SINGULAR CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR SOLUTIONS TO THE CAUCHY PROBLEM FOR THE COMPRESSIBLE EULER EQUATIONS WITH DAMPING

2002 ◽  
Vol 12 (09) ◽  
pp. 1317-1336 ◽  
Author(s):  
M. DI FRANCESCO ◽  
P. MARCATI
Keyword(s):  
Energy Estimates ◽  
Damping Term ◽  
Gas Dynamics Equations ◽  
Small Perturbations ◽  
One Dimensional ◽  
Diffusion Waves ◽  
The Cauchy Problem ◽  
Parabolic Scaling ◽  
The One ◽  
Nonlinear Diffusion Waves

We investigate the singular limit for the solutions to the compressible gas dynamics equations with damping term, after a parabolic scaling, in the one-dimensional isentropic case. In particular, we study the convergence in Sobolev norms towards diffusive prophiles, in case of well-prepared initial data and small perturbations of them. The results are obtained by means of symmetrization and energy estimates.

A Numerical Model for the Radial Flow Through Porous and Deformable Shells

2004 ◽  
Vol 4 (1) ◽  
pp. 34-47 ◽  
Author(s):  
Francisco J. Gaspar ◽  
Francisco J. Lisbona ◽  
Petr N. Vabishchevich
Keyword(s):  
Finite Difference ◽  
Finite Difference Methods ◽  
Energy Estimates ◽  
One Dimensional ◽  
Consolidation Model ◽  
Finite Difference Discretization ◽  
Flow Through ◽  
The One ◽  
Theoretical Results ◽  
Difference Discretization

AbstractEnergy estimates and convergence analysis of finite difference methods for Biot's consolidation model are presented for several types of radial ow. The model is written by a system of partial differential equations which depend on an integer parameter (n = 0; 1; 2) corresponding to the one-dimensional ow through a deformable slab and the radial ow through an elastic cylindrical or spherical shell respectively. The finite difference discretization is performed on staggered grids using separated points for the approximation of pressure and displacements. Numerical results are given to illustrate the obtained theoretical results.

GLOBAL INHOMOGENEOUS SCHRÖDINGER FLOW

2000 ◽  
Vol 11 (08) ◽  
pp. 1079-1114 ◽  
Author(s):  
HONG-YU WANG ◽  
YOU-DE WANG
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Spin System ◽  
Holomorphic Sectional Curvature ◽  
Heisenberg Spin ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Schrodinger Flow ◽  
The One ◽  
Complete Kähler Manifold

In this paper, we consider the global existence of one-dimensional nonautonomous inhomogeneous Schrödinger flow. By exploiting geometric symmetries, we prove that, given a smooth initial map, the Cauchy problem of the one-dimensional nonautonomous inhomogeneous Schrödinger flow from S1 into a complete Kähler manifold with constant holomorphic sectional curvature admits a unique global smooth solution. As a corollary, we establish the global existence for the Cauchy problem of the inhomogeneous Heisenberg spin system.

On the Cauchy problem for one dimension generalized Boussinesq equation

2015 ◽  
Vol 26 (03) ◽  
pp. 1550023 ◽  
Author(s):  
Yinxia Wang
Keyword(s):  
Cauchy Problem ◽  
Boussinesq Equation ◽  
Nonlinear Approximation ◽  
Global Solutions ◽  
One Dimension ◽  
Diffusion Waves ◽  
Self Similar Solution ◽  
The Cauchy Problem ◽  
Nonlinear Diffusion Waves ◽  
Generalized Boussinesq Equation

In this paper, we study the Cauchy problem for one dimension generalized damped Boussinesq equation. First, global existence and decay estimate of solutions to this problem are established. Second, according to the detail analysis for solution operator the generalized damped Boussinesq equation, the nonlinear approximation to global solutions is established. Finally, we prove that the global solution u to our problem is asymptotic to the superposition of nonlinear diffusion waves expressed in terms of the self-similar solution of the viscous Burgers equation as time tends to infinity.

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