Well-posedness for a monotone solver for traffic junctions

In this paper we aim at proving well-posedness of solutions obtained as vanishing viscosity limits for the Cauchy problem on a traffic junction where $m$ incoming and $n$ outgoing roads meet. The traffic on each road is governed by a scalar conservation law $\rho_{h,t} + f_h(\rho_h)_x = 0$, for $h\in \{1,\ldots, m+n\}$. Our proof relies upon the complete description of the set of road-wise constant solutions and its properties, which is of some interest on its own. Then we introduce a family of Kruzhkov-type adapted entropies at the junction and state a definition of admissible solution in the same spirit as in \cite{diehl, ColomboGoatinConstraint, scontrainte, AC_transmission, germes}

Entropy conditions for scalar conservation laws with discontinuous flux revisited

We propose new entropy admissibility conditions for multidimensional hyperbolic scalar conservation laws with discontinuous flux which generalize one-dimensional Karlsen-Risebro-Towers entropy conditions. These new conditions are designed, in particular, in order to characterize the limit of vanishing viscosity approximations. On the one hand, they comply quite naturally with a certain class of physical and numerical modeling assumptions; on the other hand, their mathematical assessment turns out to be intricate. \smallskip The generalization we propose is not only with respect to the space dimension, but mainly in the sense that the "crossing condition" of [K.H. Karlsen, N.H. Risebro, J. Towers, Skr.\,K.\,Nor.\,Vid.\,Selsk. (2003)] is not mandatory for proving uniqueness with the new definition. We prove uniqueness of solutions and give tools to justify their existence via the vanishing viscosity method, for the multi-dimensional spatially inhomogeneous case with a finite number of Lipschitz regular hypersurfaces of discontinuity for the flux function.Comment: multidimensional case is included and mistakes are correcte

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 N. 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 N. 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 1 contraction principle for the GV V-entropy solutions. Our formulation is compatible with the standard vanishing viscosity approximation u ε t + div (f(x, u ε)) = ε∆u ε, u ε |t=0 = u0, ε ↓ 0, of the conservation law. We show that, provided u ε, ε> 0, enjoy a uniform L ∞ bound and the flux f(x, ·) is non-degenerately nonlinear, vanishing viscosit