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Entropy and global existence for hyperbolic balance laws. (English) Zbl 1058.35162
The paper is concerned with multidimensional systems of balance laws $U_t+\sum\limits_{j=1}^d F_j(U)_{x_j}=Q(U), \quad U=U(x,t), \quad (x,t)\in {\mathbb R}^d\times [0,+\infty).$ Under some structural conditions, including the entropy dissipation condition and the Kawashima condition, the author proves existence and stability of global smooth solutions with initial data close to a constant equilibrium state. It is also shown that the Kawashima condition reduces to the known Kawashima condition for the hyperbolic-parabolic systems derived through the Chapman-Enskog expansion. Applications to Bouchut’s discrete velocity BGK models are given.

##### MSC:
 35L65 Hyperbolic conservation laws 35L45 Initial value problems for first-order hyperbolic systems 35B65 Smoothness and regularity of solutions to PDEs 35B45 A priori estimates in context of PDEs
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