Global Well-Posedness for the Three-Dimensional Full Compressible Viscous Non-resistive MHD System

Li, Yang

doi:10.1007/s00021-022-00668-5

Cited by 5 publications

(2 citation statements)

References 42 publications

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“…Zhong [33] constructed the local strong solutions with possible initial vacuum but without any Cho-Choe-Kim type compatibility conditions in R 2 . The global existence of smooth solutions on the horizontally infinite flat layer Ω = R 2 × (0, 1) for the isentropic and non-isentropic cases was proved by Tan-Wang [24] and Li [16], respectively. Recently, Wu and Zhai [28] proved the global well-posedness of strong solutions on T 3 under the assumptions that the initial data is close enough to an equilibrium state, see [20] for an improvement of [28] for the compressible viscous non-isentropic MHD flows without magnetic diffusion.…”

Section: Recall Some Known Resultsmentioning

confidence: 99%

Stability and exponential decay for the compressible viscous non-resistive MHD system

Dong,

Wu,

Zhai

2024

Nonlinearity

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How to construct global solutions of the compressible viscous magnetohydrodynamic (MHD) equations without magnetic diffusion even with small initial data in R 3 or T 3 is still an extremely challenging open problem. The difficulty comes from the lack of magnetic diffusion and the fact that solutions to inviscid equations generally grow in time. Motivated by this open problem, the present paper focuses on a special case of this MHD system in T 3 when the magnetic field is vertical. We establish the global existence and uniqueness of smooth solutions to this system near a steady-state solution. In addition, the solution is shown to be stable and decay exponentially in time. The proof discovers and makes use of the smoothing and stabilizing effect of the steady magnetic field on the perturbations.

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Section: Recall Some Known Resultsmentioning

confidence: 99%

Stability and exponential decay for the compressible viscous non-resistive MHD system

Dong,

Wu,

Zhai

2024

Nonlinearity

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“…Meanwhile, Liu and Zhong [39] deduced global existence of strong solutions under a smallness condition on scaling invariant quantity independent of any norms of the initial data. We refer to [32,34] for more results on global solutions to multi-dimensional full compressible nonresistive MHD equations. There are some interesting results on other studies of (1.1), such as blow-up criterion of solutions [16,45,49], asymptotic limits of solutions [19][20][21]23], and so on.…”

Section: Introductionmentioning

confidence: 99%

Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity

Liu¹,

Zhong²

2022

Preprint

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The mathematical analysis on the behavior of the entropy for viscous, compressible, and heat conducting magnetohydrodynamic flows near the vacuum region is a challenging problem as the governing equation for entropy is highly degenerate and singular in the vacuum region. In particular, it is unknown whether the entropy remains its boundedness. In the present paper, we investigate the Cauchy problem to the three-dimensional (3D) compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity only. We show that the uniform boundedness of the entropy and the L 2 regularities of the velocity and temperature can be propagated provided that the initial density decays suitably slow at infinity. The main tools are based on singularly weighted energy estimates and De Giorgi type iteration techniques developed by Li and Xin (https://arxiv.org/abs/2111.14057) for the 3D full compressible Navier-Stokes system. Some new mathematical techniques and useful estimates are developed to deduce the lower and upper bounds on the entropy.

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Time-periodic solutions for 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

Sun

Zhang

Liu

et al. 2023

Z. Angew. Math. Phys.

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Global Well-Posedness for the Three-Dimensional Full Compressible Viscous Non-resistive MHD System

Cited by 5 publications

References 42 publications

Stability and exponential decay for the compressible viscous non-resistive MHD system

Stability and exponential decay for the compressible viscous non-resistive MHD system

Entropy-bounded solutions to the 3D compressible heat-conducting magnetohydrodynamic equations with vacuum at infinity

Time-periodic solutions for 2D MHD equations with horizontal dissipation and horizontal magnetic diffusion

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