DYNAMICS AND UNIVERSALITY OF AN ISOTHERMAL COMBUSTION PROBLEM IN 2D

2006 ◽  
Vol 18 (03) ◽  
pp. 285-310 ◽  
Author(s):  
Y. W. QI
Keyword(s):  
Lewis Number ◽  
Large Time Behavior ◽  
Nonlinear Pdes ◽  
Regularity Of Solutions ◽  
Group Method ◽  
Time Behavior ◽  
The Cauchy Problem ◽  
Spatio Temporal ◽  
Non Gaussian ◽  
Isothermal Combustion

In this paper, the Cauchy problem of the system [Formula: see text] is studied, where x ∈ R2, m ≥ 1 and d > 0 is the Lewis number. This system models isothermal combustion (see [7]), and auto-catalytic chemical reaction. We show the global existence and regularity of solutions with non-negative initial values having mild decay as |x| → ∞. More importantly, we establish the exact spatio-temporal profiles for such solutions. In particular, we prove that for m = 1, the exact large time behavior of solutions is characterized by a universal, non-Gaussian spatio-temporal profile, with anomalous exponents, due to the fact that quadratic nonlinearity is critical in 2D. Our approach is a combination of iteration using Renormalization Group method, which has been developed into a very powerful tool in the study of nonlinear PDEs largely by the pioneering works of Bricmont, Kupiainen and Lin [6], Bricmont, Kupiainen and Xin, [7], (see also [9]) and key estimates using the PDE 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.

scholarly journals SHARP ASYMPTOTIC BEHAVIOR OF SOLUTIONS TO NONLINEAR SCHRÖDINGER EQUATIONS WITH REPULSIVE INTERACTIONS

2005 ◽  
Vol 07 (02) ◽  
pp. 167-176 ◽  
Author(s):  
NAOYASU KITA ◽  
TOHRU OZAWA
Keyword(s):  
Large Time Behavior ◽  
Nonlinear Schrödinger Equations ◽  
Schrodinger Equations ◽  
Asymptotic Behavior Of Solutions ◽  
Time Behavior ◽  
Schrödinger Equations ◽  
Nonlinear Schrödinger ◽  
Nonlinear Schrodinger Equations ◽  
Repulsive Interactions ◽  
The Cauchy Problem

A detailed description is given on the large time behavior of scattering solutions to the Cauchy problem for nonlinear Schrödinger equations with repulsive interactions in the short-range case without smallness condition on the data.

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.

scholarly journals Optimal Convergence Rates for the Strong Solutions to the Compressible MHD Equations with Potential Force

2019 ◽  
Vol 2019 ◽  
pp. 1-12
Author(s):  
Miao Ouyang
Keyword(s):  
Convergence Rates ◽  
Strong Solutions ◽  
Large Time Behavior ◽  
Decay Rates ◽  
Mhd Equations ◽  
Time Behavior ◽  
Potential Force ◽  
Small Initial Data ◽  
The Cauchy Problem ◽  
Compressible Mhd Equations

In this paper, the large-time behavior of solutions to the Cauchy problem for the 3D compressible MHD equations is considered with the effect of external force. We construct the global unique solution with the small initial data near the stationary profile. The optimal Lp-L2(1≤p≤2) time decay rates of the solution to the system are built in multifrequency decompositions.

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.

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