Global well-posedness and asymptotic behavior of solutions for the three-dimensional MHD equations with Hall and ion-slip effects

We study the asymptotic behavior of solutions with finite energy to the displacement problem of linear elastostatics in a three-dimensional exterior Lipschitz domain.

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On the blow-up criterion of strong solutions for the MHD equations with the Hall and ion-slip effects in $${\mathbb{R}^{3}}$$ R 3

Zeitschrift für angewandte Mathematik und Physik ◽

10.1007/s00033-016-0617-3 ◽

2016 ◽

Vol 67 (2) ◽

Cited By ~ 9

Author(s):

Sadek Gala ◽

Maria Alessandra Ragusa

Keyword(s):

Blow Up ◽

Strong Solutions ◽

Mhd Equations ◽

Slip Effects ◽

Ion Slip ◽

Blow Up Criterion

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Existence and approximation of attractors for nonlinear coupled lattice wave equations

Discrete and Continuous Dynamical Systems - Series B ◽

10.3934/dcdsb.2021272 ◽

2021 ◽

Vol 0 (0) ◽

pp. 0

Author(s):

Lianbing She ◽

Mirelson M. Freitas ◽

Mauricio S. Vinhote ◽

Renhai Wang

Keyword(s):

Asymptotic Behavior ◽

Global Attractor ◽

Wave Equations ◽

Upper Semicontinuity ◽

Asymptotic Behavior Of Solutions ◽

Well Posedness ◽

Asymptotic Compactness ◽

Finite Dimensional ◽

Lattice Wave ◽

Infinite Lattices

<p style='text-indent:20px;'>This paper is concerned with the asymptotic behavior of solutions to a class of nonlinear coupled discrete wave equations defined on the whole integer set. We first establish the well-posedness of the systems in <inline-formula><tex-math id="M1">\begin{document}$ E: = \ell^2\times\ell^2\times\ell^2\times\ell^2 $\end{document}</tex-math></inline-formula>. We then prove that the solution semigroup has a unique global attractor in <inline-formula><tex-math id="M2">\begin{document}$ E $\end{document}</tex-math></inline-formula>. We finally prove that this attractor can be approximated in terms of upper semicontinuity of <inline-formula><tex-math id="M3">\begin{document}$ E $\end{document}</tex-math></inline-formula> by a finite-dimensional global attractor of a <inline-formula><tex-math id="M4">\begin{document}$ 2(2n+1) $\end{document}</tex-math></inline-formula>-dimensional truncation system as <inline-formula><tex-math id="M5">\begin{document}$ n $\end{document}</tex-math></inline-formula> goes to infinity. The idea of uniform tail-estimates developed by Wang (Phys. D, 128 (1999) 41-52) is employed to prove the asymptotic compactness of the solution semigroups in order to overcome the lack of compactness in infinite lattices.</p>

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Global well-posedness for the three-dimensional incompressible viscous non-resistive MHD equations in an infinite slab

10.22541/au.163715715.52827183/v1 ◽

2021 ◽

Author(s):

Youyi Zhao

Keyword(s):

Initial Data ◽

Three Dimensional ◽

Mhd Equations ◽

Well Posedness ◽

Lagrangian Coordinates ◽

Mathematical Approach ◽

Compatibility Conditions ◽

Finite Height ◽

Infinite Slab ◽

Mhd System

In this paper, we investigate the global well-posedness of the system of incompressible viscous non-resistive MHD fluids in a three-dimensional horizontally infinite slab with finite height. We reformulate our analysis to Lagrangian coordinates, and then develop a new mathematical approach to establish global well-posedness of the MHD system, which requires no nonlinear compatibility conditions on the initial data.

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Well-Posedness and Asymptotic Behavior of a Nonclassical Nonautonomous Diffusion Equation with Delay

International Journal of Bifurcation and Chaos ◽

10.1142/s0218127415400210 ◽

2015 ◽

Vol 25 (14) ◽

pp. 1540021 ◽

Cited By ~ 4

Author(s):

Tomás Caraballo ◽

Antonio M. Márquez-Durán ◽

Felipe Rivero

Keyword(s):

Fixed Point ◽

Asymptotic Behavior ◽

Diffusion Equation ◽

Existence Of Solutions ◽

Local Solution ◽

Asymptotic Behavior Of Solutions ◽

Well Posedness ◽

Pullback Attractors ◽

Nonautonomous Dynamical Systems ◽

Equation With Delay

In this paper, a nonclassical nonautonomous diffusion equation with delay is analyzed. First, the well-posedness and the existence of a local solution is proved by using a fixed point theorem. Then, the existence of solutions defined globally in future is ensured. The asymptotic behavior of solutions is analyzed within the framework of pullback attractors as it has revealed a powerful theory to describe the dynamics of nonautonomous dynamical systems. One difficulty in the case of delays concerns the phase space that one needs to construct the evolution process. This yields the necessity of using a version of the Ascoli–Arzelà theorem to prove the compactness.

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