Random dynamical systems and regularization by noise for stochastic partial differential equations.

• Applicant: Professor Dr. Benjamin Gess
• Subject Area: Mathematics
• Term: from 2013 to 2015
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 247297551

Stochastic partial differential equations (SPDE) are key tools in several models arising in fluid-, continuum- and quantum-mechanics, in economics as well as in technical applications. Only rarely explicit solutions to such equations may be found, thus causing the need for mathematical, qualitative methods and numerical simulations. Already in the more classical situation of deterministic partial differential equations (PDE), the behavior of individual trajectories often is too complex to allow for a detailed mathematical description. A key part of the understanding of such complex systems hence relies on the qualitative analysis of the flow of solutions, based on a more geometrical viewpoint. Despite the complex or chaotic behavior of individual trajectories, this approach often allows understanding their qualitative behavior, such as long-time behavior. A central aim of the mathematical research on SPDE during the last 20 years has therefore been the generalization of these qualitative techniques from the field of (deterministic) dynamical systems to the stochastic case. This intriguing area of research thus lies on the borderline in between analysis, stochastics and dynamical systems.While many of the well-known, deterministic techniques developed in the field of dynamical systems have successfully been generalized to the abstract setting of random dynamical systems (RDS), there is a major gap between the assumptions of this abstract theory and the properties known for solutions to SPDE. More precisely, one basic assumption of the theory of RDS is the stochastic flow property which, however, is only known to hold for solutions to SPDE in very special cases. Therefore, so far the applicability of the methods from RDS was essentially restricted to SPDE with very simple (mostly affine-linear) diffusion coefficients. It is a key aim of this research project to further develop methods allowing proving the stochastic flow property for SPDE with more general, non-linear diffusion coefficients.So far we have encountered noise in PDE as an additional difficulty as regarding their mathematical understanding. On the other hand there are well-known effects of regularization and simplification of dynamics due to noise. For example, in a recent, ground-laying work the well-posedness of stochastic transport equations due to noise has been established. Such results are of high importance in the field of PDE, since they justify the hope to gain a better understanding of the Millennium-problem of uniqueness of solutions to the 3d-Navier-Stokes equations by including stochastic effects. On the other hand, the complex behavior and possible blow-up in fluid dynamics arises due to the nonlinear structure of these equations, while transport equations are linear. Therefore, a second main aim of this research project is the analysis of the possible regularization by noise for scalar conservation laws, as a nonlinear generalization of transport equations.

• DFG Programme: Research Fellowships
• International Connection: USA
• Host: Professor Panagiotis E. Souganidis, Ph.D.