Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion

Ren, Xiaoxia; Wu, Jiahong; Xiang, Zhaoyin; Zhang, Zhifei

doi:10.1016/j.jfa.2014.04.020

Cited by 216 publications

(113 citation statements)

References 9 publications

Supporting

Mentioning

107

Contrasting

Order By: Relevance

“…Jiu and Niu [7] established the local existence of solutions in 2D for initial data in H s for integer s 3, and more recently Ren et al [12] and Lin et al [10] have established the existence of global-in-time solutions for initial data sufficiently close to certain equilibrium solutions (again in 2D).…”

Section: Introductionmentioning

confidence: 99%

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Fefferman

McCormick

Robinson

et al. 2016

Arch Rational Mech Anal

103

View full text Add to dashboard Cite

This paper establishes the local-in-time existence and uniqueness of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations in R d , whereand any 0 < ε < 1. The proof relies on maximal regularity estimates for the Stokes equation. The obstruction to taking ε = 0 is explained by the failure of solutions of the heat equation with initial data u 0 ∈ H s−1 to satisfy u ∈ L 1 (0, T ; H s+1 ); we provide an explicit example of this phenomenon.

show abstract

Section: Introductionmentioning

confidence: 99%

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Fefferman

McCormick

Robinson

et al. 2016

Arch Rational Mech Anal

103

View full text Add to dashboard Cite

show abstract

“…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

“…That is to say, by Lemma , we may derived directly the basic energy estimates

‖ u ‖_{L^{2}}^{2} + ‖ b ‖_{L^{2}}^{2} + \int_{0}^{t} (‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla b ‖_{L^{2}}^{2}) d τ \leq ‖ u_{0} ‖_{L^{2}}^{2} + ‖ b_{0} ‖_{L^{2}}^{2},

‖ \nabla u ‖_{L^{2}}^{2} + ‖ \nabla b ‖_{L^{2}}^{2} \leq C (u_{0}, b_{0})

and

‖ {normalΛ}^{s} u ‖_{L^{2}}^{2} + ‖ {normalΛ}^{s} b ‖_{L^{2}}^{2} \leq C (u_{0}, b_{0} .

The second purpose of this study is to examine the large‐time decay rates for solutions of the MHD equations . It should be mentioned that there is a large literature on the decay rates of the full dissipative systems (see other works()). Due to the partial kinematic dissipation ( ∂ yy u 1 , ∂ xx u 2 ) and the partial magnetic diffusion ( ∂ yy b 1 , ∂ xx b 2 ), however, the classic Kato's methods (see Kato) that is mainly based on the L p − L q decay estimates of heat equations are not available any more.…”

Section: Introductionmentioning

confidence: 99%

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

Zhang

Dong

Jia

2019

Math Methods in App Sciences

View full text Add to dashboard Cite

This paper focuses on a system of the two‐dimensional (2D) magnetohydrodynamic (MHD) equations with the partial kinematic dissipation (∂yyu1,∂xxu2) and the partial magnetic diffusion (∂yyb1,∂xxb2). Based on the basic energy estimates only, we are able to show that this system always possesses a unique global smooth solution when the initial data are sufficiently smooth. Moreover, we obtain optimal large‐time decay rates of both solutions and their higher order derivatives by developing the classic Fourier splitting methods together with the auxiliary decay estimates of the first derivative of solutions and induction technique.

show abstract

Global existence and decay of smooth solution for the 2-D MHD equations without magnetic diffusion

Cited by 216 publications

References 9 publications

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

Local Existence for the Non-Resistive MHD Equations in Nearly Optimal Sobolev Spaces

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

Remarks on the global regularity and time decay of the 2D MHD equations with partial dissipation

Contact Info

Product

Resources

About