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).
Here are the main novelties expected following the same numbering of the previous part.
1. A rigorous analysis of the simplified RD in the case of two equations will be performed trying to understand if the presence of the degenerate diffusion is still compatible with the presence of a strictly positive (negative) propagation critical speed. We also intend to explore the nature of such traveling profiles will be given. In the presence of heterogeneity and for the general case, we would like to analyze both analytically and numerically the effect of periodic oscillating functions in different frequency and amplitude regimes. In the one-dimensional case, as the frequency tends to infinity (for fixed amplitude) appearance of limiting propagating fronts of some appropriate uniform effective equation is expected. Differently, the case of propagation blocking is probable for small frequency and specific form of the heterogeneity. Since the explicit formulas for the speed of wave propagation are rare, we intend to use methodically the so-called LeVeque-Yee formula (inspired by CL) to validate numerically the analytical evidences.
2. The innovative part of the project consists in determining a formulation for CL with measure-valued solutions in the most general context, in which there are no limitations for the flows chosen and for the measures considered. The case of measures with the variable sign has already been addressed in [BSTT3] but essentially in the case of limited flows for which the interaction between the measures of different sign does not take place. Furthermore, some qualitative aspects of solutions with bounded flows, which are part of the project, are still open. Regarding HJ, the innovative part of the project consists of treating the case of discontinuous initial data for the viscosity solutions. The main part of the analysis is to introduce singular Neumann boundary problems along the discontinuities. Another innovative part consists of analyzing in parallel the results obtained for CL with measure data with that stated for HJ with discontinuous data. This allows us to study at the same time qualitative behavior for the solutions of both problems.
3. The analysis carried out in [KYZ20] is based on probabilistic techniques, in particular on the Hopf-Cole transformation to reduce the problem to a convex one and on the Feynman-Kac formula to represent the solutions. We plan to revisit this approach by making more substantial use of PDE tools, aiming at simplifying the arguments and at extending the homogenization results to functions G of more general form. This strategy has been already successfully employed for the work [DK17]: the probabilistic approach provides a first insight on the phenomena taking place and on the results to be expected, but it only applies to equations of very specific form; the PDE approach provides a more general setting to frame the phenomena observed and to extend the scope of application of the analysis.
4. Following [CIS], it is crucial to put HJ in networks in the right homological frame, choosing a right parametrization for the paths of the dynamics associated to the PDE, and to study the class of corresponding Mather measures, which, roughly speaking, act as average tools in the homogenization procedure. One of the ultimate goals of this stream of research is the adaptation of first-order Mean Field game models to the discrete setting. This issue is related to the full understanding of Wasserstein spaces on graphs and has also a relevance from the point of view of applications.
5. Concerning the long time behavior MFG problem, we would like to check the validity of two new ideas to attack the issue. First, we want to adapt to the MFG setting the approach introduced in [IS] for vanishing discount problems, where the convergence result is achieved by somehow lifting the problem to a suitable functional space of measures. Secondly, we want to express the whole MFG model introducing the new notion of time-dependent Mather measures. The knowledge of the structure of such measures should help in understanding the behavior of optimal measure trajectories. The main problem is to prove local uniform convergence and to fully characterize the limit function.
6. The fluid-particle interaction model and its propagation properties can be analyzed by considering its hydrodynamical limit and the corresponding traveling wave solutions. Two different models could be considered to be explored parallelly: a Burgers-Fokker-Planck model and an Euler-Fokker-Planck one. For both existence and stability of propagation fronts can be explored with complete analytical results to be expected in the small-amplitude regime and numerical exploration for the large-amplitude case.