BV solutions for a class of viscous hyperbolic systems. (English) Zbl 0988.35109

Bianchini, Stefano; Bressan, Alberto

Indiana Univ. Math. J. 49, No. 4, 1673-1713 (2000).

This paper is devoted to the Cauchy problem for a nonlinear, stricly hyperbolic system with small viscosity: \[ u_t+A(u)u_x=\varepsilon u_{xx}, \quad u(0,x)= u_0(x).\tag{1} \] The authors assume that the integral curves of the eigenvectors \(r_i\) of the matrix \(A\) are straight lines. On the other hand they do not require the system (1) to be in conservation form, nor do they make any assumption on genuine linearity or linear degeneracy of the characteristic fields. Under very natural assumptions (see above) the authors prove existence of BV solutions for (1).

Reviewer: Messoud Efendiev (Berlin)

MSC:

35L65	Hyperbolic conservation laws
35B25	Singular perturbations in context of PDEs

Keywords:

stricly hyperbolic system; small viscosity