Fractional BV solutions for $2 \times 2$ systems of conservation laws with a genuinely nonlinear field and a linearly degenerate field - Archive ouverte HAL
Pré-Publication, Document De Travail Année : 2022

Fractional BV solutions for $2 \times 2$ systems of conservation laws with a genuinely nonlinear field and a linearly degenerate field

Résumé

The class of $2 \times 2$ nonlinear hyperbolic systems with one genuinely nonlinear field and one linearly degenerate field are considered. Existence of global weak solutions for small initial data in fractional $BV$ spaces $BV^s$ is proved. The exponent $s$ is related to the usual fractional Sobolev derivative. Riemann invariants $w$ and $z$ corresponding respectively to the genuinely nonlinear component and to the linearly degenerate component play different key roles in this work. We obtain the existence of a global weak solution provided that the initial data written in Riemann coordinates $(w_0 , z_0)$ are small in $BV^s \times L^\infty , 1/3 ≤ s < 1$. The restriction on the exponent $s$ is due to a fundamental result of P.D. Lax, the variation of the Riemann invariant $z$ on the Lax shock curve depends in a cubic way of the variation of the other Riemann invariant $w$.
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Dates et versions

hal-02532444 , version 1 (05-04-2020)
hal-02532444 , version 2 (29-03-2022)

Identifiants

Citer

Boris Haspot, Stéphane Junca. Fractional BV solutions for $2 \times 2$ systems of conservation laws with a genuinely nonlinear field and a linearly degenerate field. 2022. ⟨hal-02532444v2⟩
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