,b_0,P_0)$ is defined by

$$\begin{aligned} {\mathcal {M}}_0(w_0,b_0,P_0):=|b_0|^2\Delta P_0-(b_0\cdot \nabla )^2 P_0+(b_0\cdot \nabla )^2b_0\cdot b_0+{\mathcal {R}}_0(w_0,b_0) \end{aligned}$$

and ${\mathcal {R}}_0(w_0,b_0)$ only contains the first-order derivative of $b_0$ and $v_0$

$$\begin{aligned} {\mathcal {R}}_0(w_0,b_0):=P_0(b_0)\left( (\nabla ^{i_1} w_0)(\nabla ^{i_2} w_0)+(\nabla ^{j_1} b_0)(\nabla ^{j_2} b_0)\right) , \end{aligned}$$

where $P_0(b_0)$ is a polynomial of $b_0$ only contains cubic and quadratic terms, and $(i_1,i_2,j_1,j_2)=(1,1,0,0)\text { or }(0,0,1,1)$.

We see that the 2nd-order compatibility condition $P_{(2)}:=\partial _t^2 P|_{t=0}=0$ on $\Gamma$ is equivalent to (we use $b_0^3|_{\Gamma }=0$)

$$\begin{aligned} -\Delta P_0= & {} {\mathcal {M}}_0(w_0,b_0,P_0)+{\mathcal {N}}_0(w_0,b_0,P_0)\nonumber \\= & {} |b_0|^2\Delta P_0-({\bar{b}}_0\cdot \overline{\nabla })^2 P_0+({\bar{b}}_0\cdot \overline{\nabla })^2b_0\cdot b_0+{\mathcal {R}}(w_0,b_0) \text { on }\Gamma . \end{aligned}$$

(A.3)

Taking time derivatives in the wave equation above repeatedly we get for $k \geqq 1$

$$\begin{aligned} P_{(k+2)}-\Delta P_{(k)}={\mathcal {M}}_k(w_0,b_0,P_0)+{\mathcal {N}}_k(w_0,b_0,P_0)\text { in }\Omega , \end{aligned}$$

where, after long and tedious calculations, the functionals ${\mathcal {M}}_k,{\mathcal {N}}_k$ have the following form for $r\geqq 1$

$$\begin{aligned}&k=2r-1,~~{\mathcal {M}}_{2r-1}(w_0,b_0,P_0)\nonumber \\&\quad =-|b_0|^2\Delta ^r\text {div }w_0+({\bar{b}}_0\cdot \overline{\nabla })^2\Delta ^{r-1}\text {div }w_0\nonumber \\&\qquad +\sum _{l=2}^{r+1}\underbrace{b_0^{i_1}\cdots b_0^{i_{2l}}(\nabla ^{2r+1}w_0)}_{<2^{l}\text { terms}}+{\mathcal {R}}_{2r-1}(w_0,b_0,P_0), \end{aligned}$$

(A.4)

$$\begin{aligned}&k=2r,~~{\mathcal {M}}_{2r}(w_0,b_0,P_0)\nonumber \\&\quad =~|b_0|^2\Delta ^{r+1} P_0-({\bar{b}}_0\cdot \overline{\nabla })^2\Delta ^{r}P_0+{\mathcal {R}}_{2r}(w_0,b_0,P_0),\nonumber \\&\qquad +\sum _{l=2}^{r+1}\underbrace{({\bar{b}}_0\cdot \overline{\nabla })^{r+2}(\nabla ^rb_0) b_0^{i_1}\cdots b_0^{i_{2l}}+({\bar{b}}_0\cdot \overline{\nabla })^{2}(\nabla ^{2r} P_0) b_0^{j_1}\cdots b_0^{j_{2l}}}_{<2^{l}\text { terms }}; \end{aligned}$$

(A.5)

and the term ${\mathcal {R}}_k$, where every top-order term has $(k+1)$-th order derivative, has the following form

$$\begin{aligned} {\mathcal {R}}_k(w_0,b_0,P_0)= & {} P_k(b_0)\left( C^{k}_{i_1\cdots i_m,j_1\cdots j_n, k_1\cdots k_l}(\nabla ^{i_1} w_0)\cdots (\nabla ^{i_m} w_0)(\nabla ^{j_1} b_0)\cdots \right. \\{} & {} \left. \quad (\nabla ^{j_n} b_0)(\nabla ^{k_1} P_0)\cdots (\nabla ^{k_l} P_0)\right) , \end{aligned}$$

where $P_k(\cdot )$ is a polynomial of its arguments and the lowest power is 4 and the indices above satisfy

$$\begin{aligned} 1\leqq i_1,\cdots , i_m,j_1,\cdots , j_n\leqq k+1, 0\leqq k_1,\cdots , k_l \leqq k+1,\\ i_1+\cdots +i_m+j_1+\cdots +j_n+k_1+\cdots +k_l=k+1. \end{aligned}$$

The term ${\mathcal {N}}_k(w_0,b_0,P_0)$ has the following form

$$\begin{aligned} {\mathcal {N}}_k(w_0,b_0,P_0)=&~P_{k,1}(b_0)(\nabla ^{1+2\lfloor \frac{k}{2}\rfloor }w_0)(\nabla w_0)+P_{k,2}(b_0)(\nabla ^{2\lceil \frac{k}{2}\rceil } P_0)(\nabla w_0)\\&+P_{k,0}(b_0)(\nabla ^{k+1}b_0)(\nabla w_0)\\&+P'_k(b_0)D^{k}_{i_1\cdots i_m,j_1\cdots j_n, k_1\cdots k_l}\\&\left( (\nabla ^{i_1} w_0)\cdots (\nabla ^{i_m} w_0)(\nabla ^{j_1} b_0)\cdots (\nabla ^{j_n} b_0)(\nabla ^{k_1} P_0)\cdots (\nabla ^{k_l} P_0)\right) , \end{aligned}$$

where $P_{k,1}(\cdot ), P_{k,2}(\cdot ),P_k'(\cdot )$ are polynomials of their arguments and $P_{k,0}(\cdot )$ is a polynomial of its arguments and the lowest power is 2. The indices above satisfy

$$\begin{aligned} 1\leqq i_1,\cdots , i_m,j_1,\cdots , j_n\leqq k, 0\leqq k_1,\cdots , k_l \leqq k,\\ i_1+\cdots +i_m+j_1+\cdots +j_n+k_1+\cdots +k_l=k+1. \end{aligned}$$

So the k-th compatibility condition can be equivalently written as

$$\begin{aligned} k&=2r+1,~~\Delta ^r\text {div }w_0=\sum _{j=0}^{r}\Delta ^j({\mathcal {M}}_{2r-1-2j}(w_0,b_0,P_0)\nonumber \\&\quad +{\mathcal {N}}_{2r-1-2j}(w_0,b_0,P_0))\text { on }\Gamma , \end{aligned}$$

(A.6)

$$\begin{aligned} k&=2r,~~-\Delta ^r P_0=\sum _{j=0}^{r-1}\Delta ^j({\mathcal {M}}_{2r-2-2j}(w_0,b_0,P_0)\nonumber \\&\quad +{\mathcal {N}}_{2r-2-2j}(w_0,b_0,P_0))\text { on }\Gamma , \end{aligned}$$

(A.7)

where ${\mathcal {M}}_{-1}(w_0,b_0):=|b_0|^2\text {div }w_0-(b_0\cdot \nabla )w_0\cdot b_0$ and ${\mathcal {N}}_{-1}:=0$.

1.2 Regularization of the Given Data and Recovery of Compatibility Conditions

To construct a smooth data satisfying the compatibility conditions up to infinite order, the first step is to regularize the given data such that we get smooth functions. By the standard approximation of Sobolev function, we know for any given $\varepsilon >0$, there exists $(w_0^\varepsilon ,b_0^\varepsilon ,P_0^\varepsilon )\in C^{\infty }(\Omega )$ such that

$$\begin{aligned} \Vert w_0^\varepsilon -w_0,b_0^\varepsilon -b_0,P_0^\varepsilon -P_0\Vert _s<\varepsilon . \end{aligned}$$

However, such smooth approximation does not preserve the boundary conditions, even for the vanishing boundary conditions for $P_0$ and $b_0^3$. So we need to recover the compatibility conditions up to the same order as the given data.

From now on, we assume

• $m=8$, that is, the given data satisfies the compatibility conditions (A.6)–(A.7) up to 7-th order. This corresponds to the minimal requirement in Theorem 1.3.

• $\Vert b_0\Vert _{L^{\infty }({{\bar{\Omega }}})}<\delta _0<1$ where $\delta _0$ is a suitably small number to be determined: to absorb the terms containing $(k+2)$-th order derivative arising in ${\mathcal {M}}_k$. Note that we do not need $\Vert b_0\Vert _{L^{\infty }({{\bar{\Omega }}})}$ to be arbitrarily small in the proof.

1.2.1 Recovering the Initial Constraints

The new data should also satisfy the initial constraints: divergence-free condition of magnetic field, vanishing normal component of magnetic field on the boundary and the Rayleigh-Taylor sign condition. The Rayleigh-Taylor sign condition still holds for $P_0^\varepsilon$, as $-\partial _3P_0^\varepsilon$ is just a small perturbation of $-\partial _3 P_0$. We then modify $b_0^\varepsilon$. First, we introduce ${\tilde{b}}_0^\varepsilon$ defined by

$$\begin{aligned} {\tilde{b}}_0^{\varepsilon ,1}=b_0^{\varepsilon ,1},~~{\tilde{b}}_0^{\varepsilon ,2}=b_0^{\varepsilon ,2};~~-\Delta {\tilde{b}}_0^{\varepsilon ,3}=-\Delta b_0^{\varepsilon ,3}\text { in }\Omega ,~{\tilde{b}}_0^{\varepsilon ,3}=0\text { on }\Gamma , \end{aligned}$$

(A.8)

and then ${\tilde{b}}_0^\varepsilon \in C^{\infty }(\Omega )$ and the elliptic estimates imply $\Vert {\tilde{b}}_0^\varepsilon -b_0\Vert _s\leqq \Vert b_0^\varepsilon -b_0\Vert _s+|0-0|_{s-0.5}=O(\varepsilon )$. Next, we recover the divergence-free condition by introducing ${{\textbf {b}}}_0^\varepsilon :={\tilde{b}}_0^\varepsilon +\nabla \varphi$ with $\varphi$ determined by

$$\begin{aligned} -\Delta \varphi =\text {div }{\tilde{b}}_0^\varepsilon \text { in }\Omega ,~\partial _3\varphi =0\text { on }\Gamma . \end{aligned}$$

(A.9)

With this modification, we now have $\text {div }{{\textbf {b}}}_0^\varepsilon =\text {div }{\tilde{b}}_0^\varepsilon +\Delta \varphi =0$ in $\Omega$, and ${{\textbf {b}}}_0^3|_{\Gamma }=0$ still holds thanks to the Neumann boundary condition $\partial _3\varphi =0\text { on }\Gamma$. So, ${{\textbf {b}}}_0^\varepsilon$ is the desired magnetic field that we need, and it still satisfies a smallness assumption $\Vert {{\textbf {b}}}_0\Vert _{L^{\infty }({\bar{\Omega }})}<2\delta _0$. We’ll drop $\varepsilon$ in ${{\textbf {b}}}_0$ for the sake of clean notations.

1.2.2 Recovering the Compatibility Conditions Up to $(m-1)$-th Order

Next we focus on the modification of $w_0^\varepsilon ,P_0^\varepsilon$. After the regularization, we don’t even know if $P_0^\varepsilon =0$ on $\Gamma$ holds or not. So the first step is to recover the 0-th order compatibility condition $P_0|_\Gamma =0$. We define ${{\textbf {P}}}_0^{(1)}$ by

$$\begin{aligned} -\Delta {{\textbf {P}}}_0^{(1)}=-\Delta P_0^\varepsilon \text { in }\Omega ,~~{{\textbf {P}}}_0^{(1)}=0\text { on }\Gamma . \end{aligned}$$

(A.10)

Since $P_0|_{\Gamma }=0$, we know

$$\begin{aligned} \Vert {{\textbf {P}}}_0^{(1)}-P_0\Vert _s\leqq \Vert P_0^\varepsilon -P_0\Vert _s+|0-0|_{s-0.5}=O(\varepsilon ). \end{aligned}$$

Next we define ${{\textbf {w}}}_0^{(1)}$ to be the following function such that $({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(1)})$ satisfies the compatibility condition up to first order: ${{\textbf {w}}}_0^{(1),1}=w_0^{\varepsilon , 1},~{{\textbf {w}}}_0^{(1),2}=w_0^{\varepsilon ,2}$ and ${{\textbf {w}}}_0^{(1),3}$ is determined by the bi-harmonic system

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^2{{\textbf {w}}}_0^{(1),3}=\Delta ^2w_0^{\varepsilon ,3}~~&{}\text { in }\Omega ,\\ {{\textbf {w}}}_0^{(1),3}=w_0^{\varepsilon ,3},~~\partial _3{{\textbf {w}}}_0^{(1),3}=-\overline{\partial }{}_1 w_0^{\varepsilon ,1}-\overline{\partial }{}_2 w_0^{\varepsilon ,2}+{\mathcal {M}}_{-1}({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0)~~&{}\text { on }\Gamma , \end{array}\right. }\nonumber \\ \end{aligned}$$

(A.11)

where ${\mathcal {M}}_{-1}({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0^{(1)})$ is given by (A.1). Note that the second boundary contidition only involves $\partial _3{{\textbf {w}}}_0^{(1)}$ because the tangential components are the same of $w_0^\varepsilon$. So the elliptic estimates give us

$$\begin{aligned} \begin{aligned} \Vert {{\textbf {w}}}_0^{(1)}-w_0\Vert _s\lesssim&~\Vert \Delta ^2w_0^\varepsilon -\Delta ^2 w_0\Vert _{s-4}+|w_0^\varepsilon -w_0|_{s-0.5}+|\partial _3{{\textbf {w}}}_0^{(1)}-\partial _3 w_0|_{s-1.5}\\ \lesssim&~O(\varepsilon )+|b_0|_{L^{\infty }}^2|\nabla {{\textbf {w}}}_0^{(1)}-\nabla w_0|_{s-1.5}, \end{aligned}\nonumber \\ \end{aligned}$$

(A.12)

where the last term can be absorbed by the left side if we pick $\delta _0$ sufficiently small. Therefore, by the second boundary condition, we know $({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(1)})$ satisfies the compatibility condition up to first order.

Again, we construct ${{\textbf {P}}}_0^{(2)}$ such that $({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(2)})$ satisfies the compatibility condition (A.7) up to 2nd order. The new pressure is defined by the poly-harmonic system

$$\begin{aligned} {\left\{ \begin{array}{ll} -\Delta ^3 {{\textbf {P}}}_0^{(2)}=-\Delta ^3 {{\textbf {P}}}_0^{(1)}~~&{}\text { in }\Omega ,\\ {{\textbf {P}}}_0^{(2)}={{\textbf {P}}}_0^{(1)}=0,~~\partial _3{{\textbf {P}}}_0^{(2)}=\partial _3{{\textbf {P}}}_0^{(1)}~~&{}\text { on }\Gamma ,\\ -\Delta {{\textbf {P}}}_0^{(2)}={\mathcal {M}}_0({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,{\mathcal {P}}^{(1)})+{\mathcal {N}}_0({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0)~~&{}\text { on }\Gamma , \end{array}\right. } \end{aligned}$$

(A.13)

and thus

$$\begin{aligned} \begin{aligned} \Vert {{\textbf {P}}}_0^{(2)}-P_0\Vert _s\lesssim&~\Vert \Delta ^3{{\textbf {P}}}_0^{(1)}-\Delta ^3 P_0\Vert _{s-6}+|\partial _3{{\textbf {P}}}_0^{(1)}-\partial _3 P_0|_{s-1.5}\\&+|{\mathcal {M}}_0({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,\Vert {\textbf{Q}}_{(m+7)}^{(m-1)}\Vert _0+\Vert {\textbf{Q}}_{(m+6)}^{(m-1)}\Vert _1^{(1)})\\&+{\mathcal {M}}_0({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0)-{\mathcal {M}}_0(w_0,b_0,P_0)-{\mathcal {N}}_0(w_0,b_0)|_{s-2.5}\\ \lesssim&~O(\varepsilon )+|b_0|_{L^{\infty }}^2|\partial _3^2({{\textbf {P}}}_0^{(2)}-P_0)|_{s-2.5}, \end{aligned} \end{aligned}$$

(A.14)

where the last term is again absorbed by the left side if we choose $|b_0|\leqq \delta _0$ to be suitably small. It should also be noted that, ${\mathcal {M}}_{k}$ also has other terms containing $(k+2)$-th order derivative, but there are at least two derivatives appearing as $({\bar{b}}_0\cdot \overline{\nabla })$, and thus we can replace ${{\textbf {P}}}_0^{(2)}$ with ${{\textbf {P}}}_0^{(1)}$ using the remaining boundary conditions.

Next we construct ${{\textbf {w}}}_0^{(2)}$ via the following system such that $({{\textbf {w}}}_0^{(2)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(2)})$ satisfies the compatibility condition (A.6) up to 3rd order.

$$\begin{aligned} {\left\{ \begin{array}{ll} \Delta ^4{{\textbf {w}}}_0^{(2),3}=\Delta ^4{{\textbf {w}}}_0^{(1),3}~~&{}\text { in }\Omega ,\\ \partial _3^j{{\textbf {w}}}_0^{(2),3}=\partial _3^j{{\textbf {w}}}_0^{(1),3},(0\leqq j\leqq 2)~~&{}\text { on }\Gamma \\ \Delta \text {div }{{\textbf {w}}}_0^{(2)}={\mathcal {M}}_1({{\textbf {w}}}_0^{(2)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(2)})\\ \quad +{\mathcal {N}}_1({{\textbf {w}}}_0^{(2)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(2)})+\Delta {\mathcal {M}}_{-1}({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0)~~&{}\text { on }\Gamma , \end{array}\right. } \end{aligned}$$

(A.15)

and similarly as above we can get

$$\begin{aligned} \begin{aligned} \Vert {{\textbf {w}}}_0^{(2)}-w_0\Vert _s\lesssim&~\Vert \Delta ^4{{\textbf {w}}}_0^{(1)}-\Delta ^4 w_0\Vert _{s-8}+\sum _{j=0}^2|\partial _3^j{{\textbf {w}}}_0^{(1)}-\partial _3^jw_0|_{s-j-0.5}\\&+|({\mathcal {M}}_1+{\mathcal {N}}_1+\Delta {\mathcal {M}}_{-1})({{\textbf {w}}}_0^{(1)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(2)})\\&-({\mathcal {M}}_1+{\mathcal {N}}_1+\Delta {\mathcal {M}}_{-1})(w_0,b_0,P_0)|_{s-3.5}\\ \lesssim&~O(\varepsilon )+|b_0|_{L^{\infty }}^2|\partial _3^3({{\textbf {w}}}_0^{(2)}-w_0)|_{s-3.5}, \end{aligned} \end{aligned}$$

(A.16)

where the last term is again absorbed by the left side if we choose $|b_0|\leqq \delta _0$ to be suitably small.

So, we can repeat the above procedures such that ${{\textbf {P}}}_0^{(m)}$ is determined by the poly-harmonic equation $\Delta ^{2m-1}{{\textbf {P}}}_0^{(m)}=\Delta ^{2m-1}{{\textbf {P}}}_0^{(m-1)}$ in $\Omega$ equipped with the boundary conditions $\partial _3^j{{\textbf {P}}}_0^{(m)}=\partial _3^j{{\textbf {P}}}_0^{(m-1)}$ on $\Gamma$ for $0\leqq j\leqq 2\,m-3$ and the compatibility condition (A.7) for the case $k=2\,m-2$ with $(w_0,b_0,P_0)$ replaced by $({{\textbf {w}}}_0^{(m-1)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(m)})$.

Similarly, ${{\textbf {w}}}_0^{(m)}$ is determined by ${{\textbf {w}}}_0^{(m),1,2}={{\textbf {w}}}_0^{(m-1),1,2}$ and $\Delta ^{2m}{{\textbf {w}}}_0^{(m),3}=\Delta ^{2m}{{\textbf {w}}}_0^{(m-1),3}$ in $\Omega$ equipped with the boundary conditions $\partial _3^j{{\textbf {w}}}_0^{(m),3}=\partial _3^j{{\textbf {w}}}_0^{(m-1),3}$ on $\Gamma$ for $0\leqq j\leqq 2\,m-2$ and the compatibility condition (A.6) for the case $k=2\,m-1$ with $(w_0,b_0,P_0)$ replaced by $({{\textbf {w}}}_0^{(m)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(m)})$.

Since the given rough data $(w_0,b_0,P_0)$ satisfies the compatibility conditions up to 7-th order, we stop the above procedure after we get $({{\textbf {w}}}_0^{(4)},{{\textbf {b}}}_0,{{\textbf {P}}}_0^{(4)})$ which is a smooth data and also satisfies the compatibility conditions up to 7-th order. We rename this smooth data to be $({{\textbf {v}}}_0,{{\textbf {b}}}_0,{{\textbf {Q}}}_0)$. For any given $\varepsilon >0$, we construct a smooth data $({{\textbf {v}}}_0,{{\textbf {b}}}_0,{{\textbf {Q}}}_0)$ that satisfies the compatibility conditions up to the same order as the given rough data $(w_0,b_0,P_0)$ and has the following approximation

$$\begin{aligned} \Vert {{\textbf {v}}}_0-w_0\Vert _8+\Vert {{\textbf {b}}}_0-b_0\Vert _8+\Vert {{\textbf {Q}}}_0-P_0\Vert _8=O(\varepsilon ). \end{aligned}$$

(A.17)

1.3 Extend the Compatibility Conditions Up to Infinite Order

1.3.1 Formal Constructions

We then try to extend the initial data such that the compatibility conditions are fulfilled up to infinite order. First we briefly state some formal construction. Recall in A, for a given data $(w_0,b_0,P_0)$, the corresonding solution satisfies the wave equation

$$\begin{aligned} \partial _t^2P-\Delta _b P={\mathcal {M}}_0(v,b,P)+{\mathcal {N}}_0(v,b),~~\Delta _b:=(1+|b|^2)\Delta -(b\cdot \partial )^2, \end{aligned}$$

and ${\mathcal {M}}_0(v,b,P):=(b_0\cdot \partial )^2b\cdot b+{\mathcal {R}}(v,b)$ where ${\mathcal {R}}$ only contains the first-order derivatives of b, v. So if we start with an irrotational velocity $w_0=\nabla \psi$ and define $P_{(-1)}:=-\psi$, then since $P=p+\frac{1}{2}|b|^2$ we have

$$\begin{aligned} \partial _t P= & {} \partial _tp+b\cdot \partial _t b=-\text {div }w+b\cdot ((b\cdot \partial ) w-b\text {div }w)\\= & {} -(1+|b|^2)\text {div }w+(b\cdot \partial ) w\cdot b, \end{aligned}$$

which then gives, after restricting it to $\{t=0\}$

$$\begin{aligned} -\Delta _bP_{(-1)}=-P_{(1)}+(b_0\cdot \partial )\nabla \psi \cdot b_0, \end{aligned}$$

where the right side only depends on the given data of velocity and magnetic field. Taking more time derivatives and setting $t=0$ yields an infinite elliptic system of the form

$$\begin{aligned} -\Delta _{{{\textbf {b}}}_0}P_{(k)}=-P_{(k+2)}+{\mathcal {N}}_{k}'(P_{(-1)},\ldots , P_{(k-1)}),~~k\geqq -1, \end{aligned}$$

where ${\mathcal {N}}_k'$ is a functional that only depends on the derivatives of its arguments and $b_0$ up to a certain order. This system has similar structure as in [40, Lemma 16.1] and thus can be solved in a similar manner. The only difference comes from the appearance of magnetic field, but $(b_0\cdot \partial )$ is a tangential derivatiev and we can pick suitable $b_0$ such that its normal component vanishes in a neighborhood of the boundary.

1.3.2 Full Construction Procedure

For specific calculations, we now define the desired smooth data $({{\textbf {v}}}_0^{\infty },{{\textbf {b}}}_0^\infty ,{{\textbf {Q}}}_0^\infty )$ by

$$\begin{aligned} {{\textbf {v}}}_0^{\infty }:={{\textbf {v}}}_0-\nabla {\textbf{Q}}_{(-1)}^{\infty },~~{{\textbf {b}}}_0^\infty :={{\textbf {b}}}_0. \end{aligned}$$

(A.18)

And after a long and tedious calculation, we find ${{\textbf {Q}}}_0^\infty$ is determined by the following infinite elliptic system in $\Omega$, where $\lambda :=1+|{{\textbf {b}}}_0|^2$.

$$\begin{aligned} -\lambda \Delta {\textbf{Q}}_{-1}^{\infty }=&-({\textbf{Q}}_{(1)}^{\infty }-{\textbf{Q}}_{(1)})-({{\textbf {b}}}_0\cdot \nabla )^2 {\textbf{Q}}_{-1}^{\infty }+{\mathcal {N}}'_{-1}({{\textbf {b}}}_0,{\textbf{Q}}_{-1}^{\infty }) \end{aligned}$$

(A.19)

$$\begin{aligned} -\lambda \Delta {\textbf{Q}}_0^{\infty }=&-{\textbf{Q}}_{(2)}^{\infty }-({{\textbf {b}}}_0\cdot \nabla )^2 {\textbf{Q}}_{0}^{\infty }+({{\textbf {b}}}_0\cdot \nabla )^2{{\textbf {b}}}_0\cdot {{\textbf {b}}}_0+{\mathcal {N}}'_{0}({{\textbf {b}}}_0,{{\textbf {v}}}_0^{\infty },{\textbf{Q}}_{-1}^{\infty }) \end{aligned}$$

(A.20)

$$\begin{aligned} -\lambda \Delta {\textbf{Q}}_{(1)}^{\infty }=&-{\textbf{Q}}_{(3)}^{\infty }-({{\textbf {b}}}_0\cdot \nabla )^2 {\textbf{Q}}_{1}^{\infty }+({{\textbf {b}}}_0\cdot \nabla )^3{{\textbf {v}}}_0^{\infty }\cdot {{\textbf {b}}}_0\nonumber \\&\quad -|{{\textbf {b}}}_0|^2({{\textbf {b}}}_0\cdot \nabla )^2(\nabla \cdot {{\textbf {v}}}_0^{\infty })+{\mathcal {N}}'_{1}({{\textbf {b}}}_0,{{\textbf {v}}}_0^{\infty },{\textbf{Q}}_{-1}^{\infty },{\textbf{Q}}_0^{\infty }), \end{aligned}$$

(A.21)

$$\begin{aligned} -\lambda \Delta {\textbf{Q}}_{(2)}^{\infty }=&-{\textbf{Q}}_{(4)}^{\infty }-({{\textbf {b}}}_0\cdot \nabla )^2\Delta {\textbf{Q}}_{0}^{\infty }+{\mathcal {N}}'_{2}({{\textbf {b}}}_0,{{\textbf {v}}}_0^{\infty },{\textbf{Q}}_{-1}^{\infty },{\textbf{Q}}_0^{\infty },{\textbf{Q}}_{(1)}^{\infty }) \end{aligned}$$

(A.22)

and for $k\geqq 2$

$$\begin{aligned} -\lambda \Delta {\textbf{Q}}_{(k)}^{\infty }=&-{\textbf{Q}}_{(k+2)}^{\infty }-({{\textbf {b}}}_0\cdot \nabla )^2\Delta {\textbf{Q}}_{(k-2)}^{\infty }\nonumber \\ {}&\quad +{\mathcal {N}}'_{k}({{\textbf {b}}}_0,{{\textbf {v}}}_0^{\infty },{\textbf{Q}}_{-1}^{\infty },{\textbf{Q}}_0^{\infty },{\textbf{Q}}_{(1)}^{\infty },\cdots ,{\textbf{Q}}_{(k-1)}^{\infty }), \end{aligned}$$

(A.23)

with vanishing boundary conditions for each ${\textbf{Q}}_{(k)}$. These ${\mathcal {N}}_k'$’s have the following structure

$$\begin{aligned} \begin{aligned}&{\mathcal {N}}'_k({{\textbf {v}}}_0^\infty ,{{\textbf {b}}}_0,{\textbf{Q}}^\infty _{(-1)},\cdots ,{\textbf{Q}}^{(\infty )}_{(k-1)})\\&\quad =P({{\textbf {b}}}_0)C^{k;m_1\cdots m_r}_{i_1\cdots i_m,j_1\cdots j_n, k_1\cdots k_r}\left( (\nabla ^{i_1} {{\textbf {v}}}_0^{\infty })\cdots (\nabla ^{i_m} {{\textbf {v}}}_0^{\infty })(\nabla ^{j_1} {{\textbf {b}}}_0)\cdots \right. \\&\left. \qquad (\nabla ^{j_n} {{\textbf {b}}}_0)(\nabla ^{k_1} {\textbf{Q}}_{(m_1)})\cdots (\nabla ^{k_r} {\textbf{Q}}_{(m_l)})\right) , \end{aligned} \end{aligned}$$

(A.24)

where the indices satisfy

$$\begin{aligned} i_1+\ldots +i_m+j_1+\ldots +j_n+(k_1+m_1)+\ldots +(k_r+m_r)=k+2,\\ 1\leqq i_1,\ldots , i_m,j_1,\ldots ,j_m,k_1,\ldots ,k_r\leqq k+1,\\ -1\leqq m_1,\ldots , m_r\leqq k-1,~1\leqq k_1+m_1, \ldots , k_r+m_r\leqq k+1. \end{aligned}$$

Remark

Note that (A.22) is derived from taking two time derivatives in (A.20). On can also In fact, taking two time derivatives in (A.20), we know the right side has top-order terms $({{\textbf {b}}}_0\cdot \nabla )^2 {\textbf{Q}}_{2}-({{\textbf {b}}}_0\cdot \nabla )^2{\textbf{b}}_2\cdot {{\textbf {b}}}_0$. For the latter term $({{\textbf {b}}}_0\cdot \nabla )^2{\textbf{b}}_2\cdot {{\textbf {b}}}_0$, we recall that ${\textbf{Q}}={\textbf{q}}+\frac{1}{2}|{{\textbf {b}}}_0|^2$. Taking one time derivative and using continuity equation, we get ${\textbf{Q}}_{(1)}=-\text {div }{{\textbf {v}}}_0+{\textbf{b}}_1\cdot {{\textbf {b}}}_0$. Taking one more time derivative and using the momentum equation ${\textbf{v}}_1=({{\textbf {b}}}_0\cdot \nabla ){{\textbf {b}}}_0-\nabla {\textbf{Q}}_0$, we get $\text {div }{\textbf{v}}_1=\text {div }(-\nabla {\textbf{Q}}_0+({{\textbf {b}}}_0\cdot \nabla ){{\textbf {b}}}_0)$. Using $\text {div }{{\textbf {b}}}_0=0$ we know $\text {div }({{\textbf {b}}}_0\cdot \nabla ){{\textbf {b}}}_0$ is of lower order. So we have ${\textbf{Q}}_{(2)}\sim \Delta {\textbf{Q}}_0+{\textbf{b}}_2\cdot {{\textbf {b}}}_0$, and thus

$$\begin{aligned} ({{\textbf {b}}}_0\cdot \nabla )^2 {\textbf{Q}}_{2}-({{\textbf {b}}}_0\cdot \nabla )^2{\textbf{b}}_2\cdot {{\textbf {b}}}_0=({{\textbf {b}}}_0\cdot \nabla )^2\Delta {\textbf{Q}}_0+\text { lower order terms.} \end{aligned}$$

We choose to write in this form because it makes equations shorter. Alternatively one can differentiate (A.21) in time variable again and again to get the form $-\Delta _{{{\textbf {b}}}_0}{\textbf{Q}}_{(k)}^\infty =-{\textbf{Q}}_{(k+2)}^\infty +{\mathcal {M}}_k'({{\textbf {b}}}_0^\infty ,{\textbf{Q}}_{(-1)},\ldots ,{\textbf{Q}}_{(k-1)})+{\mathcal {N}}'_k({{\textbf {b}}}_0^\infty ,{\textbf{Q}}_{(-1)},\ldots ,{\textbf{Q}}_{(k-1)})$ where $\Delta _{{{\textbf {b}}}_0}:=(1+|{{\textbf {b}}}_0|^2)\Delta -({{\textbf {b}}}_0\cdot \nabla )^2$, and ${\mathcal {M}}_k'$ denotes the terms containing $(k+2)$-th order derivative.

This elliptic system has a parallel structure as [40, (16.11)]. Following [40, Lemma 16.2], we impose the system with boundary conditions

$$\begin{aligned} {\textbf{Q}}_{(k)}^\infty |_{\Gamma }={\textbf{Q}}_{0,k},~~-\nabla _N{\textbf{Q}}_{(k)}^\infty |_{\Gamma }={\textbf{Q}}_{1,k},~~k\geqq -1. \end{aligned}$$

Then the system has a formal power series in the distance to the boundary

$$\begin{aligned} \overline{{\textbf{Q}}}_{(k)}(r,\omega )\sim \sum {\textbf{Q}}_{n,k}(\omega )\frac{(1-r)^n}{n!}, \end{aligned}$$

where r is the distance to the boundary and $\omega$ is the angular variable. Let $0\leqq \chi (\cdot )\leqq 1$ be a smooth bump function on $\mathbb {R}$ that equals to 1 in $[-1,1]$ and vanishes outside $[-2,2]$. Then, by [40, Lemma 16.2], there exist $\varepsilon _{k,n}$ such that

$$\begin{aligned} \overline{{\textbf{Q}}}_{(k)}(r,\omega )=\sum \chi \left( \frac{1-r}{\varepsilon _{k,n}}\right) {\textbf{Q}}_{n,k}(\omega )\frac{(1-r)^n}{n!}, \end{aligned}$$

such that the above elliptic system holds to infinite order on the boundary. Note that $({{\textbf {b}}}_0\cdot \nabla )$ is tangential on the boundary and ${{\textbf {b}}}_0$ has smallness assumption, so the extra terms involving ${{\textbf {b}}}_0$ will not affect the convergence of the power series.

Now let $({\tilde{{{\textbf {v}}}_0}},{{\textbf {b}}}_0,{\tilde{{{\textbf {Q}}}_0}})$ are functions that vanish to infinite order on the boundary. Define

$$\begin{aligned} {{\textbf {w}}}_0^{\infty }&:={\tilde{{{\textbf {v}}}_0}}-\nabla \overline{{\textbf{Q}}}_{(-1)},~{{\textbf {P}}}_0^\infty :={\tilde{{{\textbf {Q}}}_0}}+\overline{{\textbf{Q}}}_0,\\ {\textbf{P}}_{(1)}&:=-(1+|{{\textbf {b}}}_0|^2)(\text {div }{\tilde{{{\textbf {v}}}_0}}+\Delta \overline{{\textbf{Q}}}_{(-1)})+({{\textbf {b}}}_0\cdot \nabla ){{\textbf {w}}}_0^{\infty }\cdot {{\textbf {b}}}_0, \end{aligned}$$

where $\overline{{\textbf{Q}}}_0,\overline{{\textbf{Q}}}_{(-1)}$ are given by the above construction. Then inductively one can show that ${\textbf{P}}_{(k)}^\infty ={\tilde{{{\textbf {Q}}}_0}}_{(k)}+\overline{{\textbf{Q}}}_{(k)}$, where $\overline{{\textbf{Q}}}_{(k)}$ are constructed above and ${\tilde{{{\textbf {Q}}}_0}}_{(k)}$ vanishes to infinite order on the boundary. Hence, choosing boundary data such that ${\textbf{Q}}_{0,k}=0$ for $k\geqq 0$ and ${\textbf{Q}}_{1,k}\geqq c_0>0$, then the Rayleigh-Taylor sign condition for ${{\textbf {P}}}_0^\infty$ is fulfilled and also we have ${\textbf{P}}^{\infty }_{(k)}|_{\Gamma }=0$ for all $k\geqq 0$.