SS28 AIMS 2016 Meeting, Orlando, Florida, USA

Title:  Recent developments related to conservation laws and Hamilton-Jacobi equations
Name: Affiliation: Country: Email Address:
Laura Caravenna
Department of Mathematics, Universit`a di Padova
Annalisa Cesaroni
Department of Statistical Sciences, University of Padova
Hung Vinh Tran
Department of Mathematics, University of Wisconsin Madison
The session focuses on some recent developments of first order nonlinear Partial Differential Equations, and in particular conservation laws, Hamilton-Jacobi equations and related topics such as dynamical properties and homogenization. Recently the joint analysis of conservation laws and Hamilton-Jacobi equations on heterogeneous structures has received an increasing attention. This includes problems on networks and their applications to modeling of traffic flows, homogenization in periodic and random media, dynamical properties of solutions, etc. One of the main motivation for problems on networks is the application to dynamic models of traffic flow, e.g. the flowing of cars on a highway or of gas along pipelines or of packages of data on telecommunication networks. The established mathematical framework of these models consists of single conservation laws, systems of conservation or balance laws running on a network modeled as a topological graph. These models are widely used also in engineering: it is an area where fundamental studies are interesting but which is also directly related to applications. More recently, a complementary analysis of the network dynamics based on Hamilton-Jacobi equations has also been developed. The current trend consists of proposing new models/problems, studying their well-posednesses, dynamical properties (optimal control formulas, large time behaviors), and related homogenization problems. Numerical approaches are as well of great interest. Homogenization problems are about finding averaged (effective) properties of solutions to inhomogeneous equations depending on small parameters and set in self averaging media. The area is moving fast in various perspectives: qualitative and quantitative properties of the effective equations, rates of convergences, stochastic homogenization of front propagations, and non-convex Hamilton-Jacobi equations, etc. Moreover, there have been a lot of developments connecting homogenization and problems on networks such as homogenization on junction framework, Hamilton-Jacobi equations on a network as a limit of a singularly perturbed problem in optimal control defined on thin strips around the network. The aim of this session is to bring together mathematicians working on conservation laws and Hamilton-Jacobi from different backgrounds and perspectives, including homogenization. It will be a great occasion to present the new advances and directions of different research groups, in order to interact and to improve the understanding of these exciting topics that have been considered intensively in the last few years.
List of approved abstract