The Inviscid Limit of Viscous Burgers through Shock Formation and Development
Abstract
We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order ν1/4 up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to t=0, we prove a pointwise error bound on a fixed interval 0≤ t ≤ ε̅, where ε̅>0 is independent of the viscosity ν. The bound consists of a term of order ν1/4 and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.
Related
- Relative Entropy and the Inviscid Limit for Planar Shocks of the Multi-D Burgers' Equation
- A Necessary and Sufficient Condition for the Existence of Global Smooth Solutions to the One-Dimensional Inviscid Burgers-Vlasov System
- NESTED DISCONTINUOUS ASYMPTOTIC PROFILES FOR THE VISCOUS BURGERS EQUATION WITH INFINITE MASS
- Inviscid Limit Problem for Radially Symmetric Stationary States of Viscous and Heat-Conductive Ideal Gas
- A Shifted One-Parameter Horadam Polynomial Framework for the Time-Fractional Supergeneralized Viscous Burgers Equation
BibTeX
@misc{wang2026inviscid,
title = {{The Inviscid Limit of Viscous Burgers through Shock Formation and Development}},
author = {Chao Wang and Yueyi Yang and Zhifei Zhang},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.16420},
eprint = {2610.16420},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.16420},
}
RIS
TY - UNPB
TI - The Inviscid Limit of Viscous Burgers through Shock Formation and Development
AU - Chao Wang
AU - Yueyi Yang
AU - Zhifei Zhang
PY - 2026
DA - 2026/10/12
PB - arXiv
DO - 10.48550/arxiv.2610.16420
UR - https://arxiv.org/abs/2610.16420
AB - We study the inviscid limit of the one-dimensional viscous Burgers equation from smooth initial data through nondegenerate shock formation and subsequent shock development. Chaturvedi and Graham \cite{CG} obtained sharp estimates of order ν^1/4 up to the shock-formation time. We extend their analysis beyond this time, comparing the viscous solution with the inviscid entropy solution evolving from the same initial data. With the shock-formation time normalized to t=0, we prove a pointwise error bound on a fixed interval 0≤ t ≤ ε̅, where ε̅>0 is independent of the viscosity ν. The bound consists of a term of order ν^1/4 and an exponentially localized contribution originating from the viscous shock layer. The proof is based on matching an outer expansion, an inner expansion near the formation point, and a viscous shock-profile expansion. Combined with the results of Chaturvedi and Graham, this yields a quantitative description of the inviscid limit from the initial time through the development of the young shock.
ER -