scholarly journals Decay of the 3D Quasilinear Hyperbolic Equations with Nonlinear Damping

2017 ◽  
Vol 2017 ◽  
pp. 1-13 ◽  
Author(s):  
Hongjun Qiu ◽  
Yinghui Zhang
Keyword(s):  
Low Temperature ◽  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Initial Perturbation ◽  
Strong Solutions ◽  
Nonlinear Damping ◽  
Decay Rates ◽  
Quasilinear Hyperbolic Equations ◽  
Global Existence And Uniqueness ◽  
Very Low Temperature

We investigate the 3D quasilinear hyperbolic equations with nonlinear damping which describes the propagation of heat wave for rigid solids at very low temperature, below about 20 K. The global existence and uniqueness of strong solutions are obtained when the initial data is near its equilibrium in the sense of H3-norm. Furthermore, if, additionally, Lp-norm (1≤p<6/5) of the initial perturbation is finite, we also prove the optimal Lp-L2 decay rates for such a solution without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

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.

scholarly journals The periodic quasigeostrophic equations: existence and uniqueness of strong solutions

1982 ◽  
Vol 91 (3-4) ◽  
pp. 185-203 ◽  
Author(s):  
A. F. Bennett ◽  
P. E. Kloeden
Keyword(s):  
Existence And Uniqueness ◽  
Hyperbolic Equations ◽  
Three Dimensional ◽  
Strong Solutions ◽  
Coupled System ◽  
Classical Solutions ◽  
Schauder Estimates ◽  
Initial Vorticity ◽  
Relative Potential ◽  
Quasigeostrophic Equations

SynopsisThe periodic quasigeostrophic equations are a coupled system of a second order elliptic equation for a streamfunction and first order hyperbolic equations for the relative potential vorticity and surface potential temperatures, on a three-dimensional domain which is periodic in both horizontal spatial co-ordinates. Such equations are used in both numerical and theoretical studies in meteorology and oceanography. In this paper Schauder estimates and a Schauder fixed point theorem are used to prove the existence and uniqueness of strong, that is classical, solutions of the periodic quasigeostrophic equations for a finite interval of time, which is inversely proportional to the sum of the norms of the initial vorticity and surface temperatures.

Boundary effects and large-time behaviour for quasilinear equations with nonlinear damping

2015 ◽  
Vol 145 (5) ◽  
pp. 959-978 ◽  
Author(s):  
Shifeng Geng ◽  
Lina Zhang
Keyword(s):  
Asymptotic Behaviour ◽  
Large Time ◽  
Convergence Rates ◽  
Hyperbolic Equations ◽  
Nonlinear Damping ◽  
Boundary Effects ◽  
Time Behaviour ◽  
Quasilinear Equations ◽  
Quasilinear Hyperbolic Equations ◽  
Quarter Plane

This paper is concerned with the asymptotic behaviour of solutions to quasilinear hyperbolic equations with nonlinear damping on the quarter-plane (x, t) ∈ ℝ+ x ∈ ℝ+. We obtain the Lp (1 ≤ p ≤ +∞) convergence rates of the solution to the quasilinear hyperbolic equations without the additional technical assumptions for the nonlinear damping f(v) given by Li and Saxton.

scholarly journals EXISTENCE AND UNIQUENESS OF STRONG SOLUTIONS FOR A COMPRESSIBLE MULTIPHASE NAVIER–STOKES MISCIBLE FLUID-FLOW PROBLEM IN DIMENSION n = 1

2009 ◽  
Vol 19 (03) ◽  
pp. 443-476 ◽  
Author(s):  
C. MICHOSKI ◽  
A. VASSEUR
Keyword(s):  
Diffusion Coefficient ◽  
Fluid Flow ◽  
Existence And Uniqueness ◽  
Flow Problem ◽  
Stokes Equations ◽  
Strong Solutions ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Global Existence And Uniqueness ◽  
Miscible Fluid

We prove the global existence and uniqueness of strong solutions for a compressible multifluid described by the barotropic Navier–Stokes equations in dim = 1. The result holds when the diffusion coefficient depends on the pressure. It relies on a global control in time of the L2 norm of the space derivative of the density, via a new kind of entropy.

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