Kinetic formulation of the isentropic gas dynamics and \(p\)-systems. (English) Zbl 0799.35151

Lions, P. L.; Perthame, B.; Tadmor, E.

Commun. Math. Phys. 163, No. 2, 415-431 (1994).

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

Keywords:

advection velocity; invariant regions

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