Well-posedness in Sobolev spaces of the two-dimensional MHD boundary layer equations without viscosity

1.   Introduction

In this work we study the existence and uniqueness of solution to the two-dimensional magnetohydrodynamic (MHD) boundary layer system without viscosity which reads, letting $\Omega: = \mathbb T\times\mathbb R_+ = \big\{ (x,y)\in\mathbb R^2;\ x\in\mathbb T, y>0\big \}$ be the fluid domain,

where $(u, v)$ and $(f, g)$ stand for velocity and magnetic fields, respectively, and $\mu$ is resistivity coefficients, and $U, B$ and $p$ are the values on the boundary of the tangential velocity, magnetic fields and pressure, respectively, in the ideal MHD system satisfying the Bernoulli's law:

Note the MHD boundary system with a nonzero hydrodynamic viscosity will reduce to the classical Prandtl equations in the absence of a magnetic field, and the main difficulty for investigating Prandtl equation lies in the nonlocal property coupled with the loss of one order tangential derivative when dealing with the terms $v \partial_y u$. The mathematical study on the Prandtl boundary layer has a long history, and there have been extensive works concerning its well/ill-posedness theories. So far the two-dimensional (2D) Prandtl equation is well-explored in various function spaces, see, e.g., [1,2,4,5,6,7,9,10,11,13,14,22,23,24] and the references therein. Compared with the Prandtl equation the treatment is more complicated since we have a new difficulty caused by the additional loss of tangential derivative in the magnetic field. So far the MHD boundary layer system is mainly explored in the two settings.

● Without any structural assumption on initial data the well-posedness for 2D and 3D MHD boundary systems was established in Gevrey space by the first author and T. Yang [15] with Gevrey index up to $3/2$, and it remains interesting to relax the Gevrey index therein to $2$ inspired the previous works of [4,12] on the well-posedness for the Prandtl equations in Gevrey space with optimal index $2$.

● Under the structural assumption that the tangential magnetic field dominates, i.e., $f\neq 0$, the well-posedness in weighted Sobolev space was established by Liu-Xie-Yang [18] and Liu-Wang-Xie-Yang [16] without Oleinik's monotonicity assumption, where the two cases that with both viscosity and resistivity and with only viscosity are considered, respectively; see also the work of Gérard-Varet and Prestipino [8] for the stability analysis of the MHD boundary layer system with insulating boundary conditions (i.e. $f|_{y = 0} = 0$). These works, together with the essential role of the Oleinik's monotonicity for well-posedness theory of the Prandtl equations (see, e.g., [1,19,20]), justify the stabilizing effect of the magnetic field on MHD boundary layer, no matter whether or not there is resistivity in the magnetic boundary layer equation.

The aforementioned works [18,16] investigated the well-posedness for MHD boundary layer system with the nonzero viscosity coefficient. This work aims to consider the case without viscosity coefficient, giving a complement to the previous works [18,16]. To simply the argument we will assume without loss of generality that $\mu = 1$ and $( U, B)\equiv 0$ in the system (1) since the result will hold true in the general case if we use some kind of the nontrivial weighted functions similar to those used in for the Prandtl equation. Hence we consider the following 2D MHD boundary layer system in the region $\Omega = \mathbb T\times\mathbb R_+$

By the boundary condition and divergence-free condition above, we have

We remark that the equation for $g$ in (3) can be derived from the one for $f$ and the main difficulty in analysis is the loss of $x$-derivatives in the two terms $v$ and $g.$ As to be seen in the next Section 2, The system (3) can be derived from the MHD system

where ${u\mathit{\boldsymbol{}}}^{\varepsilon} = ( u^{\varepsilon}, v^{\varepsilon}) , {H\mathit{\boldsymbol{}}}^{\varepsilon} = (f^{\varepsilon}, g^{\varepsilon})$ denote velocity and magnetic field, respectively. The MHD system (4) is complemented with the boundary condition that

It is an important issue in both mathematics and physics to ask the high Reynolds number limit for MHD systems, and so far it is justified mathematically by Liu-Xie-Yang [17] with the presence of viscosity and the other cases remain unclear.

Notation. Before stating the main result we first list some notation used frequently in this paper. Given the domain $\Omega = \mathbb T\times\mathbb R_+$, we will use $\Vert{\cdot}\Vert_{L^2}$ and $\left({\cdot, \cdot}\right)_{L^2}$ to denote the norm and inner product of $L^2 = L^2(\Omega)$ and use the notation $\Vert{\cdot}\Vert_{L_x^2}$ and $\left({\cdot, \cdot}\right)_{L_x^2}$ when the variable $x$ is specified. Similar notation will be used for $L^\infty$. Moreover, we use $L^p_x(L^q_y) = L^p (\mathbb T; L^q(\mathbb R_+))$ for the classical Sobolev space. Let $H^m = H^m(\Omega)$ be the standard Sobolev space and define the weighted Sobolev space $H_\ell^m$ by setting, for $\ell\in \mathbb{R}$,

where here and below $\left\langle {y} \right\rangle = {{(1+\left| y \right|^2)}^{1/2}}.$ With the above notation the well-posedness theory of (1) in weighted Sobolev space can be stated as below. Here the main assumption is that the tangential magnetic field in (1) dominates, that is, $f\neq 0$.

Theorem 1.1. Let $\ell >\frac 12$ and $\delta>\ell+\frac 12$ be two given numbers. Suppose the initial data $u_{0}, f_{0}$ of (3) lie in $H^{4}_{\ell}(\Omega)$ satisfying that there exists a constant $c_0>0$ such that for any $(x,y)\in\Omega$,

Then the MHD boundary layer system (3) admits a unique local-in-time solution

for some $T>0$. Moreover a constant $c>0$ exists such that for any $(t,x,y)\in [0,T]\times \Omega$,

Remark 1. The solution in Theorem 1.1 lies in the same Sobolev space as that for initial data, different from the previous works [15,16,18] where the loss of regularity occurs at positive time. Here the tangential magnetic field is allowed to decay polynomially at infinity, and this relaxes the condition in [16,18] where the infimum of the tangential component is strictly positive.

Remark 2. The result above confirms that the magnetic field may act as a stabilizing factor on MHD boundary layer. The stabilizing effect was justified by [18] for the case with both viscosity and resistivity, and by [16] for the case without resistivity.

2.   Derivation of the boundary layer system

This section is devoted to deriving the boundary layer system (1). We consider the MHD system in $\Omega$

where ${u\mathit{\boldsymbol{}}}^{\varepsilon} = (u^{\varepsilon},v^{\varepsilon}), {H\mathit{\boldsymbol{}}}^{\varepsilon} = (f^{\varepsilon},g^{\varepsilon})$ denote velocity and magnetic fields, respectively. The above system is complemented with the no-slip boundary condition on the normal component of velocity field and perfectly conducting boundary condition on the magnetic field, that is,

A boundary layer will appear in order to overcome a mismatch on the boundary $y = 0$ for the tangential magnetic fields between (6) and the limiting equations by letting $\varepsilon\rightarrow 0.$ To derive the governing equations for boundary layers we consider the ansatz

where we used the notation $\tilde{ y } = y/ \sqrt{\varepsilon}$. We suppose $u^b, f^b,$ and $p^b$ in the expansion (8) polynomially trend to zero as $\tilde y\rightarrow +\infty$, that is, as $\varepsilon\rightarrow 0.$ Similarly for the expansion of the initial data.

Boundary conditions. Taking trace on $y = 0$ for the second and the fourth expansions in (8) and recalling the boundary condition (7), we derive that

and using again the second and the fourth equations in (8) and letting $\varepsilon\rightarrow 0$, we get that

Moreover observe

This gives

The governing equations of the fluid behavior near and far from the boundary. We substitute the ansatz (8) into (6) and consider the order of $\varepsilon$. At the order $\varepsilon^{-1/2}$ we get

This with the assumption that $p^b$ goes to $0$ as $\tilde y\rightarrow +\infty$ implies

At the order $\varepsilon^0$, letting $\tilde y\rightarrow +\infty$ $( \varepsilon\rightarrow 0)$ and taking into account (9) and fact that $u^b, f^b,$ and $p^b$ polynomially trend to zero as $\varepsilon\rightarrow 0$, we see the limiting system is the ideal incompressible MHD system:

complemented with the boundary condition (9) and initial data ${u\mathit{\boldsymbol{}}}^0_{in}$ and ${H\mathit{\boldsymbol{}}}^0_{in}$, where ${u\mathit{\boldsymbol{}}}^{0} = (u^0, v^0), {H\mathit{\boldsymbol{}}}^{0} = ( f^0, g^0)$.

Next we will derive the boundary layer equations. Let ${u\mathit{\boldsymbol{}}}^{0} = (u^0, v^0), {H\mathit{\boldsymbol{}}}^{0} = ( f^0, g^0)$ be the solution to the ideal MHD system (13). By Taylor expansion we write $u^0(t,x,y)$ as

where here and below we use the notation $\overline{h}$ to stand for the trace of a function $h$ on the boundary ${y = 0}$. Similarly,

Now we compare the order $\varepsilon^0$ for the resulting equation by substituting the ansatz (8) as well as the above Taylor expansion of ${u\mathit{\boldsymbol{}}}^{0} = (u^0, v^0), {H\mathit{\boldsymbol{}}}^{0} = ( f^0, g^0)$ into (6); this gives, by virtue of (12) and (13),

Denoting

and recalling $u ^ { b }, f^b$ polynomially trend to $0$ as $\tilde y \rightarrow +\infty$, we combine (14) and the boundary conditions (9)–(11) to conclude that all equations except the third one in (1) are fulfilled by $u, v, f,$ and $g$. For simplicity of notation, we have replaced $\tilde{y}$ by $y$. Note that the third equation in (1) can be derived from the second one and the boundary condition $\partial_yf|_{y = 0} = 0$ by observing that

Finally we remark the Bernoulli's law (2) follows by taking trace on $y = 0$ for the ideal MHD system (13).

3.   A priori energy estimates

The general strategy for constructing solutions to (3) involves mainly two ingredients. One is to construct appropriate approximate solutions, which reserve a similar properties as (5) for initial data by applying the standard maximum principle for parabolic equations in the domain $\Omega$ (see [19,Lemmas E.1 and E.2 ] for instance). Next we need to deduce the uniform estimate for these approximate solutions. For sake of simplicity we only present the following a priori estimate for regular solutions, which is a key part to prove the main result Theorem 1.1.

Theorem 3.1. Let $\ell >\frac 12$ and $\delta>\ell+\frac 12$ be two given numbers, and let $u,f\in L^\infty([0,T]; H_\ell^4)$ solve the MHD boundary layer system (3) satisfying that a constant $c>0$ exists such that for any $(t,x,y)\in [0,T]\times \Omega$,

Then there exists a constant $C>0$ such that

where here and below

We will present the proof of Theorem 3.1 in the next two subsections, one of which is devoted to the estimates on tangential and another to the normal derivatives. To simplify the notation we will use the capital letter $C$ in the following argument to denote some generic constants that may vary from line to line, and moreover use $C_ \varepsilon$ to denote some generic constants depending on a given number $0< \varepsilon\ll 1.$

3.1.   Energy estimates: Tangential derivatives

In this part, we will derive the estimate on tangential derivatives, following the cancellation mechanism observed in the previous work of Liu-Xie-Yang [18].

Lemma 3.2. Under the same assumption as in Theorem 3.1 we have, for any $t\in[0,T],$

Recall $\mathcal E$ is defined in (15).

Proof. Without loss of generality we may consider $i = 4$, apply $\partial_x^4$ to the first second equations and $\partial_x^{3}$ to the third equation in (3), respectively, this gives

where

and

In order to eliminate the terms $\partial_x^4 v$ and $\partial_x^4 g$ where the fifth order tangential derivatives are involved, we introduce, observing $f>0$ by assumption,

Multiplying the third equation in (16) by $(\partial_y f)/f$ and then taking summation with the second one in (16), we obtain the equation solved by $\psi$, that is,

where

with $P_4$ and $Q_4$ given in (16). Similarly we multiply the third equation in (16) by $(\partial_y u)/f$ and then add the resulting equation by the first one in (16), to obtain

where

with $F_4$ and $Q_4$ given in (16). Note that (18) is complemented with the boundary condition that

Thus we perform the weighted energy estimate for (18) and (19) and use the fact that

to get

where here and below we use $[\mathcal T_1,\ \mathcal T_2]$ to denote the commutator between two operators $\mathcal T_1, \mathcal T_2$, that is

Observe the derivatives are at most up to the fourth order for the terms on the right of (20). Then by direct compute we have

Substituting the above estimate into (20) and then integrating over $[0,t]$ for any $0<t<T$ gives

Next we will derive the estimates for $f, u$ from the ones of $\psi,\varphi$. In fact in view of the representation of $\psi$ given in (17), we use Hardy-type inequality (cf. [19,Lemma B.1] for instance) to conclude

As a result, using the representation of $\psi$ and $\varphi$ given in (17) and the fact that $|(\partial_yf)/f |\lesssim \left<y\right>^{-1}$ gives

and

Moreover, using again (17),

Combining these inequality with (22) we conclude

Note the above estimate still holds true if we replace $\partial_x^4$ by $\partial_x^i$ with $i\leq 4$. This gives the desired estimate in Lemma 3.2, completing the proof.

3.2.   Estimate for normal derivatives

In this part, we perform the estimate for normal derivatives. Compared with [18] a new difficulty arises when dealing the boundary integrals because of the absence of the hydrodynamic viscosity.

Lemma 3.3. Under the same assumption as in Theorem 3.1 we have, for any $t\in[0,T]$,

Recall $\mathcal E, \mathcal D$ are defined by (15).

Proof. Step 1). We first consider the case of $i = 0$ and $j = 4$. In this step we will prove that, for any $\varepsilon>0$,

Applying $\left<y\right>^{\ell+4}\partial_y^4$ to (3) yields that

and

Recall $[\cdot, \cdot]$ is given by (21), standing for the commutator between two operators. Taking inner product with $\left<y\right>^{\ell+4} \partial_y^4 u$ to the first equation above, and with $\left<y\right>^{\ell+4} \partial_y^4 f$ to the second one, and then taking summation and observing

we obtain

with

Direct computation shows

It remains to deal with the boundary integeral on the right of (24). We first apply $\partial_y$ to the second equation in (3) and then take trace on $y = 0$; this together with the boundary condition in (3) gives

By virtue of the above representation of $\partial_y^3f|_{y = 0}$, we compute directly that

As a result we combine (27) with Sobolev's inequality to conclude

Substituting the above inequality and (25) into (24) and then integrating over $[0, t]$ for any $t\in[0, T]$ we obtain the desired estimate (23).

Step 2). In this step we will treat the first term on the right of (23) and prove that

holds true for any $\varepsilon>0$. To do so we recall some facts on the Fourier multiplier. Let $k\in\mathbb Z$ be the partial Fourier dual variable of $x\in\mathbb T$ and let $\Lambda_x^\sigma,\sigma\in\mathbb R,$ be the Fourier multiplier with symbol $\left({1+k^2}\right)^{\sigma/2}$, that is,

where $\mathcal F_x$ stands for the Fourier transform with respect to $x$ variable:

Similarly we define $|D_x|^\sigma, \sigma>0,$ by setting

Given a $C^1$ function $\rho$ of $x\in\mathbb T$ with bounded derivatives, we have, for $0<\sigma<1$,

with $C_\sigma$ a constant depending only on $\sigma$, recalling the commutator $[\left\vert{D_x}\right\vert^\sigma , \ \rho]$ is defined by (21). Note the counterpart for $x\in\mathbb R$ of (29) is clear, see, e.g., [3,Pages 702–704], the estimate (29) can be proven in a similar inspirit and we omit it for brevity and and refer to [21] and references therein for the comprehensive argument on the extension of the classical pseudo-differential calculus in $\mathbb R$ to the torus case $x\in\mathbb T$.

With the Fourier multipliers introduced above we use (29) to compute

On the other hand, using the fact that

we compute

A similar argument gives

Substituting the two inequalities above into (30) yields

This gives the desired estimate (28).

Step 3). We combine (23) and (28) to obtain

Observe the above inequality still holds true if we replace $\left<y\right>^{\ell+4} \partial_y^4$ by $\left<y\right>^{\ell+j} \partial_x^i \partial_y^j$ with $i+j\leq 4$ and use the boundary conditions (26) and $\partial_y f|_{y = 0} = 0$. Since the argument is straightforward we omit it for brevity. The proof of Lemma 3.3 is completed.

3.3.   Completing the proof of the energy estimate

Combining the estimates in Lemmas 3.2–3.3 and letting $\varepsilon$ be small enough we obtain the desired energy estimate, completing the proof of Theorem 3.1.