Abstract
In this paper, we study the well-posedness of boundary layer problems for the inhomogeneous incompressible magnetohydrodynamics (MHD) equations, which are derived from the two-dimensional density-dependent incompressible MHD equations. Under the assumption that initial tangential magnetic field is not zero and density is a small perturbation of the outer constant flow in supernorm, the local-in-time existence and uniqueness of inhomogeneous incompressible MHD boundary layer equations are established in weighted conormal Sobolev space by energy method.
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Acknowledgements
Jincheng Gao’s research was partially supported by NNSF of China (11801586) and Natural Science Foundation of Guangdong Province of China (2020B1515310004). Daiwen Huang’s research was partially supported by NNSF of China (11971067, 11631008, 11771183). Zheng-an Yao’s research was partially supported by NNSF of China (11971496, 12026244).
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Appendices
Appendix A: Calculus inequalities
In this appendix, we will introduce some basic inequalities that be used frequently in this paper. First of all, we introduce the following Hardy type inequality, which can refer to [36].
Lemma A.1
Let the proper function \(f:{\mathbb {T}}\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\), and satisfies \(f(x, y)|_{y=0}=0\) and \( \underset{{y \rightarrow +\infty }}{\lim } f(x,y) = 0\). If \(\lambda > - \frac{1}{2}\), then it holds true
Next, we will state the following Sobolev-type inequality.
Lemma A.2
Let the proper function \(f:{\mathbb {T}}\times {\mathbb {R}}^+ \rightarrow {\mathbb {R}}\), and satisfies \( \underset{{y \rightarrow +\infty }}{\lim } f(x,y) = 0\). Then there exists a universal constant \(C>0\) such that
or equivalently
Proof
Indeed, the estimate (A.3) follows directly from estimate (A.2) and the Cauchy–Schwartz inequality. Hence, we only give the proof for the estimate (A.2). On one hand, thanks to the one-dimensional Sobolev inequality for the \(y-\)variable, we get
On the other hand, we apply the following one-dimensional Sobolev inequality for \(x-\)variable to get
Therefore, substituting the estimate (A.5) into (A.4), we complete the proof of estimate (A.2). \(\square \)
Now we will state the Moser type inequality as follow:
Lemma A.3
Denote \(\Omega :={\mathbb {T}}\times {\mathbb {R}}^+\), let the proper functions \(f(t, x, y): {\mathbb {R}}^+\times \Omega \rightarrow {\mathbb {R}}\) and \(g(t, x, y): {\mathbb {R}}^+ \times \Omega \rightarrow {\mathbb {R}}\). Then, there exists a constant \(C_m>0\) such that
where \(|\beta +\gamma |=m\) and \(l_1+l_2=l\).
Proof
For any \(p\ge 2\), due to the relation \(|Z_2 f|^p=Z_2(f Z_2 f |Z_2 f|^{p-2})-(p-1)f Z_2^2 f|Z_2 f|^{p-2}\), we find
Integrating by part and applying the Hölder inequality, we find for \(0 \le \theta \le 1\) and \(0\le \theta _1 \le \frac{\theta }{2}\) that
and hence, it follows
Here \(\frac{1}{q}+\frac{1}{r}=\frac{2}{p}\). Integrating with respect to t and x variables, and applying Hölder inequality, we get
Here \(Q_T=\Omega \times [0, T]\). Similarly, it is easy to justify for \(i=0,1\),
Here \(\frac{1}{q}+\frac{1}{r}=\frac{2}{p}\). By multiple application of the above inequality, we get(proof by induction)
where \(\frac{1}{p_1}=\frac{1}{q_1}(1-\frac{|\alpha |}{m})+\frac{|\alpha |}{r_1 m}\) and \(1\le |\alpha | \le m-1\). Then, we get for \(|\beta |+|\gamma |=m\) that
Therefore, we complete the proof of this lemma. \(\square \)
Finally, we establish the following \(L^\infty -\)estimate with weight for the heat equation.
Lemma A.4
For the heat equation \(\partial _t F(t, x)-\epsilon \partial _x^2 F(t, x)=G(t, x), \ (t, x) \in {\mathbb {R}}^+\times {\mathbb {R}}^+\); with the boundary condition \(F(t, x)|_{x=0}=0\) and initial data \(F(t, x)|_{t=0}=F_0\). Then, it holds true
where C is a constant independent of the parameter \(\epsilon \).
Proof
First of all, let us consider the heat equation
with the initial data and boundary condition
In order to transform the problem (A.8) into a problem in the whole space, let us define \({\widetilde{H}}(t, x)\) by
and define the initial data \({\widetilde{H}}_0(x)\) by
It is easy to justify that the function \({\widetilde{H}}(t, x)\) solves the following evolution equation
Define \(S(t, x)=\frac{1}{\sqrt{4\pi \epsilon t}}e^{-\frac{|x|^2}{\sqrt{4\epsilon t}}}\), then the solution of evolution (A.9) can be expressed as
which implies directly
In view of the relation \(x \partial _x S(t, x-\xi )=(x-\xi )\partial _x S(t, x-\xi )+\xi \partial _x S(t, x-\xi )\), we get
Due to \(\int _{{\mathbb {R}}}|(x-\xi ) \partial _x S(t, x-\xi )|d\xi \le C\), it follows
Using the equality \(\partial _x S(t, x-\xi )=-\partial _{\xi } S(t, x-\xi )\), the integration by part yields directly
and hence, we get
This, along with representation (A.10) and the well-known Duhamel formula, we complete the proof of this lemma. \(\square \)
Appendix B: Almost equivalence of weighted norms
In this subsection we will use the quantity \(h^\epsilon _m\) in weighted norm, \(h^\epsilon \) and its derivatives in \(L^\infty \) norm to control the quantities \(Z_\tau ^{\alpha _1}h^\epsilon \) and \(Z_\tau ^{\alpha _1}\psi ^\epsilon \) in weighted norm. To derive these estimates, we shall apply the Lemma A.1, which has been introduced previously in Appendix A.
Lemma B.1
Let the stream function \(\psi ^\epsilon (t, x, y)\) satisfies \(\partial _y \psi ^\epsilon =h^\epsilon , \partial _x \psi ^\epsilon = -g^\epsilon , \psi ^\epsilon |_{y=0}=0\). There exists a constant \(\delta \in (0, 1)\), such that \(h^\epsilon (t, x, y)+1\ge \delta , \forall (t, x, y)\in [0, T]\times \Omega \). Then, for \(l \ge 1\) and \(|\alpha _1|=m\), we have the following estimates:
where the constant \(C_l\) depends only on l.
Proof
(i) By virtue of the definition \(h^\epsilon _m=Z_\tau ^{\alpha _1}h^\epsilon -\frac{\partial _y h^\epsilon }{h^\epsilon +1} Z_\tau ^{\alpha _1} \psi ^\epsilon \), it is easy to obtain \(h^\epsilon _m=(h^\epsilon +1)\partial _y(\frac{Z_\tau ^{\alpha _1} \psi ^\epsilon }{h^\epsilon +1})\). Integrating over [0, y] and applying the boundary condition \(\psi ^\epsilon |_{y=0}=0\), we have
and along with the Hardy inequality (A.1), yields directly
where we have used the fact \(h^\epsilon +1\ge \delta \) in the last inequality.
(ii) In view of the relation \(Z_\tau ^{\alpha _1}h^\epsilon =h^\epsilon _m+\frac{\partial _y h^\epsilon }{h^\epsilon +1} Z_\tau ^{\alpha _1} \psi ^\epsilon \), we get
which, together with estimate (B.6), yields directly
(iii) Differentiating the equality \({Z_\tau ^{\alpha _1} \psi ^\epsilon }=(h^\epsilon +1)\int _0^y \frac{h^\epsilon _m}{h^\epsilon +1}d\xi \) with respect to x variable, we find
which, implies that
and hence, we apply the Hardy inequality (A.1) and \(h^\epsilon +1 \ge \delta \) to get
(iv)Differentiating the equality \(Z_\tau ^{\alpha _1}h^\epsilon =h^\epsilon _m+\frac{\partial _y h^\epsilon }{h^\epsilon +1} Z_\tau ^{\alpha _1} \psi ^\epsilon \) with the y variable, it follows
which, together with the relation (B.5), yields
where \(\varphi (y)=\frac{y}{1+y}\) and \(\eta _h=\frac{\partial _y h^\epsilon }{h^\epsilon +1}\). The application of Hardy inequality (A.1) and \(h^\epsilon +1\ge \delta \) yields
Therefore, the estimates (B.6)–(B.9) imply the estimates (B.1)–(B.4). \(\square \)
Let us define
and hence we will establish the following almost equivalent relation.
Lemma B.2
Let \((\varrho ^\epsilon , u^\epsilon , v^\epsilon , h^\epsilon , g^\epsilon )\) be sufficiently smooth solution, defined on \([0, T^\epsilon ]\), to the regularized MHD boundary layer equations (3.2)–(3.3). There exists a constant \(\delta \in (0, 1)\), such that \(h^\epsilon (t, x, y)+1\ge \delta , \forall (t, x, y)\in [0, T]\times \Omega \). Then, for \(m \ge 4\) and \(l \ge 1\), it holds true
and
where \(\Theta _{m,l}(t)\) and \({\mathcal {N}}_{m,l}(t)\) are defined in (3.5) and (3.114) respectively.
Proof
By virtue of the definition \(\varrho ^\epsilon _m=Z_\tau ^{\alpha _1} \varrho ^\epsilon -\frac{\partial _y \varrho ^\epsilon }{h^\epsilon +1}Z_\tau ^{\alpha _1}\psi ^\epsilon \) and the estimate (B.1), we find
Similarly, we can obtain for \(|\alpha _1|=m\) that
The combination of the above two estimates yields directly
and hence, we have for \(m \ge 4, l \ge 1\)
Due to the definition of \(X_{m,l}(t)\) and \(Y_{m,l}(t)\) in (3.56) and (B.10) respectively, we get from (B.14) that
On the other hand, by virtue of the definition of \(\varrho ^\epsilon _m(t)\) and the estimate (B.1), we find
and hence, we also have
Then, the combination of the above estimates yields directly
where \(m \ge 4, l \ge 1\). According to the definition of \(X_{m,l}(t)\) and \(Y_{m,l}(t)\), we get from (B.16) that
Next, by virtue of the definition \(\varrho ^\epsilon _m(t)\) and estimate (3.89), we find
Similarly, by routine checking, we may conclude that
for \(m\ge 5, l \ge 1\), and hence it follows
where \(D_x^{m,l}(t)\) is defined in (3.115). By virtue of the definition \(\varrho ^\epsilon _m(t)\) and estimate (3.86), we get
Similarly, by routine checking, we may conclude that
and hence, it follows
where \(D_x^{m,l}(t)\) is defined in (3.115). Similarly, we can justify the estimates
and
Therefore, the combination of estimates (B.15), (B.17), (B.18), (B.19), (B.20) and (B.21) can establish the estimates (B.11) and (B.12). \(\square \)
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Gao, J., Huang, D. & Yao, Za. Boundary layer problems for the two-dimensional inhomogeneous incompressible magnetohydrodynamics equations. Calc. Var. 60, 67 (2021). https://doi.org/10.1007/s00526-021-01958-y
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DOI: https://doi.org/10.1007/s00526-021-01958-y