Global cauchy problem of 2D generalized MHD equations

Fan, Jishan; Malaikah, Hunida; Monaquel, Satha; Nakamura, Gen; Zhou, Yong

doi:10.1007/s00605-014-0652-0

Cited by 114 publications

(55 citation statements)

References 8 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…There have been significant recent developments on the MHD equations with partial or fractional dissipation. One can refer to previous studies (see, eg,) for details and the references therein.…”

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global regularity for the 2D magneto‐micropolar system with partial and fractional dissipation

Liu

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper focuses on the 2D incompressible magneto‐micropolar sysytem with the kinematic dissipation given by the fractional operator (−Δ)α, the magnetic diffusion by the fractional operator (−Δ)β and the spin dissipation by the fractional operator (−Δ)γ. α,β, and γ are nonnegative constants. We proved that this system with any α+β=2,1 ≤ α ≤ 2,γ=0, and α+γ ≥ 1,β=1 always possesses a unique global smooth solution false(boldu,.2emboldb,.2emnormalwfalse)∈Hsfalse(R2false)false(s>2false) if the initial data is sufficiently smooth. In addition, we also obtained the global regularity results for several partial dissipation cases.

show abstract

Section: Introduction and Main Resultsmentioning

confidence: 99%

Global regularity for the 2D magneto‐micropolar system with partial and fractional dissipation

Liu

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

show abstract

“…Because the result in was the case ν , η > 0, κ = 0 and α = β = 1, g 1 ≡ g 2 ≡1, our result indicates that the powers of α may be shifted to γ and therefore represents a complete generalization in this sense. The proof of Theorem was inspired by the recent developments on the two‐dimensional generalized MHD system with β = 1, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν , η > 0, β = 1, α > 0, g 1 ≡ g 2 ≡1 admits the global regularity result. Therefore, Theorem also represents an extension of the result in to the magnetic Bénard problem, interestingly without requiring −Δ θ but only Λ 2 γ , γ = 1− α , α > 0. The proof of Theorem was inspired by the recent work on the two‐dimensional generalized MHD system with ν = 0, and in particular in which the authors showed that the two‐dimensional generalized MHD system with ν = 0, η > 0, β > 1, g 2 ≡1 allows the global regularity result to hold.…”

Section: Introduction and Statement Of Main Resultsmentioning

confidence: 99%

Global regularity of generalized magnetic Benard problem

Yamazaki

2016

Math Methods in App Sciences

View full text Add to dashboard Cite

We study the magnetic Bénard problem in two‐dimensional space with generalized dissipative and diffusive terms, namely, fractional Laplacians and logarithmic supercriticality. Firstly, we show that when the diffusive term for the magnetic field is a full Laplacian, the solution initiated from data sufficiently smooth preserves its regularity as long as the power of the fractional Laplacians for the dissipative term of the velocity field and the diffusive term of the temperature field adds up to 1. Secondly, we show that with zero dissipation for the velocity field and a full Laplacian for the diffusive term of the temperature field, the global regularity result also holds when the diffusive term for the magnetic field consists of the fractional Laplacian with its power strictly bigger than 1. Finally, we show that with no diffusion from the magnetic and the temperature fields, the global regularity result remains valid as long as the dissipation term for the velocity field has its strength at least at the logarithmically supercritical level. These results represent various extensions of previous work on both Boussinesq and magnetohydrodynamics systems. Copyright © 2016 John Wiley & Sons, Ltd.

show abstract

“…Fan and his collaborators [5] proved the global existence of smooth solutions with α > 0, β = 1. Global regularity for the case α = 0, β > 1 was established by Jiu and Zhao [8] which improves the result in [1].…”

Section: Introductionmentioning

confidence: 98%

“…This work is partially motivated by the recent progress on the 2D incompressible generalized MHD system with fractional diffusion, we refer to [1,4,5,8,14,16,17] and the references therein. In [14], Tran, Yu and Zhai obtained the global regularity of 2D GMHD equations for the following three cases:…”

Section: Introductionmentioning

confidence: 99%