Liouville-type theorems for the stationary MHD equations in 2D

Wendong Wang; Yuzhao Wang

doi:10.1088/1361-6544/ab32a6

Liouville-type theorems for the stationary MHD equations in 2D

Wendong Wang, Yuzhao Wang

Mathematics

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Abstract

This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some L p norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.

Original language	English
Pages (from-to)	4483-4505
Number of pages	23
Journal	Nonlinearity
Volume	32
Issue number	11
DOIs	https://doi.org/10.1088/1361-6544/ab32a6
Publication status	Published - 10 Oct 2019

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This is an author-created, un-copyedited version of an article published in Nonlinearity. IOP Publishing Ltd is not responsible for any errors or omissions in this version of the manuscript or any version derived from it. The Version of Record is available online at https://doi.org/10.1088/1361-6544/ab32a6
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title = "Liouville-type theorems for the stationary MHD equations in 2D",

abstract = "This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some L p norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.",

author = "Wendong Wang and Yuzhao Wang",

year = "2019",

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doi = "10.1088/1361-6544/ab32a6",

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TY - JOUR

T1 - Liouville-type theorems for the stationary MHD equations in 2D

AU - Wang, Wendong

AU - Wang, Yuzhao

PY - 2019/10/10

Y1 - 2019/10/10

N2 - This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some L p norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.

AB - This paper is devoted to investigating Liouville type properties of the two-dimensional stationary incompressible Magnetohydrodynamics equations. More precisely, we show that there are no non-trivial solutions to MHD equations either the Dirichlet integral or some L p norm of the velocity-magnetic fields is suitably bounded, which generalize the well-known results for the 2D Navier–Stokes equations by Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404; Koch et al 2009 Acta Math. 203 83–105). Compared to the Navier–Stokes equations, there is no maximum principle for solutions to the MHD equation. To overcome this difficulty, we develop a different approach, which does not appeal to the special structure of the vorticity equation as Gilbarg and Weinberger (1978 Ann. Scuola Norm. Super. Pisa Cl. Sci. 5 381–404) did.

UR - https://arxiv.org/abs/1808.06234

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DO - 10.1088/1361-6544/ab32a6

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VL - 32

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EP - 4505

JO - Nonlinearity

JF - Nonlinearity

IS - 11

ER -

Liouville-type theorems for the stationary MHD equations in 2D

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