Kinetic Formulation of the Isentropic Gas Dynamics and p-Systems

We consider the 2 x 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°° estimates using moments lemma and prove L°° — w* stability for 7 > 3. Introduction We consider the equations of isentropic gas dynamics. In the Eulerian coordinates these equations form a 2 x 2 hyperbolic system of nonlinear conservation laws dtρ + dxρu = 0, dt(ρu) + dx(ρu 2 + p(ρ)) = 0, t > 0, x e R, p(ρ) = κp, 7 > 1, κ= — , 47 where the unknowns ρ(t,x) and q := ρu(t,x) are respectively the density and the momentum of the gas. They are given at time t — 0 by the initial data ρ°(x) and q° = ρ°u°(x). And of course, ^ > 0 o n l + x l . We will also consider another 2 x 2 system, dtw -|dxp(υ) = 0, t > 0, x e R, endowed with the pressure law p(^) = ^ 7 , 7 > 0 , ft = ( 7 ~ 1 } . (2b) 47 416 P. L. Lions, B. Perthame, E. Tadmor The system (2a)-(2b) governs the isentropic gas dynamics written in Lagrangian coordinates. In general Eqs. (2a)-(2b) will be referred to as the p-system (see Lax [7], Smoller [14]...). They are also complemented by initial conditions υ°(x), w°(x). In this paper we construct and analyze a kinetic formulation of these systems. By this we mean a formulation which is based on an appropriate transport equation such that • it involves an additional variable, ξ, the so-called kinetic velocity variable; • its ^-moments recover the original equations and their augmenting entropy conditions. This approach was already used by the authors for scalar conservation laws in [9, 10]. The scalar conservation law and all of its associated entropy inequalities were formulated in terms of a single BGK-type kinetic transport equation. Here, we provide kinetic formulations for the isentropic system (1) and the p-system (2). These kinetic formulations are not of BGK-type \ but instead they involve a limit collision term. This enables us to represent the corresponding 2 x 2 systems and their associated family of so-called weak entropies. (Strong entropies could be handled by a different kinetic formulation.) It is evident in both cases of a scalar equation or the present 2 x 2 systems, that the kinetic formulation is in fact a way to represent a "rich" enough family of entropy inequalities. This seems to give the limits of the method, but also explains its power and why many properties of the system can be proved or recovered so easily using this formulation: these properties are in fact obtained with the use of certain particular entropies which here can be handled easily in the context of our kinetic formulation. As a first illustration of this advantage, we recover immediately the invariant regions. This provides, as it is well-known, see for instance Dafermos [2], Serre [13], 7-1 a maximum principle on the Riemann invariants u ± ρ 2 for weak solutions. A second illustration is the derivation of a new estimate for weak solutions, which, in the Eulerian case gives for some C > 0, + OO >\u\ + ρ 2 )(: < C ί(ρ°\u°\ + (ρ°))(y)dy, VxGR. (3) E The proof relies on the moments lemma for transport equations (see Perthame [12]) in the form in which they were set recently by Lions and Perthame [6]. Again it can be interpreted here as a choice of appropriate entropies. A (relatively!) surprising feature is that, with this method, (3) appears very close to dispersive effects for the Schrodinger equation. Our last illustration is a proof of the strong convergence of families of solutions corresponding to initial data ρ®, vPε bounded in L°°(W), in the Eulerian case (1) for 7 > 3. Let us recall that this result was proved by DiPerna [3,4] for values of 7 given by 7 = (N + 2)/N, N > 3 and extended by Chen [1] to 1 < 7 < 5/3. The main difficulty lies with the degeneracy of the system close to the vacuum (ρ = 0). This restricts the allowed entropies to the so-called weak family, and it is precisely this family of weak entropies that is represented by our kinetic formulation. 1 It is possible to formulate the isentropic system (1) as a BGK-type kinetic equation but this formulation recovers just one entropy the mechanical energy Kinetic Formulation of Isentropic Gas Dynamics 417 We note that unlike the scalar case our current formulations are not "purely" kinetic, in the sense that the advection velocity of the underlying transport equation involves a combination of the kinetic velocity, £, as well as the macroscopic velocities, u and w\ consult (18) (with 7 ^ 3 ) and (44) below. Kinetic formulations of such "non-local" type are familiar from kinetic modeling in other contexts. This particular formulation is compatible with moments lemma, but we do not seem to be able to use here the averaging lemmas (see Golse, Lions, Perthame, Sentis [6]; DiPerna, Lions, Meyer [5]). Instead, our convergence proof employs compensated compactness arguments, as in [3, 4, 1] (see Murat [11], Tartar [15]), and its relative simplicity is due to some algebraic properties of the kinetic formulation. We will also give details about the case 7 = 3 for the Eulerian case, announced by the authors in [9]. In this case, the kinetic formulation is particularly simple (since it becomes purely local), and we can prove regularizing effects in Sobolev spaces, similar to the scalar case (see [10]). The rest of this paper is organized as follows. We first give the kinetic formulation in the Eulerian case, from which we derive in a second section the a priori estimate (3). The third section is devoted to the strong convergence of bounded families of solutions for 7 > 3, a result that yields existence theorems by the vanishing viscosity method. Such existence results will be presented elsewhere. Treatment of the case j = 3 concludes the third section. Finally, in Sect. IV we give the kinetic formulation for the Lagrangian case, and as before we recover invariant regions and the L°° estimate of averages in time similar to (3). Let us finally mention that the case 1 < 7 < 3 can also be studied, as far as strong convergence and existence results are concerned, and we shall come back on this in a future publication. I. Kinetic Formulation in the Eulerian Case In this section, we consider weak solutions of the system (1) and we will give its kinetic formulation. This requires the knowledge of a complete family of "supplementary conservation laws" or, more precisely, the weak entropy inequalities. This is achieved in Subsect. 1. Then, we present our kinetic formulation and we conclude this section with several remarks in order to connect this formulation with classical notions like the eigenvalues of the system or Riemann invariants. I.I. Entropy Inequalities Smooth solutions of (1) satisfy the additional conservation laws ) = 0, (4) if and only if (77, H) satisfies Q = % + ^yVu> u = QVρ + uηu , (5)