Local Existence of Solutions to the Non-Resistive 3D MHD Equations with Power-law Type

Abstract

We investigate the local-in-time existence results of solutions to non-resistive 3D MHD equations with power-law type nonlinear viscous fluid when magnetic resistance is vanished. Furthermore, if the existence of the global in time regular solution is guaranteed, the large time behavior result for solutions is obtained under the suitable assumption for the stress tensor part.

A Appendix

In this section, for the convenience of readers, we only give a proof for $H^3$-solution and the we refer [8] for the higher solution)

First of all, we estimate

$$\begin{aligned} \langle -\partial ^l\Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ),\partial ^l u\rangle _{L^2}. \end{aligned}$$

For this,

$$\begin{aligned}{} & {} \langle -\partial ^l\Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ),\partial ^l u\rangle _{L^2}=\langle -\Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ), u\rangle _{L^2}\\{} & {} +\langle -\nabla \Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ),\nabla u\rangle _{L^2}+\langle -\nabla ^2 \Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ),\nabla ^2 u\rangle _{L^2}\\{} & {} +\langle -\nabla ^s \Big (\nabla \cdot \big ( T[|Du(t)|^2] Du(t) \big )\Big ),\nabla ^s u\rangle _{L^2}:=I_0+I_1+I_2++I_3. \end{aligned}$$

1.1 A.1 A Priori Estimate

$\bullet$ ($I_1$-estimate)  Noting that

$$\begin{aligned} \begin{aligned} \partial _{x_i}(T[|Du|^2] Du): \partial _{x_i} Du&=\big [ \partial _{x_i}T[|Du|^2] Du + T[|Du|^2]\partial _{x_i} Du \big ]: \partial _{x_i} Du\\&= 2 T^{'}[|Du|^2](D u: \partial _{x_i}Du)( Du: \partial _{x_i} Du) + T[|Du|^2]|\partial _{x_i} Du|^2\\&= 2 T^{'}[|Du|^2]| Du: \partial _{x_i} Du|^2 + T[|Du|^2]|\partial _{x_i} Du|^2, \end{aligned} \end{aligned}$$

we have

$$\begin{aligned} I_1=\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2]|\partial _{x_i} Du|^2\,dx + \int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]| Du: \partial _{x_i} Du|^2\,dx:=good\ term. \end{aligned}$$

$\bullet$ ($I_2$-estimate)  We observe that

$$\begin{aligned}{} & {} \int \limits _{{{\mathbb {R}} }^3}\partial _{x_{j}}\partial _{x_{i}}\Big [T[|Du|^2] D u\Big ]:\partial _{x_{j}}\partial _{x_{i}}D u\,dx\\{} & {} \quad =\underbrace{\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2]|\partial _{x_{j}}\partial _{x_{i}}D u|^2}_{good \ term}, +\underbrace{\sum _{\sigma }\int \limits _{{{\mathbb {R}} }^3}\partial _{x_{\sigma (i)}}T[|Du|^2] (\partial _{x_{\sigma (j)}}D u: \partial _{x_{j}}\partial _{x_{i}}D u)\,dx}_{bad \ term}\\{} & {} \qquad +\int \limits _{{{\mathbb {R}} }^3}\partial _{x_{j}}\partial _{x_{i}}T[|Du|^2](D u: \partial _{x_{j}}\partial _{x_{i}}D u)\,dx=:I_{21}+I_{22}+I_{23}, \end{aligned}$$

where $\sigma :\{ i,j\}\rightarrow \{i,j \}$ is a permutation of $\{i,j \}$. We separately estimate terms $I_{22}$ and $I_{23}$. Using Hölder, Young’s and Gagliardo-Nirenberg inequalities, we have for $I_{22}$

$$\begin{aligned} |I_{22}|= & {} \left| \int \limits _{{{\mathbb {R}} }^3} 2T^{'}[|Du|^2](Du: \partial _{x_{\sigma (i)}}D u)(\partial _{x_{\sigma (j)}}D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,dx \right| \\\lesssim & {} \Vert T[|Du|^2]\Vert _{L^\infty }\Vert \nabla Du\Vert ^2_{L^4}\Vert \nabla ^2 Du\Vert _{L^2}. \end{aligned}$$

Using the condition (1.3), it follows

$$\begin{aligned} |I_{22}|\lesssim \Vert T[|Du|^2]\Vert _{L^\infty }\Vert Du\Vert _{L^\infty }\Vert \nabla ^2 Du\Vert ^2_{L^2}. \end{aligned}$$

For $I_{23}$, using Lemma 2.1, we compute

$$\begin{aligned} \begin{aligned} I_{23}&= \int \limits _{{{\mathbb {R}} }^3}2\big ( T^{'}[|Du|^2](Du: \partial _{x_{j}}\partial _{x_{i}}Du) + E_2 \big ) (D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,dx\\&=\underbrace{\int \limits _{{{\mathbb {R}} }^3}E_2(D u: \partial _{x_{j}}\partial _{x_{i}}Du)\,dx}_{=I_{231}:\ bad \ term} + 2\underbrace{\int \limits _{{{\mathbb {R}} }^3}T^{'}[|Du|^2] |Du: \partial _{x_{j}}\partial _{x_{i}}Du|^2\,dx}_{good \ term}. \end{aligned} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} \left| I_{231} \right|&\le C\Vert T[|Du|^2]\Vert _{L^\infty } \Vert Du\Vert _{L^\infty }\Vert \nabla Du\Vert ^2_{L^4}\Vert \nabla ^2 Du\Vert _{L^2}\\&\le C\Vert T[|Du|^2]\Vert _{L^\infty } \Vert Du\Vert _{L^\infty }^2\Vert \nabla ^2 Du\Vert _{L^2}^2, \end{aligned} \end{aligned}$$

And thus, it yields

$$\begin{aligned} I_2=\underbrace{\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2]|\partial _{x_{j}}\partial _{x_{i}}D u|^2 + \int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]|Du: \partial _{x_{j}}\partial _{x_{i}}Du|^2}_{good \ term}+\underbrace{I_{22}+I_{231}}_{bad \ term}, \end{aligned}$$

where

$$\begin{aligned} \underbrace{|I_{22}+I_{231}|}_{bad \ term}\le C\Vert T[|Du|^2]\Vert _{L^\infty }(\Vert Du\Vert _{L^\infty } + \Vert Du\Vert _{L^\infty }^2)\Vert \nabla ^2 Du\Vert ^2_{L^2}. \end{aligned}$$

$\bullet$ ($\Vert \nabla ^3 u \Vert _{L^2}$-estimate) Direct computations show that

$$\begin{aligned} \int \limits _{{{\mathbb {R}} }^3}\partial ^3\Big [T[|Du|^2] D u\Big ]: \partial ^3Du\,dx=\underbrace{\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2]|\partial ^3D u|^2\,dx}_{good \ term}\\ +\underbrace{\sum _{\sigma _3}\int \limits _{{{\mathbb {R}} }^3}\partial _{x_{\sigma _3(i)}}T[|Du|^2](\partial _{x_{\sigma _3(k)}}\partial _{x_{\sigma _3(j)}}Du:\partial ^3D u)\,dx}_{bad \ term}\\ +\underbrace{\sum _{\sigma _3}\int \limits _{{{\mathbb {R}} }^3}\partial _{x_{\sigma _3(j)}}\partial _{x_{\sigma _3(i)}}T[|Du|^2](\partial _{x_{\sigma _3(k)}}Du: \partial ^3Du)\,dx}_{bad \ term}\\ \quad +\int \limits _{{{\mathbb {R}} }^3}\partial ^3T[|Du|^2](D u:\partial ^3Du)\,dx=I_{31}+I_{32}+I_{33}+I_{34}, \end{aligned}$$

where $\sigma _3=\pi _3\circ {\tilde{\sigma }}_3$ such that ${\tilde{\sigma }}_3:\{i,j,k\}\rightarrow \{i,j,k\}$ is a permutation of $\{i,j,k \}$ and $\pi _3$ is a mapping from $\{i,j,k\}$ to $\{1,2,3\}$.

We separately estimate terms $I_{32}$, $I_{33}$ and $I_{34}$. We note first that

$$\begin{aligned} \begin{aligned} |I_{32}|&\le \int \limits _{{{\mathbb {R}} }^3}|2(T^{'}[|Du|^2]| |Du||\partial _{x_{\sigma _3(i)}}Du||\partial _{x_{\sigma _3(k)}}\partial _{x_{\sigma _3(j)}} D u| |\partial ^3 D u|\,dx\\&\le C\Vert T[|Du|^2]\Vert _{L^\infty }\Vert \nabla Du\Vert _{L^6}\Vert \nabla ^2 Du\Vert _{L^3}\Vert \nabla ^3 Du\Vert _{L^2}\\&\le C\Vert T[|Du|^2]\Vert _{L^\infty }\Vert \nabla ^2 Du\Vert _{L^2} \Vert Du\Vert _{L^\infty }^{\frac{1}{3}} \Vert \nabla ^3 Du\Vert ^{\frac{2}{3}}_{L^2}\Vert \nabla ^3Du\Vert _{L^2}, \end{aligned} \end{aligned}$$

and

$$\begin{aligned} |I_{33}|{} & {} =\left| \int \limits _{{{\mathbb {R}} }^3}( 2T^{'}[|Du|^2]Du:\partial _{x_{\sigma _3(j)}} \partial _{x_{\sigma _3(i)}} D u + E_2 )(\partial _{x_{\sigma _3(k)}}D u: \partial ^3Du)\,dx \right| \\{} & {} \le \int \limits _{{{\mathbb {R}} }^3}2(T^{'}[|Du|^2]|Du||\nabla ^2 D u| + T[|Du|^2]|\nabla D u|^2 )|\nabla D u| |\nabla ^3 Du|\,dx\\{} & {} \le C \Vert T^{'}[|Du|^2]Du\Vert _{L^\infty } \Vert \nabla ^2 Du\Vert _{L^3}\Vert \nabla Du\Vert _{L^6}\Vert \nabla ^3 Du\Vert _{L^2} +\Vert T[|Du|^2]\Vert _{L^\infty }\Vert \nabla Du\Vert ^3_{L^6}\Vert \nabla ^3 Du\Vert _{L^2} \end{aligned}$$

Finally, for $I_{34}$, using Lemma 2.1, we note that

$$\begin{aligned} \begin{aligned} I_{34}=&\int \limits _{{{\mathbb {R}} }^3}\big ( 2T^{'}[|Du|^2]Du:\partial ^3 D u) +E_3\big )(D u:\partial ^3D u)\,dx\\ =&\,2\underbrace{\int \limits _{{{\mathbb {R}} }^3} T^{'}[|Du|^2]|Du:\partial ^3 D u|^2\,dx}_{good \ term}+ \underbrace{\int \limits _{{{\mathbb {R}} }^3} E_3(D u:\partial ^3D u)\,dx}_{bad \ term}. \end{aligned} \end{aligned}$$

(A.1)

The second term in (A.1) is estimated as follows:

$$\begin{aligned}{} & {} \int \limits _{{{\mathbb {R}} }^3}E_3(D u:\partial ^3D u)\,dx \le \int \limits _{{{\mathbb {R}} }^3}|E_3||D u||\nabla ^3 D u|\,dx\\{} & {} \le C\int \limits _{{{\mathbb {R}} }^3} T[|Du|^2]\big (|\nabla Du|^3 +|\nabla ^2 Du||\nabla Du| \big )|D u||\nabla ^3 D u|\,dx\\{} & {} \le C\Vert T[|Du|^2]\Vert _{L^\infty }\Vert Du\Vert _{L^\infty } \big ( \Vert \nabla Du\Vert _{L^6}^3 + \Vert \nabla Du\Vert _{L^6}\Vert \nabla ^2 Du\Vert _{L^3} \big )\Vert \nabla ^3 D u\Vert _{L^2} \end{aligned}$$

Adding up the estimates above, we obtain

$$\begin{aligned} I_3=\underbrace{\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2]|\partial ^3D u|^2\,dx +\int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]|Du: \partial ^3Du|^2\,dx}_{good \ term}+\underbrace{I_{32}+I_{33}+I_{342}}_{bad \ term} \end{aligned}$$

(A.2)

where

$$\begin{aligned} |I_{32}+I_{33}+I_{342}|\le C(\Vert T[|Du|^2]\Vert ^2_{L^\infty }+ \Vert T[|Du|^2]\Vert ^6_{L^\infty } )(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}. \end{aligned}$$

We combine the estimates abovec to conclude

$$\begin{aligned}{} & {} \underbrace{\int \limits _{{{\mathbb {R}} }^3}T[|Du|^2] (|\nabla ^3 Du|^2+|\nabla ^2 Du|^2 + |\nabla Du|^2 + | Du|^2)\,dx}_{good \ term}\\{} & {} + \underbrace{\int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]| Du: \partial _{x_i} Du|^2\,dx}_{good \ term}+ \underbrace{\int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]|Du: \partial _{x_{j}}\partial _{x_{i}}Du|^2}_{good \ term}+\underbrace{\int \limits _{{{\mathbb {R}} }^3}2T^{'}[|Du|^2]|Du: \partial ^3Du|^2}_{good \ term}\\{} & {} +\underbrace{I_{22}+I_{231}+I_{32}+I_{33}+I_{342}}_{bad \ term} \end{aligned}$$

where

$$\begin{aligned} \begin{aligned} |bad \ term|\lesssim&|T[|Du|^2]\Vert _{L^\infty }(\Vert Du\Vert _{L^\infty } + \Vert Du\Vert _{L^\infty }^2)\Vert \nabla ^2 Du\Vert ^2_{L^2}\\&+ (\Vert T[|Du|^2]\Vert ^2_{L^\infty }+ \Vert T[|Du|^2]\Vert ^6_{L^\infty } )(\Vert Du\Vert ^2_{L^\infty }+\Vert Du\Vert ^4_{L^\infty })\Vert \nabla ^2 Du\Vert ^{6}_{L^2}. \end{aligned} \end{aligned}$$

Hence, set $X(t):= \Vert u(t)\Vert _{H^3({{\mathbb {R}} }^3)}$ and it then follows $\frac{d}{dt} X^2\le g_3(X)X^a,\quad a>2.$ Therefore, it immediately implies that there exists $T_3>0$ such that

$$\begin{aligned} \sup _{0\le t\le T_3}X(t) < \infty . \end{aligned}$$