Quadratic Interaction Estimate for Hyperbolic Conservation Laws: an Overview
- Authors: Modena S.1
-
Affiliations:
- S.I.S.S.A.
- Issue: Vol 233, No 6 (2018)
- Pages: 905-929
- Section: Article
- URL: https://journal-vniispk.ru/1072-3374/article/view/241719
- DOI: https://doi.org/10.1007/s10958-018-3972-0
- ID: 241719
Cite item
Abstract
In a joint work with S. Bianchini [8] (see also [6, 7]), we proved a quadratic interaction estimate for the system of conservation laws
where u : [0, ∞) × ℝ → ℝn, f : ℝn → ℝn is strictly hyperbolic, and Tot.Var.(u0) ≪ 1. For a wavefront solution in which only two wavefronts at a time interact, such an estimate can be written in the form
where αj and \( {\alpha}_j^{\prime } \) are the wavefronts interacting at the interaction time tj, σ(·) is the speed, |·| denotes the strength, and C(f) is a constant depending only on f (see [8, Theorem 1.1] or Theorem 3.1 in the present paper for a more general form).
The aim of this paper is to provide the reader with a proof for such a quadratic estimate in a simplified setting, in which:
• all the main ideas of the construction are presented;
• all the technicalities of the proof in the general setting [8] are avoided.
About the authors
S. Modena
S.I.S.S.A.
Author for correspondence.
Email: smodena@sissa.it
Italy, Trieste
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