Ricerc@Sapienza

Our research proposal is concerned with the mathematical analysis of nonlinear evolutive PDEs focussing on well-posedness, large time-behavior and other qualitative properties determined by the presence of an underlying heterogeneous, possibly non-smooth, structure, either at the level of a persistent internal structure (random media, network, anisotropicity), at the level of an initial/boundary described by some highly non-smooth and singular data (discontinuous or Radon measures).

Depending on the applications, four prototype equations appear: Reaction-Diffusion systems (RD), Conservation Laws (CL), Hamilton-Jacobi equations (HJ), and Mean Field Games (MFG). The common denominator is the study of the appearance of coherent heterogeneous patterns in the presence of an underlying low-regularity general framework.

The topics considered are
a. Front propagation in heterogeneous and anisotropic environments (RD and CL);
b. Singular and Radon measure initial data (CL and HJ);
c. Homogenization in random media in the case of non-convex hamiltonians (HJ);
d. Networks and graphs structures (HJ);
e. Long time behaviour in the first-order case (MFG).
f. Fluid-particle interaction model (CL).