scholarly journals Global existence and large-time behavior of solutions to the Cauchy problem of one-dimensional viscous radiative and reactive gas

2018 ◽  
Vol 265 (5) ◽  
pp. 2076-2120 ◽  
Author(s):  
Yongkai Liao ◽  
Huijiang Zhao
Keyword(s):  
Cauchy Problem ◽  
Global Existence ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
One Dimensional ◽  
The Cauchy Problem ◽  
Reactive Gas

scholarly journals SMOOTHING EFFECT AND LARGE TIME BEHAVIOR OF SOLUTIONS TO SCHRÖDINGER EQUATIONS WITH NONLINEARITY OF INTEGRAL TYPE

2004 ◽  
Vol 06 (04) ◽  
pp. 681-703 ◽  
Author(s):  
T. OZAWA ◽  
Y. YAMAZAKI
Keyword(s):  
Cauchy Problem ◽  
Large Time ◽  
Large Time Behavior ◽  
Time Behavior ◽  
Smoothing Effect ◽  
Range Effect ◽  
Integral Type ◽  
Phase Modification ◽  
The Cauchy Problem ◽  
Two Space Dimensions

We study the smoothing effect in space and asymptotic behavior in time of solutions to the Cauchy problem for the nonlinear Schrödinger equation with interaction described by the integral of the intensity with respect to one direction in two space dimensions. A detailed description is given on the phase modification of scattering solutions by taking into account the long range effect of the interaction.

Stability of the composite wave for compressible Navier–Stokes/Allen–Cahn system

2019 ◽  
Vol 30 (02) ◽  
pp. 343-385
Author(s):  
Ting Luo ◽  
Haiyan Yin ◽  
Changjiang Zhu
Keyword(s):  
Cauchy Problem ◽  
Large Time ◽  
Large Time Behavior ◽  
Navier Stokes ◽  
Time Behavior ◽  
Rarefaction Waves ◽  
Small Perturbations ◽  
The Cauchy Problem ◽  
Composite Wave ◽  
Contact Wave

This paper is devoted to the study of the nonlinear stability of the composite wave consisting of two rarefaction waves and a viscous contact wave for the Cauchy problem to a one-dimensional compressible non-isentropic Navier–Stokes/Allen–Cahn system which is a combination of the classical Navier–Stokes system with an Allen–Cahn phase field description. We first construct the composite wave through Euler equations under the assumption of [Formula: see text] for the large time behavior, and then prove that the composite wave is time asymptotically stable under small perturbations for the corresponding Cauchy problem of the non-isentropic Navier–Stokes/Allen–Cahn system. The proof is mainly based on a basic energy method.

POINTWISE ESTIMATES FOR THE WAVE EQUATION WITH DISSIPATION IN ODD SPATIAL DIMENSION

2008 ◽  
Vol 05 (02) ◽  
pp. 477-486 ◽  
Author(s):  
HONGMEI XU ◽  
WEIKE WANG
Keyword(s):  
Wave Equation ◽  
Large Time ◽  
Pointwise Estimate ◽  
Spatial Dimension ◽  
Large Time Behavior ◽  
Time Behavior ◽  
The Cauchy Problem ◽  
Explicit Analysis ◽  
Huygens Principle ◽  
The Green Function

We study the pointwise estimate of solution to the Cauchy problem for the wave equation with viscosity in odd spatial dimension. Through the explicit analysis of the Green function, we obtain the large time behavior of solution, and the solution exhibit the generalized Huygens principle.

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