Quasi-linear SPDEs via a rough path approach

Culminating in Hairer's regularity structures, there has been much recent progress in developing a robust solution theory for nonlinear stochastic partial differential equations (SPDEs). This progress is inspired by Lyons' treatment of stochastic ordinary differential equations, which is much more deterministic than Ito's approach. The main deterministic ingredient can be seen as an extended Schauder theory, so a maximal regularity theory for constant-coefficient parabolic equations in Hölder spaces, where polynomials are supplemented by more general, "rough" functions. The sole stochastic ingredient is to give an "off-line" sense to a finite number of singular products of rough functions and their distributional derivatives.
So far, this approach has been limited to SPDEs where the leading-order part is the constant-coefficient diffusion operator. In this course, we will present a treatment of the quasi-linear SPDE $$ \partial_tu+a(u)\partial_x^2u=\sigma(u)f $$ with a noise \(f\in C^{\alpha-2}\) on the parabolic Hölder scale. Provided \(\alpha>\frac{2}{3}\) (which includes white noise in time) and giving an "off-line" sense to products of the form \(v(\cdot,a_0)\partial_x^2v(\cdot,a_0')\) with \(v(\cdot,a_0)\) solving the constant-coefficient SPDE \(\partial_tv-a_0\partial_x^2v=f\), we obtain a (small-data) solution theory \(C^\alpha\) for u. Loosely speaking, we extend the treatment of the singular product \(\sigma(u)f\), in the spirit of Gubinelli, to the product \(a(u)\partial_x^2u\), which has the same degree of singularity but is more nonlinear since the solution u appears in both factors.
Next to treating a wider class of non-linear equations, the merit is that we introduce some simpler tools. More specifically, we treat the singular product \(a(u)\partial_1^2u\) by controlling the commutator \([a(u),(\cdot)_\epsilon]\partial_x^2u\) of multiplication with the first factor a(u) and convolution \((\cdot)_\epsilon\), where the family of symmetric convolution operators \(\{(\cdot)_\epsilon\}_\epsilon\) satisfies a semi-group property and respects the parabolic scaling. Controlling such commutators is reminiscent of the DiPerna-Lions theory for rough transport equations. The PDE ingredient mimics the (kernel-free) Krylov-Safanov approach to ordinary Schauder theory.
This is joint work with Hendrik Weber.

Date and time info
Thursday 09:15 - 11:00

Keywords
nonlinear SPDEs, Schauder theory

Audience
MSc students, PhD students, Postdocs

Language
English