Contemporary Topics in Conservation Laws – CoToCoLa

Continuation of works n subjects where we already possess good training - experts' invitation to learn new techniques and questions related to the studies we want to develop. See .odt for details.

New results were obtained for the following problems:
error and structural stability estimates for fractional conservation laws with hyperbolic degeneracy, semigroup techniques for conservation laws with discontinuous flux, Neumann problem for scalar dedenerate parabolic equation, general boundary conditions for conservation laws, vanishing capillarity limits for two-rocks' porous medium, transport equations with minimal regularity in bounded domain, numerical simulations of a hyperbolic equation with nonlocal boundary conditionand on a Saint-Venant problem with an original viscosity term. An important work on fractional conservation laws in bounded domain is in final stage of completion.

New problems emerged from interactions with our guests (Colombo, Ghoshal, Cancès). We continue to advance on most of the project's tasks simultaneously. See .odt for details

The proposal aims, firstly, at solving several concrete questions in the modern theory of conservation laws and related convection-diffusion problems. These questions (Tasks) are concerned with actual techniques of nonlinear analysis (some of the questions seemed not accessible only a few years ago); they originate from recently identified of from long-standing important applications. The Tasks are all relevant of the domains of expertise of one of the members of the project and at the same time, there is a strong connection to at least one of the other members and to our national and international collaborations.

The second goal is to make advance our understanding of the theory and the crucial tools (including the most modern ones) for analysis of conservation laws and related nonlinear PDEs. Indeed, the tasks we have selected are representative of several fundamental issues relevant to the problems in hand, such as: notions of solution and well-posedness; analysis of non-local terms in conservation laws; cooperation of semigroups in evolution equations; convection-diffusion problems of mixed type; boundary and interface problems for conservation laws; (ir)regularity and qualitative behaviour of solutions; converging and efficient numerical approximations.

Our third goal is, while advancing on the solution of these concrete questions, to form progressively an internationally recognized
group in Besancon working on a wide spectrum of modern approaches to conservation laws problems.

Indeed, the proposal team consists of five MCF, all of us belong to the Laboratoire de Mathematiques de Besancon CNRS UMR 6623. For three of us, conservation laws or systems and related nonlinear PDEs constitute the main field of research. Yet at the present time, everyone focuses on a different domain and specific subjects: nonlocal (fractional) laws / degenerate parabolic equations, boundary and interface problems for scalar laws / systems and viscous limits. The project is born from our intention to bring our different cultures to interaction. These three persons share the main load of work on the project. The implication of the two other team members is a starting point for more ample collaboration in the future; they joined the group because of their recent interest to the subject, and they contribute an expertise of different kind, the one on linear PDE techniques and the one on numerical analysis and scientific computing. We ask for funding for a post-doc to join the team during the second year of the project and bring an experience of his own in the quickly growing field of control of hyperbolic conservation laws.

The topics we treat were all developed in strong interaction between french, italian, american, russian, norwegian, polish, german, spanish, chinese researchers. We already have strong connections with some of the leading groups in the subjects we treat, and we are intended to maintain and develop such contacts.

The work on the Tasks will be complemented by three workshops or schools (one per year), plus a final meeting with participation of exterior experts and of many collaborators of the project ,on selected aspects of the modern theory of nonlinear PDEs including actual topics in control of systems of conservation laws.