Global well-posedness to the 2D Cauchy problem of nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

2021 ◽  
Vol 60 (2) ◽  
Author(s):  
Xin Zhong
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Heat Conducting ◽  
Well Posedness ◽  
Magnetohydrodynamic Equations ◽  
Large Initial Data ◽  
Nonhomogeneous Heat

Global well-posedness of the Cauchy problem for the equations of a one-dimensional viscous heat-conducting gas with Lebesgue initial data

2004 ◽  
Vol 134 (5) ◽  
pp. 939-960 ◽  
Author(s):  
Song Jiang ◽  
Alexander Zlotnik
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Heat Conducting ◽  
Well Posedness ◽  
Global Weak Solutions ◽  
One Dimensional ◽  
Lipschitz Continuous ◽  
Viscous Heat ◽  
The Cauchy Problem ◽  
The One

We study the Cauchy problem for the one-dimensional equations of a viscous heat-conducting gas in the Lagrangian mass coordinates with the initial data in the Lebesgue spaces. We prove the existence, the uniqueness and the Lipschitz continuous dependence on the initial data of global weak solutions.

scholarly journals Global well-posedness to the Cauchy problem of 2D inhomogeneous incompressible magnetic Bénard equations with large initial data and vacuum

2021 ◽  
Vol 6 (11) ◽  
pp. 12085-12103
Author(s):  
Zhongying Liu ◽  
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Initial Density ◽  
Well Posedness ◽  
Far Field ◽  
Large Initial Data ◽  
Global Strong Solution ◽  
Vacuum States ◽  
The Cauchy Problem ◽  
Field Decay

<abstract><p>In this paper, we are concerned with the Cauchy problem of inhomogeneous incompressible magnetic Bénard equations with vacuum as far-field density in $ \Bbb R^2 $. We prove that if the initial density and magnetic field decay not too slowly at infinity, the system admits a unique global strong solution. Note that the initial data can be arbitrarily large and the initial density can contain vacuum states and even has compact support. Moreover, we extend the result of [16, 17] to the global one.</p></abstract>

Global existence and large time behavior of strong solutions to the nonhomogeneous heat conducting magnetohydrodynamic equations with large initial data and vacuum

2021 ◽  
pp. 1-27
Author(s):  
Xin Zhong
Keyword(s):  
Initial Data ◽  
Large Time ◽  
Initial Density ◽  
Decay Rates ◽  
Energy Estimates ◽  
Heat Conducting ◽  
Time Behavior ◽  
Two Dimensional ◽  
Magnetohydrodynamic Equations ◽  
Nonhomogeneous Heat

We investigate an initial boundary value problem of two-dimensional nonhomogeneous heat conducting magnetohydrodynamic equations. We prove that there exists a unique global strong solution. Moreover, we also obtain the large time decay rates of the solution. Note that the initial data can be arbitrarily large and the initial density allows vacuum states. Our method relies upon the delicate energy estimates and Desjardins’ interpolation inequality (B. Desjardins, Regularity results for two-dimensional flows of multiphase viscous fluids, Arch. Rational Mech. Anal. 137(2) (1997) 135–158).

BLOW-UP OF THE SOLUTION FOR SEMILINEAR DAMPED WAVE EQUATION WITH TIME-DEPENDENT DAMPING

2012 ◽  
Vol 14 (05) ◽  
pp. 1250034
Author(s):  
JIAYUN LIN ◽  
JIAN ZHAI
Keyword(s):  
Cauchy Problem ◽  
Wave Equation ◽  
Initial Data ◽  
Upper Bound ◽  
Blow Up ◽  
Time Dependent ◽  
Damped Wave Equation ◽  
Large Initial Data ◽  
Power Type ◽  
The Cauchy Problem

We consider the Cauchy problem for the damped wave equation with time-dependent damping and a power-type nonlinearity |u|ρ. For some large initial data, we will show that the solution to the damped wave equation will blow up within a finite time. Moreover, we can show the upper bound of the life-span of the solution.

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