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Remarks on the kinetic formulation of scalar conservation laws. (Remarques sur la formulation cinétique des lois de conservation scalaires.) (French) Zbl 0797.35120
Sémin. Équ. Dériv. Partielles, Éc. Polytech., Cent. Math., Palaiseau 1990-1991, No.VI, 13 p. (1991).
This paper presents results obtained in collaboration with P.-L. Lions and E. Tadmor about entropy solutions of conservation laws \[ {\partial u \over \partial t} + \sum^ n_{i=1} {\partial A_ i (u) \over \partial x_ i} = 0,\qquad t \geq 0,\;x \in \mathbb{R}^ n \quad \text{ with } \quad u(x,0) = u_ 0(x) \in L^ 1 \cap L^ \infty (\mathbb{R}^ n). \tag{1} \] The method also applies to the system of monoatomic isentropic gas dynamics \[ {\partial \rho \over \partial t} + {\partial \over \partial x} (\rho u) = 0,\quad {\partial \over \partial t} (\rho u) + {\partial \over \partial x} (\rho u^ 2 + \rho^ 3/12) = 0,\quad x \in \mathbb{R},\;t \geq 0. \] First, it is shown that \(u\) is an entropy solution of (1) if and only if \[ f(x,v,t) = \chi_{u(x,t)} (v) = \begin{cases} + 1 & \text{ if } 0 \leq v \leq u(x,t) \\ -1 & \text{ if } u(x,t) \leq v \leq 0 \\ 0 & \text{ elsewhere } \end{cases} \] is a solution of the kinetic equation \(\partial f/ \partial t (x,v,t) + \sum^ n_{i=1} A_ i'(v) \partial f/ \partial x_ i (x,v,t) = \partial m/\partial v\), \(t \geq 0\), \(x \in \mathbb{R}^ n\), \(v \in \mathbb{R}\), where \(m\) is a nonnegative bounded measure [see P.-L. Lions, B. Perthame and E. Tadmor, C. R. Acad. Sci., Paris, Sér. I 312, No. 1, 97-102 (1991; Zbl 0729.49007) for more details]. Then using averaging lemmas, which have been introduced by F. Golse, B. Perthame and R. Sentis [C. R. Acad. Sci., Paris, Sér. I 301, 341-344 (1985; Zbl 0591.45007)], and F. Golse, P.-L. Lions, B. Perthame and R. Sentis [J. Funct. Anal. 76, No. 1, 110-125 (1988; Zbl 0652.47031)], compactness and regularity results for an approximating semilinear hyperbolic problem of Boltzmann type are proven. The convergence of the approximating problem, which has been studied by B. Perthame and E. Tadmor [Commun. Math. Phys. 136, No. 3, 501-517 (1991; Zbl 0729.76070)] proves the existence of entropy solutions of the multi-dimensional scalar conservation law and gives regularity and compactness results for the solutions, under the nondegeneracy condition \[ \forall (\tau,\xi) \in \mathbb{R} \times \mathbb{R}^ n/ \{0,0\},\text{ meas} \{v/ \tau + \sum^ n_{i=1} A_ i'(v) \xi_ i = 0\} = 0 \] which is a generalization of the condition introduced by L. Tartar.
A more general form of equation (1) and the relation of averaging lemmas with the hypoelliptic regularization properties of (1) is also studied. The use of a variational principle due to Y. Brenier [J. Differ. Equations 50, 375-390 (1983; Zbl 0549.35055)] on kinetic entropies suggests equivalent formulations for the isentropic gas dynamics model (general case) and the compressible Euler system, but the question of the convergence (fluid limit) of the solution of the approximating kinetic equation to the minimizer of the variational problem is still open in these cases.
35L65 Hyperbolic conservation laws
35L45 Initial value problems for first-order hyperbolic systems
82C40 Kinetic theory of gases in time-dependent statistical mechanics
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