Zero Diffusion-Dispersion-Smoothing Limits for a Scalar Conservation Law with Discontinuous Flux Function

We consider multidimensional conservation laws with discontinuous flux, which are regularized with vanishing diffusion and dispersion terms and with smoothing of the flux discontinuities. We use the approach of ？？-measures to investigate the zero diffusion-dispersion-smoothing limit. 1. Introduction We consider the convergence of smooth solutions ？？=？？？？(？？,？？) with (？？,？？)∈？？+×？？？？ of the nonlinear partial differential equation ？？？？？？+div？？？？？？(？？,？？,？？)=？？div？？？？(？？？)+？？？？？？=1？？3？？？？？？？？？？？？？？(1.1) as ？？→0 and ？？=？？(？？),？？=？？(？？)→0. Here ？？∈？？(？？;？？？？(？？+？？×？？？？？？)) is the Caratheodory flux vector such that max|？？|≤？？||？？？？||(？？,？？,？？)？？？(？？,？？,？？)？0,？？？0,in？？？？loc？？+×？？？？,(1.2) for ？？>2 and every ？？>0. The aim is to show convergence to a weak solution of the corresponding hyperbolic conservation law: ？？？？？？+div？？？(？？,？？,？？)=0,？？=？？(？？,？？),？？∈？？？？,？？≥0.(1.3) We refer to this problem as the zero diffusion-dispersion-smoothing limit. In the case when the flux ？？ is at least Lipschitz continuous, it is well known that the Cauchy problem corresponding to (1.3) has a unique admissible entropy solution in the sense of Kru？hkov [1] (or measure valued solution in the sense of DiPerna [2]). The situation is more complicated when the flux is discontinuous and it has been the subject of intensive investigations in the recent years (see, e.g., [3] and references therein). The one-dimensional case of the problem is widely investigated using several approaches (numerical techniques [3, 4], compensated compactness [5, 6], and kinetic approach [7, 8]). In the multidimensional case there are only a few results concerning existence of a weak solution. In [9] existence is obtained by a two-dimensional variant of compensated compactness, while in [10] the approach of ？？-measures [11, 12] is used for the case of arbitrary space dimensions. Still, many open questions remain such as the uniqueness and stability of solutions. A problem that has not yet been studied in the context of conservation laws with discontinuous flux, and which is the topic of the present paper, is that of zero diffusion-dispersion limits. When the flux is independent of the spatial and temporal positions, the study of zero diffusion-dispersion limits was initiated in [13] and further addressed in numerous works by LeFloch et al. (e.g., [14–17]). The compensated compactness method is the basic tool used in the one-dimensional situation for the so-called limiting case in which the diffusion and dispersion parameters are in an appropriate balance. On the other hand, when diffusion dominates