On vanishing viscosity approximation of conservation laws with discontinuous flux

This note is devoted to a characterization of the vanishing viscosity limit for multi-dimensional conservation laws of the form ut + div f(x, u) = 0, u|t=0 = u0 in the domain R × R . The flux f = f(x, u) is assumed locally Lipschitz continuous in the unknown u and piecewise constant in the space variable x; the discontinuities of f(·, u) are contained in the union of a locally finite number of sufficiently smooth hypersurfaces of R . We define “GV V -entropy solutions” (this formulation is a particular case of the one of [3]); the definition readily implies the uniqueness and the L contraction principle for the GV V -entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation ut + div (f(x, u )) = ε∆u, u|t=0 = u0, ε ↓ 0, of the conservation law. We show that, provided u, ε > 0, enjoy a uniform L∞ bound and the flux f(x, ·) is non-degenerately nonlinear, vanishing viscosity approximations u converge to the unique GV V -entropy solution of the conservation law with discontinuous flux. Introduction. The study of conservation laws and related degenerate parabolic problems with space-time discontinuous flux has been intense during the last fifteen years. It is stimulated by applications such as sedimentation, porous medium flows in discontinuous media, road traffic models. We refer to [1]–[11], [13, 14], [16]– [18], [23]–[26] and references therein for some of the applications and known results. Notice that only very few studies treat the multidimensional case. However, most of the interesting phenomena appear already in the model onedimensional case, with the discontinuity along Σ = {x = 0}: ut + (f(x, u))x = 0, f : (x, z) ∈ R ×R 7→ { f (z) , x < 0 f (z) , x > 0. (1) From the purely mathematical viewpoint, the problem is quite challenging because of the possibility to give various non-equivalent generalizations of the Kruzhkov’s notion of entropy solution; moreover, different entropy solutions to the same equation may correspond to different applicative contexts. This phenomenon was discovered by Adimurthi, Mishra and Veerappa Gowda in [1]. In [7], Bürger, Karlsen and Towers proved well-posedness for (1) for a whole class of different solution notions. Date: July 28, 2009. 2000 Mathematics Subject Classification. Primary: 35L65.