scholarly journals Global Strong Solution to the Density-Dependent 2-D Liquid Crystal Flows

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Yong Zhou ◽  
Jishan Fan ◽  
Gen Nakamura
Keyword(s):  
Strong Solution ◽  
Initial Density ◽  
Regularity Criterion ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Density Dependent ◽  
Global Strong Solution ◽  
Dependent Flow ◽  
Density Dependent Flow ◽  
Initial Boundary Value

The initial-boundary value problem for the density-dependent flow of nematic crystals is studied in a 2-D bounded smooth domain. For the initial density away from vacuum, the existence and uniqueness is proved for the global strong solution with the large initial velocityu0and small∇d0. We also give a regularity criterion∇d∈Lp(0,T;Lq(Ω))  (2/q)+(2/p)=1, 2<q≤∞of the problem with the Dirichlet boundary conditionu=0,d=d0on∂Ω.

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.

Global well-posedness and decay estimates of strong solutions to the nonhomogeneous Boussinesq equations for magnetohydrodynamics convection

2020 ◽  
pp. 1-25
Author(s):  
Xin Zhong
Keyword(s):  
Initial Data ◽  
Strong Solutions ◽  
Boussinesq Equations ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Well Posedness ◽  
Decay Estimates ◽  
Two Dimensional ◽  
Global Strong Solution ◽  
Initial Boundary Value

We deal with an initial boundary value problem of nonhomogeneous Boussinesq equations for magnetohydrodynamics convection in two-dimensional domains. We prove that there is a unique global strong solution. Moreover, we show that the temperature converges exponentially to zero in H1 as time goes to infinity. In particular, the initial data can be arbitrarily large and vacuum is allowed. Our analysis relies on energy method and a lemma of Desjardins (Arch. Rational Mech. Anal. 137:135–158, 1997).

scholarly journals Global Existence of the Cylindrically Symmetric Strong Solution to Compressible Navier-Stokes Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-8
Author(s):  
Jian Liu ◽  
Ruxu Lian
Keyword(s):  
Strong Solution ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Cylindrically Symmetric ◽  
Initial Boundary Value ◽  
Dependent Viscosity

This paper is concerned with the initial boundary value problem for the three-dimensional Navier-Stokes equations with density-dependent viscosity. The cylindrically symmetric strong solution is shown to exist globally in time and tend to the equilibrium state exponentially as time grows up.

scholarly journals An Initial Boundary Value Problem for the Zakharov Equation

2013 ◽  
Vol 2013 ◽  
pp. 1-10
Author(s):  
Quankang Yang ◽  
Charles Bu
Keyword(s):  
Boundary Value Problem ◽  
Existence And Uniqueness ◽  
Boundary Value ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Zakharov Equation ◽  
One Dimensional ◽  
Global Strong Solution ◽  
The One ◽  
Initial Boundary Value

This paper studies an inhomogeneous initial boundary value problem for the one-dimensional Zakharov equation. Existence and uniqueness of the global strong solution are proved by Galerkin’s method and integral estimates.

Convergence of Finite Fifference Method for Parabolic Problem

2003 ◽  
Vol 3 (1) ◽  
pp. 45-58 ◽  
Author(s):  
Dejan Bojović
Keyword(s):  
Convergence Rate ◽  
Boundary Value Problem ◽  
Heat Equation ◽  
Parabolic Problem ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Variable Coefficients ◽  
Rate Estimate ◽  
Convergence Rate Estimate ◽  
Initial Boundary Value

Abstract In this paper we consider the first initial boundary-value problem for the heat equation with variable coefficients in a domain (0; 1)x(0; 1)x(0; T]. We assume that the solution of the problem and the coefficients of the equation belong to the corresponding anisotropic Sobolev spaces. Convergence rate estimate which is consistent with the smoothness of the data is obtained.

ON THE SOLVABILITY OF A MIXED PROBLEM WITH DEGREE LAPLACE OPERATORS WITH NONLOCAL BOUNDARY CONDITIONS

2020 ◽  
pp. 85-104
Author(s):  
Shakirbai G. Kasimov ◽  
◽  
Mahkambek M. Babaev ◽  
◽  
Keyword(s):  
Boundary Conditions ◽  
Spectral Problem ◽  
Nonlocal Boundary ◽  
Initial Boundary ◽  
Initial Boundary Value Problem ◽  
Nonlocal Boundary Conditions ◽  
Laplace Operators ◽  
Initial Boundary Problem ◽  
Initial Boundary Value ◽  
Delayed Time

The paper studies a problem with initial functions and boundary conditions for partial differential partial equations of fractional order in partial derivatives with a delayed time argument, with degree Laplace operators with spatial variables and nonlocal boundary conditions in Sobolev classes. The solution of the initial boundary-value problem is constructed as the series’ sum in the eigenfunction system of the multidimensional spectral problem. The eigenvalues are found for the spectral problem and the corresponding system of eigenfunctions is constructed. It is shown that the system of eigenfunctions is complete and forms a Riesz basis in the Sobolev subspace. Based on the completeness of the eigenfunctions system the uniqueness theorem for solving the problem is proved. In the Sobolev subspaces the existence of a regular solution to the stated initial-boundary problem is proved.

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