Global large solutions to planar magnetohydrodynamics equations with temperature-dependent coefficients

2019 ◽  
Vol 16 (03) ◽  
pp. 443-493 ◽  
Author(s):  
Yachun Li ◽  
Zhaoyang Shang
Keyword(s):  
Heat Conductivity ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Mhd Equations ◽  
Heat Conducting ◽  
Magnetic Diffusion ◽  
Viscosity Coefficients ◽  
Large Solutions ◽  
Mhd System ◽  
Initial Boundary Value

We consider the planar compressible magnetohydrodynamics (MHD) system for a viscous and heat-conducting ideal polytropic gas, when the viscosity, magnetic diffusion and heat conductivity depend on the specific volume [Formula: see text] and the temperature [Formula: see text]. For technical reasons, the viscosity coefficients, magnetic diffusion and heat conductivity are assumed to be proportional to [Formula: see text] where [Formula: see text] is a non-degenerate and smooth function satisfying some natural conditions. We prove the existence and uniqueness of the global-in-time classical solution to the initial-boundary value problem when general large initial data are prescribed and the exponent [Formula: see text] is sufficiently small. A similar result is also established for planar Hall-MHD equations.

scholarly journals Global large solutions to the planar magnetohydrodynamics equations with constant heat conductivity

2019 ◽  
Vol 2019 (1) ◽  
Author(s):  
Wei Li ◽  
Zhaoyang Shang
Keyword(s):  
Global Existence ◽  
Heat Conductivity ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Constant Heat ◽  
Magnetic Diffusion ◽  
New Methods ◽  
Large Solutions ◽  
Initial Boundary Value

Abstract This paper is concerned with global existence of large solutions to the initial-boundary value problem of the planar magnetohydrodynamic compressible flow. Under the assumptions that viscosity and heat conductivity coefficients are constants, magnetic diffusion is a function of the specific volume, we obtain the global existence of strong solutions. Some new methods are developed to deal with the complex interaction between the hydrodynamic and magnetodynamics effects.

scholarly journals On vanishing limits of the shear viscosity and Hall coefficients for the planar compressible Hall-MHD system

2021 ◽  
Author(s):  
Xia Ye ◽  
Zejia Wang
Keyword(s):  
Boundary Layer ◽  
Shear Viscosity ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Mhd Equations ◽  
Layer Function ◽  
Mhd System ◽  
Initial Boundary Value ◽  
Hall Mhd

This paper deals with an initial-boundary value problem of the planar compressible Hall-magnetohydrodynamic (for short, Hall-MHD) equations. For the fixed shear viscosity and Hall coefficients, it is shown that the strong solutions of Hall-MHD equations and corresponding MHD equations are global. As both the shear viscosity and the Hall coefficients tend to zero, the convergence rate for the solutions from Hall-MHD equations to MHD equations is given. The thickness of boundary layer is discussed by spatially weighted estimation and the characteristic of boundary layer is described by constructing a boundary layer function.

scholarly journals On the Asymptotic Behavior of the Conjugate Problem Describing a Creeping Axisymmetric Thermocapillary Motion

2020 ◽  
pp. 26-36
Author(s):  
Victor K. Andreev ◽  
Evgeniy P. Magdenko
Keyword(s):  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Conjugate Problem ◽  
Cylinder Wall ◽  
Heat Conducting ◽  
Solid Cylinder ◽  
Viscous Heat ◽  
Initial Boundary Value ◽  
Heat Conducting Fluids ◽  
Creeping Motion

In this paper the conditions for the law of temperature behavior on a solid cylinder wall describes, under which the solution of a linear conjugate inverse initial-boundary value problem describing a two-layer axisymmetric creeping motion of viscous heat-conducting fluids tends to zero exponentially with increases of time

scholarly journals Filtration of Liquid in a Non-isothermal Viscous Porous Medium

2020 ◽  
pp. 763-773
Author(s):  
Alexander A. Papin ◽  
Margarita A. Tokareva ◽  
Rudolf A. Virts
Keyword(s):  
Porous Medium ◽  
Boundary Value Problem ◽  
Fluid Motion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Heat Conducting ◽  
System Of Equations ◽  
One Dimensional ◽  
Unsteady Fluid ◽  
Initial Boundary Value

The solvability of the initial-boundary value problem is proved for the system of equations of one-dimensional unsteady fluid motion in a heat-conducting viscous porous medium

scholarly journals Global strong solution to the 2D inhomogeneous incompressible magnetohydrodynamic fluids with density-dependent viscosity and vacuum

2021 ◽  
Vol 2021 (1) ◽  
Author(s):  
Menglong Su
Keyword(s):  
Strong Solution ◽  
Blow Up ◽  
Strong Solutions ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Density Dependent ◽  
Global Strong Solution ◽  
Mhd System ◽  
Initial Boundary Value ◽  
Dependent Viscosity

AbstractIn this paper, we investigate an initial boundary value problem for two-dimensional inhomogeneous incompressible MHD system with density-dependent viscosity. First, we establish a blow-up criterion for strong solutions with vacuum. Precisely, the strong solution exists globally if $\|\nabla \mu (\rho )\|_{L^{\infty }(0, T; L^{p})}$ ∥ ∇ μ ( ρ ) ∥ L ∞ ( 0 , T ; L p ) is bounded. Second, we prove the strong solution exists globally (in time) only if $\|\nabla \mu (\rho _{0})\|_{L^{p}}$ ∥ ∇ μ ( ρ 0 ) ∥ L p is suitably small, even the presence of vacuum is permitted.

ASYMPTOTIC BEHAVIOR OF THE MOTION OF A VISCOUS HEAT-CONDUCTING ONE-DIMENSIONAL GAS WITH RADIATION: THE PURE SCATTERING CASE

2013 ◽  
Vol 11 (01) ◽  
pp. 1350003 ◽  
Author(s):  
BERNARD DUCOMET ◽  
ŠÁRKA NEČASOVÁ
Keyword(s):  
Large Time ◽  
Transport Coefficients ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Large Time Behavior ◽  
Heat Conducting ◽  
Time Behavior ◽  
Static State ◽  
Viscous Heat ◽  
Initial Boundary Value

We study the large-time behavior of the solution of an initial-boundary value problem for the equations of 1D motions of a compressible viscous heat-conducting gas coupled with radiation through a radiative transfer equation. Assuming only scattering processes between matter and photons (neglecting absorption and emission) and suitable hypotheses on the transport coefficients, we prove that the unique weak solution of the problem converges toward the static state.

scholarly journals Global Well-posedness and Asymptotics of Full Compressible Non-resistive MHD System with Large External Potential Forces

2021 ◽  
Author(s):  
Wanrong Yang ◽  
Xiaokui Zhao
Keyword(s):  
Stationary Solution ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Unique Existence ◽  
Background Magnetic Field ◽  
External Forces ◽  
Mhd System ◽  
Initial Boundary Value

We consider the global well-posedness and asymptotic behavior of compressible viscous, heat-conductive, and non-resistive magnetohydrodynamics (MHD) fluid in a field of external forces over three-dimensional periodic thin domain $\Omega=\mathbb{T}^2\times(0,\delta)$. The unique existence of the stationary solution is shown under the adhesion and the adiabatic boundary conditions. Then, it is shown that a solution to the initial boundary value problem with the same boundary and periodic conditions uniquely exists globally in time and converges to the stationary solution as time tends to infinity. Moreover, if the external forces are small or disappeared in an appropriate Sobolev space, then $\delta$ can be a general constant. Our proof relies on the two-tier energy method for the reformulated system in Lagrangian coordinates and the background magnetic field which is perpendicular to the flat layer. Compared to the work of Tan and Wang (SIAM J. Math. Anal. 50:1432–1470, 2018), we not only overcome the difficulties caused by temperature, but also consider the big external forces.

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