Felix Otto: Singular quasi-linear stochastic PDEs I

The lecture was held within the of the Hausdorff Junior Trimester Program: Randomness, PDEs and Nonlinear Fluctuations

Abstract:
We are interested in parabolic differential equations δ_tu-a(u)δ^2_xu = ξ with rough, typically random, forcing ξ, and a local non-linearity a(u) in the leading order term. In terms of scaling, the
interaction of this non-linearity with the roughness of ξ is as singular as for the non-linearity given by multiplicative noise, i. e. δ_tu-a(u)δ^2_xu = σ(u)ξ. However, its treatment, for instance in the framework of controlled rough paths or of regularity structures, requires more care and some twists.
Within the setting of regularity structures, we present a framework that on the deterministic side is able to treat exponents down to a just positive Hölder exponent α of u. Key ingredients are the
identification of the (infinite-dimensional) structure group and the connection to Safonov's kernel-free, jet-based approach to Schauder theory. This is joint work with H. Weber, J. Sauer, and S. Smith.