Joint USYD-UNSW Seminar on Stochastic PDEs: Erika Hausenblas -- Stochastic Schauder theorem

https://uni-sydney.zoom.us/j/87047585294?from=addon
Meeting ID:
870 4758 5294

Nonlinear partial differential equations naturally arise in a wide range of biological
and chemical systems.  Classical examples include activator–inhibitor models in
morphogenesis, which are capable of producing complex spatial patterns, as well as
systems with cross-diffusion or aggregation mechanisms.  In realistic settings, noisy
random fluctuations are ubiquitous, and the presence of stochastic perturbations often
leads to qualitatively new phenomena.  Such randomness can significantly influence the
long-term behaviour of solutions, improve the descriptive power of the model, and
provide a more faithful representation of the underlying processes.  A common strategy
for proving the existence of probabilistic weak solutions is to apply a fixed-point
argument, typically a stochastic version of the Schauder-Tychonoff theorem.  In
practice, this method must be combined with cut-off techniques, stopping times, and
uniform a priori bounds to control the nonlinearities and to ensure tightness of the
approximate laws.  We have summarised these steps in a general “meta-theorem,” which
encapsulates the abstract framework frequently used in such existence proofs.  

In the talk, we first introduce a stochastic Schauder-Tychonoff theorem and outline
the main ingredients of its proof.  Afterwards, to demonstrate its applicability, we
consider a coupled chemotaxis-fluid system perturbed by noise.  Using this example, we
work out in detail which assumptions are required and how the choice of underlying
Banach spaces, the construction of cutoff functions, the derivation of uniform bounds,
and the stopping-time arguments interact.