scholarly journals Asymptotic Stability of the Rarefaction Wave for the Non-Viscous and Heat-Conductive Ideal Gas in Half Space

2019 ◽  
Vol 39 (4) ◽  
pp. 1195-1212
Author(s):  
Meichen Hou
Keyword(s):  
Asymptotic Stability ◽  
Rarefaction Wave ◽  
Half Space ◽  
Ideal Gas

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.

Stability of planar rarefaction wave to the 3D bipolar Vlasov–Poisson–Boltzmann system

2019 ◽  
Vol 30 (01) ◽  
pp. 23-104 ◽  
Author(s):  
Shu Wang ◽  
Teng Wang
Keyword(s):  
Wave Propagation ◽  
Boltzmann Equation ◽  
Asymptotic Stability ◽  
Rarefaction Wave ◽  
Stability Result ◽  
Three Dimensions ◽  
Wave Patterns ◽  
Shock Profiles ◽  
Poisson Boltzmann ◽  
Math Phys

We investigate the time-asymptotic stability of planar rarefaction wave for the 3D bipolar Vlasov–Poisson Boltzmann (VPB) system, based on the micro–macro decompositions introduced in [T. P. Liu and S. H. Yu, Boltzmann equation: Micro–macro decompositions and positivity of shock profiles, Comm. Math. Phys. 246 (2004) 133–179; Energy method for the Boltzmann equation, Physica D 188 (2004) 178–192] and our new observations on the underlying wave structures of the equation to overcome the difficulties due to the wave propagation along the transverse directions and its interactions with the planar rarefaction wave. Note that this is the first stability result of basic wave patterns for bipolar VPB system in three dimensions.

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