scholarly journals Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Miao Ouyang
Keyword(s):  
Convergence Rates ◽  
Strong Solutions ◽  
Large Time Behavior ◽  
Decay Rates ◽  
Mhd Equations ◽  
Time Behavior ◽  
Potential Force ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Compressible Mhd Equations

In this paper, the large-time behavior of solutions to the Cauchy problem for the 3D compressible MHD equations is considered with the effect of external force. We construct the global unique solution with the small initial data near the stationary profile. The optimal Lp-L2(1≤p≤2) time decay rates of the solution to the system are built in multifrequency decompositions.

scholarly journals On global well-posedness and decay of 3D Ericksen-Leslie system

2021 ◽  
Vol 6 (11) ◽  
pp. 12660-12679
Author(s):  
Xiufang Zhao ◽  
◽  
Ning Duan ◽  
Keyword(s):  
Strong Solutions ◽  
Decay Rates ◽  
Well Posedness ◽  
Small Initial Data ◽  
Optimal Decay Rates ◽  
Optimal Decay ◽  
Liquid Crystal System ◽  
The Cauchy Problem ◽  
Derivatives Of ◽  
Leslie System

<abstract><p>In this paper, the small initial data global well-posedness and time decay estimates of strong solutions to the Cauchy problem of 3D incompressible liquid crystal system with general Leslie stress tensor are studied. First, assuming that $ \|u_0\|_{\dot{H}^{\frac12+\varepsilon}}+\|d_0-d_*\|_{\dot{H}^{\frac32+\varepsilon}} $ ($ \varepsilon &gt; 0) $ is sufficiently small, we obtain the global well-posedness of strong solutions. Moreover, the $ L^p $–$ L^2 $ ($ \frac32\leq p\leq2 $) type optimal decay rates of the higher-order spatial derivatives of solutions are also obtained. The $ \dot{H}^{-s} $ ($ 0\leq s &lt; \frac12 $) negative Sobolev norms are shown to be preserved along time evolution and enhance the decay rates.</p></abstract>

scholarly journals Global well-posedness of the full compressible Hall-MHD equations

2021 ◽  
Vol 10 (1) ◽  
pp. 1235-1254
Author(s):  
Qiang Tao ◽  
Canze Zhu
Keyword(s):  
Global Solution ◽  
Large Time ◽  
Initial Temperature ◽  
Large Time Behavior ◽  
Initial Energy ◽  
Mhd Equations ◽  
Well Posedness ◽  
Time Behavior ◽  
Magnetohydrodynamic Flows ◽  
Hall Mhd

Abstract This paper deals with a Cauchy problem of the full compressible Hall-magnetohydrodynamic flows. We establish the existence and uniqueness of global solution, provided that the initial energy is suitably small but the initial temperature allows large oscillations. In addition, the large time behavior of the global solution is obtained.

scholarly journals Global existence of strong solutions for incompressible hydrodynamic flow of liquid crystals with vacuum

Filomat ◽  
2013 ◽  
Vol 27 (7) ◽  
pp. 1247-1257 ◽  
Author(s):  
Shijin Ding ◽  
Jinrui Huang ◽  
Fengguang Xia
Keyword(s):  
Cauchy Problem ◽  
Liquid Crystals ◽  
Global Existence ◽  
Existence And Uniqueness ◽  
Strong Solutions ◽  
Hydrodynamic Flow ◽  
Three Dimensions ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Global Existence And Uniqueness

We consider the Cauchy problem for incompressible hydrodynamic flow of nematic liquid crystals in three dimensions. We prove the global existence and uniqueness of the strong solutions with nonnegative p0 and small initial data.

POINTWISE ESTIMATES FOR THE WAVE EQUATION WITH DISSIPATION IN ODD SPATIAL DIMENSION

2008 ◽  
Vol 05 (02) ◽  
pp. 477-486 ◽  
Author(s):  
HONGMEI XU ◽  
WEIKE WANG
Keyword(s):  
Wave Equation ◽  
Large Time ◽  
Pointwise Estimate ◽  
Spatial Dimension ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Cauchy Problem ◽  
Explicit Analysis ◽  
Huygens Principle ◽  
The Green Function

We study the pointwise estimate of solution to the Cauchy problem for the wave equation with viscosity in odd spatial dimension. Through the explicit analysis of the Green function, we obtain the large time behavior of solution, and the solution exhibit the generalized Huygens principle.

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