Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations
Abstract
We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, |Ri(x,t)|≤ C ENlog(2+t)Ψi(x,t), supx∈ℝΨi(x,t)≤ C(1+t)−1. Here EN measures the size of the initial data and Ψi is the cone-resolved weight; both are defined precisely in the main theorem below. Thus |Ri(t)|L∞≤ C EN(1+t)−1log(2+t). The key new idea is to apply a familywise Cole--Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.
Related
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BibTeX
@misc{li2026limiting,
title = {{Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations}},
author = {Rui Li and Xu Song and Guanghui Wang and Lingda Xu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.11707},
eprint = {2609.11707},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.11707},
}
RIS
TY - UNPB
TI - Limiting Pointwise Decay for the compressible isentropic Navier-Stokes equations
AU - Rui Li
AU - Xu Song
AU - Guanghui Wang
AU - Lingda Xu
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.11707
UR - https://arxiv.org/abs/2609.11707
AB - We study the long-time pointwise behavior of small localized perturbations of a constant state for the one-dimensional compressible isentropic Navier-Stokes equations. After subtracting the two Burgers diffusion waves, the convergent sum of all higher-order diffusion waves, and the cross-family viscous corrections, we prove a cone-preserving pointwise estimate for the exact physical remainder and, in particular, |R_i(x,t)|≤ C E_Nlog(2+t)Ψ_i(x,t), sup_x∈ℝΨ_i(x,t)≤ C(1+t)^−1. Here E_N measures the size of the initial data and Ψ_i is the cone-resolved weight; both are defined precisely in the main theorem below. Thus |R_i(t)|_L^∞≤ C E_N(1+t)^−1log(2+t). The key new idea is to apply a familywise Cole--Hopf transformation to the spatial antiderivative of the remainder, which exactly eliminates the critical same-family first-order feedback. We further construct an approximate Green function adapted to the two characteristic families and combine it with Gaussian-mode extraction and a Kawashima-type energy argument. This yields a cone-preserving estimate at the limiting decay rate, up to a logarithmic loss.
ER -