On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

Zijin Li; Xinghong Pan

doi:10.1007/s00021-019-0457-y

Journal of Mathematical Fluid Mechanics

December 2019, 21:52 | Cite as

On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

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Zijin Li
Xinghong Pan

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First Online: 09 September 2019

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Abstract

In this paper, we study the stationary magnetohydrodynamics system in $\mathbb {R}^2\times \mathbb {T}$. We prove trivialness of D-solutions (the velocity field u and the magnetic field h) when they are swirl-free. Meanwhile, this Liouville type theorem also holds provided u is swirl-free and h is axially symmetric, or both u and h are axially symmetric. Our method is also valid for certain related boundary value problems in the slab $\mathbb {R}^2\times [-\pi ,\,\pi ]$.

Keywords

Incompressible Magnetohydrodynamics system Swirl-free Axially symmetric

Mathematics Subject Classification

35Q30 76N10

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Notes

Acknowledgements

The authors wish to thank Prof. Qi S. Zhang in UC Riverside for his constant encouragement on this topic. Z. Li is supported by the Startup Foundation for Introducing Talent of NUIST (No. 2019r033). X. Pan is supported by Natural Science Foundation of Jiangsu Province (No. SBK2018041027) and National Natural Science Foundation of China (No. 11801268).

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Conflict of interest

The authors declare that they have no conflict of interest.

Appendix: Some Details of the Boundary Conditions

This Appendix is devoted to some explanations of the boundary conditions in Corollary 1.2. As we mentioned in the Corollary 1.2, instead of the periodic condition for the velocity field, our method is also valid for a certain Navier slip boundary condition with a slight modification. That is

$$\begin{aligned} u\cdot n=0,\quad \left( \mathbb {D}u\cdot n\right) _{\tau }=0, \quad \forall x\in \partial \Omega . \end{aligned}$$

(6.1)

Here n is the outward unit normal to $\Omega $. $\mathbb {D}$ is the strain tensor

$$\begin{aligned} \mathbb {D}u=\frac{1}{2}\left( \nabla u+\nabla ^{T}u\right) . \end{aligned}$$

(6.2)

And for a vector filed v, $v_\tau $ stands for its tangential part: $v_\tau =v-(v\cdot n)n$. In our case, since $\Omega =\mathbb {R}^2\times [-\pi ,\,\pi ]$, we have $n=(0,0,\pm 1)$. Therefore, (6.1) is reduced to

$$\begin{aligned} u^z=0,\quad \partial _zu_1=0,\quad \partial _zu_2=0,\quad \forall z=-\pi \text { or }\pi . \end{aligned}$$

(6.3)

In the cylinder coordinate, (6.3) equals to

$$\begin{aligned} \left\{ \begin{array}{ll} u^z=0, &{}\\ \partial _zu^r\cos \theta -\partial _zu^{\theta } \sin \theta =0, &{} \qquad \quad \forall z=-\pi \text { or }\pi .\\ \partial _ru^z\sin \theta +\partial _zu^{\theta } \cos \theta =0,&{} \\ \end{array}\right. \end{aligned}$$

(6.4)

That is,

$$\begin{aligned} \partial _zu^r\Big |_{z=-\pi ,\,\pi }=\partial _zu^{\theta } \Big |_{z=-\pi ,\,\pi }=u^z\Big |_{z=-\pi ,\,\pi }\equiv 0. \end{aligned}$$

(6.5)

Meanwhile, for magnetic field h, our method is valid for the Dirichlet condition

$$\begin{aligned} h=0,\quad \forall z=-\pi \text { or }\pi , \end{aligned}$$

(6.6)

and the following two physical conditions, which are widely used in the research of the boundary value problem or the initial-boundary value problems to the MHD system. See [6, 17], etc.

$$\begin{aligned} \text {[PC1]}\left\{ \begin{array}{l} h\cdot n=0,\\ \nabla \times h\times n=0,\\ \end{array}\right. \quad \quad \text {[PC2]}\left\{ \begin{array}{l} h\cdot n=0,\\ \nabla \times h=0,\\ \end{array}\right. \quad \quad \forall x\in \partial \Omega . \end{aligned}$$

(6.7)

In our cases, similarly to (6.5) before, [PC1] and [PC2] are equivalent and both of them can be simplified to

$$\begin{aligned} \partial _zh^r\Big |_{z=-\pi ,\,\pi }=\partial _zh^{\theta } \Big |_{z=-\pi ,\,\pi }=h^z\Big |_{z=-\pi ,\,\pi }\equiv 0. \end{aligned}$$

(6.8)

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Zijin Li
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Xinghong Pan
- 2
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1.College of Mathematics and StatisticsNanjing University of Information Science and TechnologyNanjingChina
2.Department of MathematicsNanjing University of Aeronautics and AstronauticsNanjingChina

On the Vanishing of Some D-Solutions to the Stationary Magnetohydrodynamics System

Abstract

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Mathematics Subject Classification

Notes

Acknowledgements

Compliance with ethical standards

Conflict of interest

References

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