Nonlinear partial differential equations in heterogeneous frameworks

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My proposed research deals with the deterministic analysis of certain partial differential equations (PDEs) in heterogeneous frameworks. The first part of the proposal is devoted to reaction-advection-diffusion equations with coefficients that depend on space/time variables in domains with perforations. Although some mathematical milestones in the theory of parabolic PDEs date back to the 1930's, the setting considered until the year 2000 was relatively homogeneous: semi-linear parabolic equations with constant diffusion and no drift. That is where those PDEs exhibit traveling wave solutions. *** Heterogeneous domains and coefficients appear in reaction-advection-diffusion equations when intended to model the evolution of quantities such as densities of chemicals or populations subject to diffusion, a reactive process as well as transport by an underlying flow. However, traveling wave solutions no longer exist in such settings. A remarkable advancement in the early 2000's generalized the notion of traveling waves to pulsating traveling fronts. In 2002, Berestycki et al. proved that, in the case of KPP nonlinearity (named after Kolmogorov, Petrovskii and Piscunov), pulsating traveling fronts exist beyond a threshold known as the KPP minimal speed. *** The first part of the proposed research is three-fold. We first focus on the role of turbulent advection: a strong flow may enhance the rate of reaction or, in some situations, block the propagation. Our goal is to find sharp criteria to characterize the flows that speed-up the propagation. Existing theory is far from complete in this regard, especially in 3-dimensional settings. The complexity inherited from the domain of the PDE and the chaotic behaviours exhibited by 3-dimensional flows bring tools from functional analysis, dynamical systems, and measure theory to the study. The second set of questions in part one of the proposed research deals with homogenization. Namely, the asymptotics of the minimal speed and solutions as the volume of the domain's periodicity-cell goes to zero. The third line in this part focuses on the interaction between the geometry of the domain and the coefficients of the PDE. Variational formulations of the KPP speed show its continuity as a function of the direction of propagation and diffusion/reaction coefficients. This initiates a search for the optimal direction(s) in which pulsating fronts propagate fastest.*** The second part of the proposal is devoted to a class of nonlinear elliptic PDEs with perturbed geometries and/or coefficients. The PDEs are non-self-adjoint and have no variational structure. Examples of these are “perturbed Lane-Emden” equations in infinite cylinders. The existence and regularity of solutions for this class of PDEs is an important study on its own. Moreover, the techniques that we develop in this part will be useful for the study of reaction-diffusion equations in random domains.