scholarly journals Time decay estimates of solutions to the mixed problem for heat equations in a half space

2013 ◽  
Vol 37 (2) ◽  
pp. 271-305
Author(s):  
Akio Baba ◽  
Kunihiko Kajitani
Keyword(s):  
Half Space ◽  
Mixed Problem ◽  
Heat Equations ◽  
Decay Estimates ◽  
Time Decay ◽  
Estimates Of Solutions

scholarly journals Global Existence and Decay Estimates of Energy of Solutions for a New Class of p -Laplacian Heat Equations with Logarithmic Nonlinearity

2021 ◽  
Vol 2021 ◽  
pp. 1-11
Author(s):  
Salah Mahmoud Boulaaras ◽  
Abdelbaki Choucha ◽  
Abderrahmane Zara ◽  
Mohamed Abdalla ◽  
Bahri-Belkacem Cheri
Keyword(s):  
Global Existence ◽  
Sufficient Conditions ◽  
Research Paper ◽  
Efficient Technique ◽  
Heat Equations ◽  
Decay Estimates ◽  
Source Terms ◽  
Estimates Of Solutions ◽  
New Class ◽  
Analytical Studies

The present research paper is related to the analytical studies of p -Laplacian heat equations with respect to logarithmic nonlinearity in the source terms, where by using an efficient technique and according to some sufficient conditions, we get the global existence and decay estimates of solutions.

Asymptotic stability of rarefaction wave for the compressible Navier‐Stokes‐Korteweg equations in the half space

2021 ◽  
pp. 1-24
Author(s):  
Yeping Li ◽  
Jing Tang ◽  
Shengqi Yu
Keyword(s):  
Asymptotic Stability ◽  
Rarefaction Wave ◽  
Half Space ◽  
Compressible Fluid ◽  
Boundary Data ◽  
Initial Perturbation ◽  
Navier Stokes ◽  
Asymptotic State ◽  
Decay Estimates ◽  
Time Decay

In this study, we are concerned with the asymptotic stability towards a rarefaction wave of the solution to an outflow problem for the Navier-Stokes Korteweg equations of a compressible fluid in the half space. We assume that the space-asymptotic states and the boundary data satisfy some conditions so that the time-asymptotic state of this solution is a rarefaction wave. Then we show that the rarefaction wave is non-linearly stable, as time goes to infinity, provided that the strength of the wave is weak and the initial perturbation is small. The proof is mainly based on $L^{2}$ -energy method and some time-decay estimates in $L^{p}$ -norm for the smoothed rarefaction wave.

scholarly journals Time-Decay Estimates for Linearized Two-Phase Navier–Stokes Equations with Surface Tension and Gravity

Mathematics ◽  
2021 ◽  
Vol 9 (7) ◽  
pp. 761
Author(s):  
Hirokazu Saito
Keyword(s):  
Surface Tension ◽  
Stokes Equations ◽  
Navier Stokes ◽  
Dimensional Euclidean Space ◽  
Decay Estimates ◽  
Time Decay ◽  
Two Phase ◽  
Navier Stokes Equations ◽  
Linearized Equations ◽  
Estimates Of Solutions

The aim of this paper is to show time-decay estimates of solutions to linearized two-phase Navier-Stokes equations with surface tension and gravity. The original two-phase Navier-Stokes equations describe the two-phase incompressible viscous flow with a sharp interface that is close to the hyperplane xN=0 in the N-dimensional Euclidean space, N≥2. It is well-known that the Rayleigh–Taylor instability occurs when the upper fluid is heavier than the lower one, while this paper assumes that the lower fluid is heavier than the upper one and proves time-decay estimates of Lp-Lq type for the linearized equations. Our approach is based on solution formulas for a resolvent problem associated with the linearized equations.

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