Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations
Abstract
While the asymptotic stability of rarefaction waves for the compressible Navier--Stokes equations has been well established since the pioneering works of Matsumura--Nishihara \cite{MatsumuraNishihara1986} and Liu--Xin \cite{LiuXin1988}, the classical L2 energy method yields only qualitative convergence in L∞(R) as t→∞ without any explicit decay rate. In this paper, we establish for the first time a standalone L4 energy estimate for perturbations of a weak rarefaction wave in the one-dimensional isentropic compressible Navier--Stokes system, thereby obtaining the first explicit algebraic time-decay rates for this model. Specifically, for small initial perturbations in H2(R) and small wave strength δ, we prove that $(v-V,u-U)$ decays in L4(R) at the rate |(v−V,u−U)(t)|L4≤ C(|(v0−V(0),u0−U(0))|H2+√δ)(1+t)−1/8, where (V,U) denotes the smooth rarefaction wave, while the relative Riemann invariants z± of the perturbed system decay in L∞(R) at the rate (1+t)−1/6. The proof relies on a new local quartic energy estimate for z± and z±−Q, where Q=μ(v)vx−μ(V)Vx, as well as a nonlocal half-space wave-interaction functional for corrected characteristic modes and a weighted higher-order derivative energy estimate coupled at the scale of the one-dimensional Nash inequality.
Related
- Large time behavior of global solutions to one-dimensional non-isentropic compressible Navier-Stokes-Poisson equations with large initial data
- Global weak solutions with vacuum for isentropic compressible Navier-Stokes and Navier-Stokes-Poisson equations in the unit disk under Dirichlet boundary conditions
- Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations
- Incompressible limit for non-isentropic compressible Euler-Navier-Stokes two-phase flow model in R^3
- Global regularity and sharp decay rates for the one-dimensional hypo-viscous compressible Navier-Stokes equations in critical Besov spaces
BibTeX
@misc{he2026timedecay,
title = {{Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations}},
author = {Mutong He and Feimin Huang and Lingjun Liu and Lingda Xu},
howpublished = {arXiv},
year = {2026},
month = nov,
doi = {10.48550/arxiv.2611.01956},
eprint = {2611.01956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2611.01956},
}
RIS
TY - UNPB
TI - Time-Decay Rates toward Rarefaction Waves for the One-Dimensional Isentropic Compressible Navier--Stokes Equations
AU - Mutong He
AU - Feimin Huang
AU - Lingjun Liu
AU - Lingda Xu
PY - 2026
DA - 2026/11/01
PB - arXiv
DO - 10.48550/arxiv.2611.01956
UR - https://arxiv.org/abs/2611.01956
AB - While the asymptotic stability of rarefaction waves for the compressible Navier--Stokes equations has been well established since the pioneering works of Matsumura--Nishihara \cite{MatsumuraNishihara1986} and Liu--Xin \cite{LiuXin1988}, the classical L^2 energy method yields only qualitative convergence in L^∞(R) as t→∞ without any explicit decay rate. In this paper, we establish for the first time a standalone L^4 energy estimate for perturbations of a weak rarefaction wave in the one-dimensional isentropic compressible Navier--Stokes system, thereby obtaining the first explicit algebraic time-decay rates for this model. Specifically, for small initial perturbations in H^2(R) and small wave strength δ, we prove that $(v-V,u-U)$ decays in L^4(R) at the rate |(v−V,u−U)(t)|_L^4≤ C(|(v_0−V(0),u_0−U(0))|_H^2+√δ)(1+t)^−1/8, where (V,U) denotes the smooth rarefaction wave, while the relative Riemann invariants z_± of the perturbed system decay in L^∞(R) at the rate (1+t)^−1/6. The proof relies on a new local quartic energy estimate for z_± and z_±−Q, where Q=μ(v)v_x−μ(V)V_x, as well as a nonlocal half-space wave-interaction functional for corrected characteristic modes and a weighted higher-order derivative energy estimate coupled at the scale of the one-dimensional Nash inequality.
ER -