Oscillations non linéaires des systèmes hyperboliques: méthodes et résultats qualitatifs

Abstract

One studies the Cauchy problem for hyperbolic systems of first order conservation laws ${\partial }_{t}u+{\partial }_{x}f(u)=0$, with an oscillating sequence of initial data. The oscillations have an $O(1)$ amplitude; one may think to $u^{\epsilon }(x,0)=a(x;\frac{x}{\epsilon })$, where $a(x,.)$ is periodic.

It turns out that one part of the initial oscillations are killed because of genuine non-linearity. But the most of the physically relevant systems are endowed with at least one linearly degenerate characteristic field, which allows oscillations of a part of the field $u$.

Our goal is to describe these oscillations and their propagation. There are several steps:

• to determine the possible oscillations, $x$ and $t$ being fixed;

• to parametrize them via a set of new variables $(v(x,t),w(x,t;y))$; here $y$ plays the role of a “fast variable”;

• to find an integro-differential system describing the evolution of $(v,w)$;

• to find appropriate initial data.

Of particular interest is the computation of the group speeds, which are nothing but the characteristic values of the relaxed system. Formulas are given, especially for the full gas dynamics, which are all but trivial.

The description of oscillations has been asked by Ronald DiPerna in 1985, who introduced the similar concept of “measure-valued solutions”. An equivalent problem is to determine whether the set of solutions is closed or not in the weak-star topology of $L^{\infty }$. Thus this work is an attempt to understand well-posedness (or illposedness, the both occur depending on the system) in $L^{\infty }$, one of the two classes where a ferv existence results are known.

This study has also implications for numerical analysis. It suggests multi-scale methods which would give accurate values of speed when mild oscillations occur.

The paper is organized into three parts. Section II-V are devoted to a rigourous analysis with applications. Sections VI-VIII concerns speculative (but efficient) ideas; this heuristic procedure gives the first description of the oscillations in gas dynamics and elastic string. The end of the article is concerned with a conjecture I made in 1983 about the compensated compactness method.

Résumé

On considère les oscillations de grande amplitude des systèmes hyperboliques de lois de conservation. Les champs vraiment non linéaires en tuent une partie tandis que les champs linéairement dégénérés en laissent une autre part se propager selon des vitesses de groupe obéissant à des formules compliquées. Cet article rassemble des résultats rigoureux via la compacité par compensation et des calculs heuristiques.