In this paper, we are concerned with the asymptotic behavior of $L^{\infty}$ weak-entropy solutions to the compressible Euler equations with a vacuum and time-dependent damping $-\frac{m}{(1+t)^{\lambda}}$. As $\lambda \in (0,\frac17]$, we prove that the $L^{\infty}$ weak-entropy solution converges to the nonlinear diffusion wave of the generalized porous media equation (GPME) in $L^2(\mathbb R)$. As $\lambda \in (\frac17,1)$, we prove that the $L^{\infty}$ weak-entropy solution converges to an expansion around the nonlinear diffusion wave in $L^2(\mathbb R)$, which is the best asymptotic profile. The proof is based on intensive entropy analysis and an energy method.
Shifeng GENG
,
Feimin HUANG
,
Xiaochun WU
. L2-CONVERGENCE TO NONLINEAR DIFFUSION WAVES FOR EULER EQUATIONS WITH TIME-DEPENDENT DAMPING[J]. Acta mathematica scientia, Series B, 2022
, 42(6)
: 2505
-2522
.
DOI: 10.1007/s10473-022-0618-6
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