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Kinetic formulation of the isentropic gas dynamics and \(p\)-systems. (English) Zbl 0799.35151
Summary: We consider the \(2\times 2\) hyperbolic system of isentropic gas dynamics, in both Eulerian or Lagrangian variables (also called the \(p\)-system). We show that they can be reformulated as a kinetic equation, using an additional kinetic variable. Such a formulation was first obtained by the authors in the case of multidimensional scalar conservation laws. A new phenomenon occurs here, namely that the advection velocity is now a combination of the macroscopic and kinetic velocities. Various applications are given: we recover the invariant regions, deduce new \(L^ \infty\) estimates using moments lemma and prove \(L^ \infty- w*\) stability for \(\gamma\geq 3\).

MSC:
35L65 Hyperbolic conservation laws
76N15 Gas dynamics (general theory)
35B45 A priori estimates in context of PDEs
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