Well-posedness of the MHD boundary layer equations with small initial data in Sobolev space

1.   Introduction and main result

In this paper, we investigate the 2D magnetohydrodynamic (MHD) boundary layer equations on the upper half plane $\mathbb{R}_+^2 = \{(x, y):x\in\mathbb{R}, \ y\in\mathbb{R}_+\}$, which reads as:

where $(u, v)$ represents the velocity field of the boundary layer flow, $(h, g)$ stands for the magnetic field and functions $U(t, x), H(t, x)$, and $p(t, x)$ denote the trace of the tangential fluid and magnetic, the pressure of the outflow, respectively, which satisfy Bernoulli's law

System (1.1) is a boundary layer model, which is derived from the 2D incompressible MHD system with a non-slip boundary condition on the velocity and a perfectly conducting condition on the magnetic field ^[1,2].

Before exhibiting the main result in this paper, let us recall some known results to system (1.1). Especially, when the magnetic field $(h, g)$ are some constants in system (1.1), it reduces to the classical Prandtl equations, which were first introduced formally by Prandtl ^[3] in 1904. The Prandtl equations are the foundation of the boundary layer theory. It describes that the fluid near the boundary of a solid body can be divided into two regions: a very thin layer in the neighborhood of the body (the boundary layer) where viscous friction plays an essential part, and the remaining region outside this layer where friction may be neglected (the outer flow). The well-posedness theory of the Prandtl equations was well understood in ^[4,5,6] and the references therein for the recent progress.

When the velocity field equation is coupled with the magnetic field equation, the phenomenon of the boundary layer is different since the boundary layers of the magnetic field may exist ^[1] and they are more complicated than the classical Prandtl equations. It is worth pointing out that some results have been obtained about the well-posedness of the MHD boundary layer equations in weighed Sobolev space. Liu et al. ^[2] proved the local existence and uniqueness of solutions for the 2D nonlinear MHD boundary layer equations without monotonicity in weighted Sobolev space by using energy methods. Liu et al. ^[7] investigated the local well-posedness of the 2D MHD boundary layer equations without resistivity in Sobolev spaces. Finally, they also got the linear instability of the 2D MHD boundary layer when the tangential magnetic field is degenerate at one point. Besides, there are some well-posedness results for the MHD equations ^[8,9].

There are some results in the analytic framework for the 2D MHD boundary layer equations, Xie and Yang ^[10] considered the global existence of solutions to the 2D MHD boundary layer equations in the mixed Prandtl and Hartmann regime when the initial datum is a small perturbation of the Hartmann profile and obtained the solution in analytic norm as an exponential decay in time. Recently, by using the cancellation mechanism, Xie and Yang ^[11] investigated the existence and uniqueness of solutions to the 2D MHD boundary layer system in an analytic space.

Besides, there are some known results about the well-posedness of boundary layer equations by using the paralinearization method. Chen et al. ^[12] obtained the local well-posedness to the classical Prandtl equations when the weighted function $\mu$ is an exponential function in $H^{3, 1}_\mu(\mathbb{R}_+^2)\cap H^{1, 2}_\mu(\mathbb{R}_+^2)$. Wang and Zhang ^[13] proved the local well-posedness of the classical Prandtl equation for monotonic data in a polynomial weighted Sobolev space. Chen et al. ^[14] studied the long-time well-posedness of the MHD equations for small initial data in an exponentially weighted Sobolev space $H^{3, 0}_\mu(\mathbb{R}_+^2)\cap H^{1, 2}_\mu(\mathbb{R}_+^2)$, and obtained the lifespan of solutions to depend on the initial data. Wang and Wang ^[15] investigated the global well-posedness of the 2D MHD equations in striped domain with small data, and proved the solutions of the anisotropic MHD equations convergence to the solutions of the hydrostatic MHD equations in $L^\infty$. Chen and Li ^[16] obtained the long-time well-posedness of the 2D MHD boundary layer equations with small initial data in an exponentially weighted Sobolev space $H^{3, 0}_\mu(\mathbb{R}_+^2)\cap H^{1, 2}_\mu(\mathbb{R}_+^2)\cap H^{2, 1}_\mu(\mathbb{R}_+^2)$, and proved the lifespan of solutions depends on the initial data. Inspired by the ideas in ^[13,14], the aim of this paper is to investigate the well-posedness of the problem (1.1) by using the paralinearization method and an abstract bootstrap argument. Similar to the Prandtl equation, the difficulty of solving the problem (1.2) in the Sobolev framework is the loss of $x$-derivative in the terms like $v\partial_y u$, $v\partial_y h$, $g\partial_y u$ and $g\partial_y h$. To overcome this essential difficulty, inspired by recent results in ^[14], we will first paralinearize system (1.2) and introduce two new good functions to symmetrize the system, then establish the estimates of solutions to system (3.3).

Finally, the rest of the paper is arranged as follows. In Section 2, we introduce the Littlewood-Paley decomposition and paraproduct and some lemmas which that be used frequently. In Section 3, we paralinearize the system (1.2) and introduce the good unknown functions to symmetrize the system. In Section 4, we prove the Sobolev estimate in the horizontal direction. In Section 5, we get the high order energy estimate in the $y$ variable and give the proof of Theorem 1.1.

Hereafter, let letter $C$ be a general positive constant independent of $\varepsilon$, which may vary from line to line at each step.

For simplicity's sake, we consider a uniform outflow $(U, H) = (0, 1)$. Let $h(t, x, y) = 1+\tilde{h}(t, x, y)$. Then $(u, \tilde{h})$ satisfies the following system:

As in ^[14], we first introduce the following weighted Sobolev space. For $m, n, \alpha, \beta \in \mathbb{N}$, $\ell\geq\frac{3}{2}$, the space $H_\ell^{m, n}(\mathbb{R}_+^2)$ consists of all functions $f\in L^2_\ell$ satisfying

where $\|f\|_{L^2_\ell} = \|\langle y\rangle^\ell f(x, y)\|_{L^2}$ with $\langle y\rangle = (1+y)$.

We are now in a position to state the main result as follows:

Theorem 1.1. Let $\ell\geq\frac{3}{2}$, $m, \beta\in \mathbb{N}$. For small enough $\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]$, assume that initial data $(u_0, \tilde{h}_0)$ satisfy

then for any given time $T$ independent of $\varepsilon$ such that the problem (1.2) has a unique solution $(u, \tilde{h})$ satisfies

for any $t\in[0, T]$. Here $C$ is a positive constant independent of $\varepsilon$.

Remark 1.1. Theorem 1.1 obtains the local well-posedness of the solution, while the global well-posedness of the solution is still an open problem.

2.   Littlewood-Paley decomposition and paraproduct

As in ^[14], we first introduce the Littlewood-Paley decomposition in the horizontal direction $x\in {\mathbf R}$. Choose two smooth functions $\varphi(\tau)$ and $\chi(\tau)$ that satisfy

Then we define

Bony's paraproduct $T_fg$ is defined by

Then we have the following Bony's paraproduct

where the remainder term $R(f, g)$ is defined by

Next, let us introduce some classical paraproduct estimates and paraproduct calculus in Sobolev space ^[17], Chapter 2.

Lemma 2.1. Let $s\in \mathbb{R}$, it holds that

If $s > 0$, then we have

Lemma 2.2. Let $s\in \mathbb{R}$ and $\sigma\in(0, 1]$, it holds that

Especially, we have

here $T^\ast_a$ is the adjoint of $T_a$.

Lemma 2.3. Let $s\in \mathbb{N}$, it holds that

In this position, we will show the Agmon inequality, whose proof is given in ^[18].

Lemma 2.4. Let $f\in H^1(\mathbb{R}^2_+)$, then

Before proving the main Theorem 1.1 by using Lemmas 2.1–2.4, we first paralinearize and symmetrize the system (1.2) according to the method in ^[14].

3.   Paralinearization and symmetrization

Similar to the Prandtl equations, the difficulty of solving problem (1.2) in the Sobolev framework is the loss of $x$-derivative in the terms $v\partial_yu-g\partial_y\tilde{h}$ and $v\partial_y\tilde{h}-g\partial_yu$ in the first and second equations of (1.2), respectively. In other words, $v = -\partial^{-1}_y\partial_xu$ and $g = -\partial^{-1}_y\partial_x\tilde{h}$ by the divergence-free conditions and boundary conditions. Thus, it creates a loss of the $x$-derivative and a $y$-integration to the $y$-variable. Then the standard energy estimates do not work. To overcome this essential difficulty, inspired by recent results in ^[14], we will first paralinearize the system (1.2) and then introduce the good unknown functions to symmetrize the system following the idea in ^[14].

Applying Bony's decomposition (2.1), we derive

where

We now define

From system $(3.1)_2$, we deduce that

Motivated by ^[14], we introduce the two good unknown functions

Consequently, we can rewrite the system (3.1) as

where

and

Moreover, it is easy to check $(u_\beta, \tilde{h}_\beta)$ satisfies the following boundary conditions:

The investigation of the local well-posedness of the solution to system (1.2) is equivalent to studying the local well-posedness of systems (3.3)–(3.6). We will establish a priori estimates of the solution to systems (3.3)–(3.6) in Sections 4 and 5.

4.   Sobolev estimate in horizontal direction

In this section, we will establish the estimates of solutions. Let us first define the following energy functionals

and

We assume that $(u, \tilde{h})$ is a smooth solution of (1.2) on $[0, T^\ast]$ and

where the positive constant $c_1 = \frac{1}{\sqrt{10C}}$.

4.1.   Some technical lemmas

We first give the proof of the lower bound of $h(t, x, y)$.

Lemma 4.1. Let ${\ell\geq\frac{3}{2}}$. For small enough ${\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]}$ and any small $\delta\in(0, 1)$, it holds that

Proof. As $h = \tilde{h}+1$, we can derive the following inequality from Lemma 2.4 and the condition $\lim\limits_{y\rightarrow +\infty}\tilde{h} = 0$

The proof is thus complete. □

The following lemma can be obtained by the Hölder inequality (^[19], Theorem 1.4.3).

Lemma 4.2. Let ${\ell\geq\frac{3}{2}}$, it holds that

Especially, thanks to $\partial_xu+\partial_yv = 0$, $\partial_x\tilde{h}+\partial_yg = 0$, it holds that for $n\in \mathbb{N}$,

Lemma 4.3. Let ${\ell\geq\frac{3}{2}}$. For ${\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]}$ is small enough and any small $\delta\in(0, 1)$, it holds that

Proof. Integrating it over $[0, y]$ and using the boundary condition $u(t, x, 0) = 0$ and Lemma 2.4, we have

and

Applying the Gagliardo–Nirenberg inequality (^[19], Theorem 1.1.18), the Young inequality (^[19], Corollary 1.4.1) and Lemma 2.4, we deduce that

Using the boundary condition $\partial^2_y u(t, x, 0) = 0$, Young's inequality (^[19], Corollary 1.4.1) and Lemma 2.4, we conclude

In the same way, using the condition $\partial_y\tilde{h}|_{y = 0} = 0$, we derive

or

By virtue of the Gagliardo-Nirenberg inequality (^[19], Theorem 1.1.18), Lemmas 2.4 and 4.3 again yield

The proof is now complete. □

The following lemma introduces the relationship of norms between good functions $(u_\beta, \tilde{h}_\beta)$ and $(u, \tilde{h})$.

Lemma 4.4. Let ${\ell\geq\frac{3}{2}}$. For sufficient small ${\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]}$, then for any $t\in [0, T^\ast]$,

Proof. Using Lemmas 2.1, 2.4, 4.1 and 4.2, we can conclude that

Analogously, we have

Therefore, by taking a small enough $\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]$, we obtain

Besides, we also obtain

Applying Lemmas 2.1, 2.4, and 4.1–4.3, we deduce

Similarly, we also obtain

The proof is therefore complete. □

Lemma 4.5. Let ${\ell\geq\frac{3}{2}}$. For any ${\varepsilon\in(0, \frac{1}{2\sqrt{10C}}]}$, it holds that,

Proof. By virtue of the Gagliardo–Nirenberg inequality (^[19], Theorem 1.1.18), Lemmas 2.4 and 4.4 yielding

The proof is hence complete. □

4.2.   The estimates of the nonlinear terms $G_1$ and $G_2$

In this subsection, we establish the estimate of the nonlinear terms $G_1$ and $G_2$.

Lemma 4.6. Let ${\ell\geq\frac{3}{2}}$, it holds that

Proof. We only establish the estimate of $G_1$. The estimate of $G_2$ could be derived in a similar way. By Lemmas 2.2, 2.4, and 4.1–4.3, we obtain

It is easy to check that

Using Lemmas 2.2, 2.4, and 4.1–4.3 again, we attain

We now apply Lemmas 2.2, 2.4, and 4.1–4.3 again, with $G_{11}$ replaced by $G_{13}$, to obtain

Recall Lemmas 2.1 and 4.1–4.4; we derive

By Lemmas 2.1 and 4.1–4.4, we have

In the same manner, we can obtain

Next, we will deal with the term $G_{14}$. From Eq (1.2), we can conclude that

Using Lemmas 2.1 and 4.1–4.4, we are led to

In the same way, we deduce

and

Thus, we can derive that

Applying Lemmas 2.1, 2.4, and 4.1–4.5, we are led to

which implies

Analogously, we obtain

Adding all the above estimates together gives the desired result.

The proof is now complete. □

4.3.   Tangential energy estimate

In this subsection, we show the high-order derivative estimates of the solutions in the horizontal variable $x$.

Lemma 4.7. Let ${\ell\geq\frac{3}{2}}$, it holds that

Proof. Multiplying Eq (3.3) by $(u_\beta, \tilde{h}_\beta)$ in $H^{3, 0}_{\ell}$, respectively, we derive

First of all, using (3.6) and integrating it by parts, we obtain

An easy computation shows that

where

By Lemmas 2.1, 2.2, 4.3, and 4.5, we conclude

and

and by Lemmas 2.3 and 4.3 gives,

Thus, we have

Likewise, we can also obtain

A simple calculation yields

Therefore, we also obtain

It follows from Lemma 4.6 that

Summing up all the above estimates, we deduce

The proof is then complete. □

5.   High order energy estimate in $y$ variable

In Lemma 4.7, we just prove the high-order derivative estimates of the solutions in the horizontal variable $x$. To make the energy estimate more complete, we need to derive the high-order derivative estimates in variable $y$. We again assume that $(u, \tilde{h})$ is a smooth solution of (1.2) on $[0, T^\ast]$ satisfying (4.1).

Lemma 5.1. Let ${\ell\geq\frac{3}{2}}$, it holds that for any $t\in[0, T^\ast]$,

Proof. The proof will be divided into the following three steps.

Step 1. ($H_{\ell-1}^{1, 0}$ estimate) Taking $H_{\ell-1}^{1, 0}$-inner product between (1.2) with $(\partial_yu, \partial_y\tilde{h})$, we obtain

First of all, integrating it by parts and using the Hölder inequality (^[19], Theorem 1.4.3), we deduce

By integration by parts and applying Lemma 4.3, we have

It infers from Lemmas 4.3 and 4.5 that

In the same way, we can obtain

This shows that

In view of ${v = \int_0^y\partial_xud\tilde{y}}$, from Lemmas 4.2 and 4.4, we can derive that

The estimate of $B_1$ could be deduced in a similar way. Then we have

Therefore, we conclude that

Step 2.($H_{\ell}^{1, 1}$ estimate) Taking $\partial_y$ to (1.2) and then taking $H_{\ell}^{1, 0}$-inner product with $(\partial_yu, \partial_y\tilde{h})$, we attain

For the estimate of term $A_2$, by integration by parts and using Lemma 4.3, we have

and

On account of the above calculations, we can obtain

For the estimate of term $B_2$, using Lemma 4.2, we thus obtain

The remainder terms of $B_2$ can be handled in much the same way, which gives

Therefore, we obtain

For the estimate of term $C_2$, we have

Thanks to Lemmas 4.3 and 4.4, we conclude

It is obvious that

Consequently, we deduce

Step 3. ($H_{\ell+1}^{1, 2}$ estimate) Taking $\partial^2_y$ to (1.2) and then taking $H_{\ell+1}^{1, 0}$-inner product with $(\partial^2_yu, \partial^2_y\tilde{h})$, we attain

We establish the estimates of the nonlinear terms as follows:

For the estimates of the term $A_3$, a direct calculation yields

and

Therefore, applying the Hölder inequality (^[19], Theorem 1.4.3), Lemmas 4.1, 4.3 and 4.5 for the above equality, we have

For the estimates of the term $B_3$, applying the Hölder inequality (^[19], Theorem 1.4.3), Lemmas 2.4, 4.2 and 4.4, we can conclude that

Analogously, using the Hölder inequality (^[19], Theorem 1.4.3), Lemmas 2.4, 4.3 and 4.4, for ${\ell\geq\frac{3}{2}}$, we can deduce that

where we used the following facts:

and

For the estimates of the term $D_3$, applying $(1.2)_3$, the Hölder inequality (^[19], Theorem 1.4.3), Lemmas 2.4 and 4.3–4.5 again, we can also derive that

The estimate of the term $E_3$ is obvious that

Thus, we obtain

Summarizing all the above estimates (5.1)–(5.3), the proof is thus complete.

The proof of Theorem 1.1

According to the initial data condition (1.3) and Lemma 4.3, we can get

Therefore, $E(0)\leq 5\varepsilon^2$.

Based on the classical bootstrap argument ^[20], we can obtain the uniform estimates of solutions to problem (1.2). First, we assume that $[0, T^\ast]$ is the maximal time interval such that

where the positive constant $c_1$ is determined later.

It follows from Lemmas 4.7 and 5.1 that

Using the assumption condition $E(t)\leq (c_1 \varepsilon)^2$ and the smallness property of $\varepsilon$, (5.5) implies

Then using the Gronwall's inequality, we can conclude that for any $t\in [0, T^\ast]$,

if we take $c_1 = \sqrt {10C}$, the theorem 1.1 follows by a bootstrap argument. □

6.   Conclusions

This paper mainly investigates the well-posedness of the 2D MHD boundary layer equations for small initial data in Sobolev space of polynomial weight and low regularity. The main steps include the following two parts: (ⅰ) We first obtain the systems (3.3)–(3.6) by paralinearizing and symmetrizing the system (1.2). (ⅱ) We establish the estimates of the solution in horizontal direction and vertical direction, respectively. In addition, the method in this article can also be used to investigate the well-posedness of the other boundary layer equations.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Acknowledgments

This paper was in part supported by the Natural Science Foundation of Henan with contract number 242300420673 and the Zhoukou Normal University high level talents research start funding project with contract number ZKNUC2022008.

Conflict of interest

The author declare there is no conflicts of interest.