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Analysis von Partiellen Differentialgleichungen mit Kreuzdiffusion und stochastischen Termen

Fachliche Zuordnung Mathematik
Förderung Förderung von 2017 bis 2021
Projektkennung Deutsche Forschungsgemeinschaft (DFG) - Projektnummer 370099393
 
Erstellungsjahr 2021

Zusammenfassung der Projektergebnisse

Differential equations are among the most common mathematical tools used across all sciences. In this project, we have focused on a particular class of differential equations, namely modelling cross-diffusion partial differential equations with stochastic forcing. A particularly important classes of concrete examples of these equations are predator-prey systems, where the different specifies diffuse/move spatially depending on the presence of other species at certain locations, i.e., the spatial spread is depending on the densities of the species, while species interact via quite classical predator-prey dynamics. The stochastic aspect of the differential equations can arise due external random forcing on the system or via intrinsic fluctuations. Mathematically, this class of problems cross-diffusion stochastic partial differential equations (SPDEs) was not understood at the beginning of this project. A main challenge for the mathematical analysis was the interplay between the complicated cross-diffusion term and the stochastic forcing. For systems without noise efficient tools have been developed in the last decades, e.g., so-called entropy methods, which use the fact that there is a global algebraic structure of the equations. Entropies provide a-priori bounds, which can be used to prove mathematically global-in-time existence of solutions for many cross-diffusion PDEs. From the perspective of dynamical systems, entropies are also very useful as they can often be used to show convergence to equilibrium, e.g., to a co-existence state in the predator-prey example. In this project, we managed to answer several fundamental questions regarding cross-diffusion SPDEs. First, we managed to prove that large classes of SPDEs, including cross-diffusion SPDEs as a special case, do have local solutions in time. This means that we can guarantee that the mathematical model is well-defined. In fact, we even managed to define and show the existence of a dynamically rather useful class of solutions, so-called pathwise mild solutions, which have a useful representation formula that was exploited in the project several times. We also proved that these solutions are equivalent to another solution concept, so-called weak solutions, which are often more difficult to handle dynamically but which are often easier to construct within a framework for existence theory. Secondly, we investigated whether the solutions we constructed also exist globally in time, which means that they cannot go to infinity in finite time. This is important since such blow-up solutions would be often unphysical, particularly in the context of mathematical biology. A key technical contribution is to prove that one can still use entropy methods from the deterministic setting in the stochastic one if a number of approximation procedures of the SPDE are applied as preliminary steps. Via these delicate approximation procedures, we managed to prove the global in time existence of a substantial number of cross-diffusion SPDEs. Thirdly, we proceeded to study dynamical properties of the obtained solutions. In particular, we were interested in attractors and invariant sets/manifolds. An attractor effectively contains the long-term dynamics of the system. Using our work on pathwise mild solutions, we proved the existence of attractors for certain classes of SPDEs. This work provides the key step to ensure well-defined long-term dynamics, which can then be analyzed further using dynamical systems techniques. To set up these dynamical methods, we employed the recently developed theory of pathwise analysis of stochastic differential equations (via so-called rough paths) to prove the existence of invariant manifolds for rough differential equations. These manifolds organize the dynamics in the state space, and our methods are set up in such a way that the mathematical techniques apply to very broad classes of stochastic differential equations, even cross-diffusion SPDEs or even very singular cross-diffusion SPDEs. As a concrete and main motivating example, the project results have been repeatedly valiadated in the stochastic version of one of the most famous cross-diffusion differential equations: the Shigesada, Kawasaki, and Teramoto (or SKT) model.

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