Analysis of PDEs with cross-diffusion and stochastic driving

Contents: Second-order evolution partial differential equations (PDEs) involving cross- diffusion terms are frequently used in mathematical biology and present a challenge for mathematical analysis. An efficient technical tool to prove the existence of global-in-time solutions are entropy methods. From the viewpoint of mathematical biology, cross-diffusion deterministic PDEs are a mean-field model and it is highly desirable to include stochastic effects due to finite-size effects or random external forcing. This naturally leads one to consider stochastically-forced partial differential equations (SPDEs). For cross-diffusion SPDEs, basically no mathematical analysis exists and one major goal of this project is to fill this substantial gap in our knowledge. We propose a multi-faceted approach to fill this gap. Methods: For local-in-time existence theory, we aim to study several different solution concepts for cross-diffusion SPDEs. For global-in-existence, we propose to transfer entropy method techniques from PDEs to the analysis of SPDEs. A main theme in this context will be the use of entropy variables in combination with global a-priori bounds. Novelty: We aim to merge and significantly extend, for the first time, theories from several different mathematical areas within the framework of cross-diffusion PDEs.

The main project`s achievements concern the study of some important mathematical properties of so-called "Stochastic Partial Differential Equations" (SPDEs). These objects are a generalization of Partial Differential Equations (PDEs), which are widely employed in the applied sciences in order to describe in a quantitative way countless physical phenomena. While PDEs are deterministic objects, that is, given an input in the form of data, one obtains, by solving the equations, a unique result, the SPDEs include random effects, which is often required in order to describe in a more fitting way the physical phenomena. However, f or the equations to correctly describe reality, one has to make sure that their solution exists in a suitable mathematical sense. That was one of our goals for this project: to show that a certain class of SPDEs admits indeed suitable solutions. This goal was achieved by showing first the existence of a cleverly defined type of solution ("mild"), which does not necessarily have all the properties that the physics requires, then to prove that such a solution does indeed satisfy the aforementioned properties and is therefore a proper solution of the system. We proved first that such solutions exist for sufficiently small time, and then showed that they exist for every time. After dealing with the problem of existence of solutions, we turned our attention to the "dynamics", that is, what happens to the solutions once we let the time flow. We proved the existence of special states that "attract" every solution of the system (that is, all solutions get closer and closer to such state, as the system evolves in time), aptly called "attractors". Such results are relevant for applications, since they tell us that we can expect the system to forget, after a certain time, about (almost) all the particular features of its state at initial time; one can even get some estimate of the time it takes to the system to get close to the attractor. Among the large family of SPDEs, we considered those presenting so-called "cross-diffusion", a particularly strong coupling mechanism between the equations themselves, typically employed to describe the complex behaviour of mixtures, which makes the analysis of the equations much more challenging. The mathematical properties that were crucial in our analysis came as a consequence of what is called "the entropy structure" of the equations. Indeed, the equations we considered mirror physics, in the sense that there is a structure embedded into them, related to the well-known Second Law of Thermodynamics, which dictates that the "disorder" of the system (measured by the function called "entropy") always increases in time. Such structure is like the skeleton of the system, which enabled us to overcome the difficult represented by cross-diffusion.