Weak solutions for two-dimensional magnetohydrodynamics equations with partial dissipation and magnetic diffusion

Shaoliang Yuan; Shaoliang Yuan

doi:10.3934/math.2026363

AIMS Mathematics

2026, Volume 11, Issue 3: 8832-8841. doi: 10.3934/math.2026363

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Research article

Weak solutions for two-dimensional magnetohydrodynamics equations with partial dissipation and magnetic diffusion

Shaoliang Yuan ^,

School of Big Data and Artificial Intelligence, Fujian Polytechnic Normal University, Fuzhou 350300, Fujian, China

Received: 23 December 2025 Revised: 14 February 2026 Accepted: 10 March 2026 Published: 31 March 2026
MSC : 35Q35, 35D30

In this article, we study the existence of weak solutions to two-dimensional incompressible magnetohydrodynamics (MHD) equations. For the case with only magnetic diffusion, the global existence of solutions in $ L^\infty(0, \infty; H^1) $ was established by Kozono, whereas the existence of $ L^\infty(0, \infty; H^1) $ solutions for the case with only dissipation is totally unknown. The purpose here is to consider the mixed case, that is, a system with partial dissipation and partial magnetic diffusion. We show that there exists a unique local solution with initial data in $ H^1 $ space.
- MHD equations,
- local existence,
- uniqueness,
- partial dissipation and magnetic diffusion
Citation: Shaoliang Yuan. Weak solutions for two-dimensional magnetohydrodynamics equations with partial dissipation and magnetic diffusion[J]. AIMS Mathematics, 2026, 11(3): 8832-8841. doi: 10.3934/math.2026363

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Abstract

In this article, we study the existence of weak solutions to two-dimensional incompressible magnetohydrodynamics (MHD) equations. For the case with only magnetic diffusion, the global existence of solutions in $ L^\infty(0, \infty; H^1) $ was established by Kozono, whereas the existence of $ L^\infty(0, \infty; H^1) $ solutions for the case with only dissipation is totally unknown. The purpose here is to consider the mixed case, that is, a system with partial dissipation and partial magnetic diffusion. We show that there exists a unique local solution with initial data in $ H^1 $ space.

References

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