Local Asymptotic Patterns for Viscous Approximations of Conservation Laws
Fuente:
arXiv
Saved in:
| Main Authors: | , , |
|---|---|
| Format: | Preprint |
| Published: |
2026
|
| Subjects: | |
| Online Access: | |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866914523895562240 |
|---|---|
| author | Bressan, Alberto Caravenna, Laura Shen, Wen |
| author_facet | Bressan, Alberto Caravenna, Laura Shen, Wen |
| contents | Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$\ldots$ For some of these methods, general ${\bf L}^1$ convergence results are available. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of point where the hyperbolic solution has a singularity. Specifically: a point along a shock, or where two shocks interact, or where a new shock is formed.
Given a sequence of $ε$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ε\to 0$ a well defined limit is obtained. This corresponds to a specific ``eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results this direction are here given, in the case of vanishing viscosity. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_00189 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Local Asymptotic Patterns for Viscous Approximations of Conservation Laws Bressan, Alberto Caravenna, Laura Shen, Wen Analysis of PDEs 35L65, 35L67 Solutions to hyperbolic conservation laws can be approximated in many different ways: by vanishing viscosity, relaxations, discrete or semi-discrete numerical schemes, approximation with a nonlocal flux, etc$\ldots$ For some of these methods, general ${\bf L}^1$ convergence results are available. Aim of this paper is to understand the local behavior of these approximations, in a neighborhood of point where the hyperbolic solution has a singularity. Specifically: a point along a shock, or where two shocks interact, or where a new shock is formed. Given a sequence of $ε$-approximate solutions, a general expectation is that, by a suitable local rescaling of coordinates, as $ε\to 0$ a well defined limit is obtained. This corresponds to a specific ``eternal solution" (globally defined both in space and in time) to the approximating equation. Precise results this direction are here given, in the case of vanishing viscosity. |
| title | Local Asymptotic Patterns for Viscous Approximations of Conservation Laws |
| topic | Analysis of PDEs 35L65, 35L67 |
| url | https://arxiv.org/abs/2605.00189 |