Pointwise decay estimates for multidimensional Navier-Stokes diffusion waves

1997 ◽  
Vol 48 (4) ◽  
pp. 597-614 ◽  
Author(s):  
D. Hoff ◽  
K. Zumbrun
Keyword(s):  
Navier Stokes ◽  
Decay Estimates ◽  
Diffusion Waves ◽  
Pointwise Decay

scholarly journals DECAY PROPERTIES OF SOLUTIONS TO THE LINEARIZED COMPRESSIBLE NAVIER–STOKES EQUATION AROUND TIME-PERIODIC PARALLEL FLOW

2012 ◽  
Vol 22 (07) ◽  
pp. 1250007 ◽  
Author(s):  
JAN BŘEZINA ◽  
YOSHIYUKI KAGEI
Keyword(s):  
Periodic Function ◽  
Stokes Equation ◽  
Parallel Flow ◽  
Navier Stokes ◽  
Decay Estimates ◽  
Convective Term ◽  
Navier Stokes Equation ◽  
Decay Properties ◽  
Linearized Problem ◽  
Time Periodic

Decay estimates on solutions to the linearized compressible Navier–Stokes equation around time-periodic parallel flow are established. It is shown that if the Reynolds and Mach numbers are sufficiently small, solutions of the linearized problem decay in L2 norm as an (n - 1)-dimensional heat kernel. Furthermore, it is proved that the asymptotic leading part of solutions is given by solutions of an (n - 1)-dimensional linear heat equation with a convective term multiplied by time-periodic function.

scholarly journals Large Time Behavior for Weak Solutions of the 3D Globally Modified Navier-Stokes Equations

2014 ◽  
Vol 2014 ◽  
pp. 1-5
Author(s):  
Junbai Ren
Keyword(s):  
Weak Solutions ◽  
Large Time ◽  
Stokes Equations ◽  
Three Dimensional ◽  
Large Time Behavior ◽  
Navier Stokes ◽  
Heat Semigroup ◽  
Time Behavior ◽  
Decay Estimates ◽  
Navier Stokes Equations

This paper is concerned with the large time behavior of the weak solutions for three-dimensional globally modified Navier-Stokes equations. With the aid of energy methods and auxiliary decay estimates together withLp-Lqestimates of heat semigroup, we derive the optimal upper and lower decay estimates of the weak solutions for the globally modified Navier-Stokes equations asC1(1+t)-3/4≤uL2≤C2(1+t)-3/4,  t>1.The decay rate is optimal since it coincides with that of heat equation.

scholarly journals Lp ASYMPTOTIC BEHAVIOR OF PERTURBED VISCOUS SHOCK PROFILES

2005 ◽  
Vol 02 (03) ◽  
pp. 595-644 ◽  
Author(s):  
MOHAMMADREZA RAOOFI
Keyword(s):  
Asymptotic Behavior ◽  
Stokes Equations ◽  
One Dimension ◽  
Navier Stokes ◽  
Navier Stokes Equations ◽  
Diffusion Waves ◽  
Shock Wave Solution ◽  
Shock Profiles ◽  
Shock Position ◽  
Shock Profile

We investigate the Lp asymptotic behavior (1 ≤ p ≤ ∞) of a perturbation of a Lax or overcompressive type shock wave solution to a system of conservation law in one dimension. The system of the equations can be strictly parabolic, or partially parabolic (real viscosity case, e.g. compressible Navier–Stokes equations or equations of Magnetohydrodynamics). We use known pointwise Green function bounds for the linearized equation around the shock to show that the perturbation of such a solution can be decomposed into a part corresponding to shift in shock position or shape, a part which is the sum of diffusion waves, i.e. the solutions to a viscous Burger's equation, conserving the initial mass and convecting away from the shock profile in outgoing modes, and another part which is more rapidly decaying in any Lp norm.

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.

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