## Project FONDECYT 1090456

## Fundamental Areas

## March 2009 - March 2013

## Raimund Bürger [P]:

### Non-Standard Conservation Laws and Related Equations: Theory, Numerics and Applications

### Summary:

PROPOSAL ABSTRACT: Must be clear and informative. Describe the main issues you plan to address, including goals, methodology and expected outcomes. A good summary facilitates an adequate description and understanding of what you intend to achieve. If selected, this abstract may be published in CONICYT’s web page. The maximum length for this section is 1 page. (Arial or Verdana, font size 10). It is well known that first-order, nonlinear conservation laws and some related partial differential equations, such as strongly degenerate parabolic equations, possess discontinuous solutions, even for smooth initial data. These solutions need to be defined in a weak sense, and require an additional selection criterion, or entropy condition, to select the so-called entropy solution among several weak solutions. The commonly accepted Kruˇzkov-type entropy solution framework includes well-posedness (existence, uniqueness and stability) of scalar equations with flux functions that depend smoothly on the data, time, and spatial coordinates; several types of numerical schemes (including finite volumes and front tracking) are known to converge to the entropy solution. This well-documented body of knowledge forms the standard entropy solution theory. Motivated by models from several applications, it is proposed to investigate the well-posedness, to develop numerical schemes, and to simulate applications of conservation laws that are bot covered by the standard theory. The planned research is subdivided into three research topics (RTI, II and III). The first research topic (RTI) deals with conservation laws and related equations whose flux function depends discontinuously on the spatial variable. This problem occurs e.g. in models of traffic flow with heterogeneous road surface conditions, clarifier-thickeners, and two-phase flow in heterogeneous media. The key issue is the treatment of jumps of the solution at flux discontinuities, which requires different entropy jump conditions for different applications. Here, the appropiate theory does not emerge as a limiting case of the standard theory of equations whose fluxes are smooth functions of their arguments. Recent results for a spatially one-dimensional scalar equation involving two different fluxes defined to either side of a given spatial position, and which intersect once in a particular simple way, include wellposedness and convergence of an Enguist-Osher type numerical scheme. It is planned to obtain similar results for flux curves that intersect in a more general way. Furthermore, it is planned to investigate entropy solutions and develop numerical schemes for one-dimensional hyperbolic systems of conservation laws with discontinuous flux (starting from a model of multi-species vehicular traffic), and for scalar conservation laws with discontinuous flux in two space dimensions. In the second research topic (RTII), conservation laws and degenerate parabolic equations whose flux does not depend on only the local value of the solution, but non-locally the function solution as a whole, will be considered. Two variants of the problem, which arise from a model of sedimentation and an aggregation model in mathematical biology, respectively, will be analyzed. In the sedimentation model, the non-locality is introduced by convolution with a kernel of compact support, and the resulting equation is approximated by a (standard, i.e., local) diffusive-dispersive PDE. The properties of the limiting model arising from letting the compact support tend to zero are of interest, since non-classical shocks are possibly produced. In the aggregation model, the non-locality arises from evaluating the flux at the total mass of the sought quantity. The third research topic (RTIII) is focused on scalar conservation laws with a non-convex flux function with “non-entropy” solutions that satisfy conditions other than standard entropy conditions. This problem is associated with non-entropic solutions and possibly non-classical shocks. The prototype equation that gives rise to this problem is an that appears in models of traffic flow, and where a particular behavioristic principle, the “driver’s ride impulse”, compels solutions that differ from the standard theory. A related problem arises in a model of sedimentation. We intend to conduct a well-posedness analysis for these applications (utilizing the mathematical theory of non-classical shocks), and to develop numerical schemes, starting from experiments with front tracking and the Artifical Compression Method.