Existence, uniqueness and stability of semi-linear rough partial differential equations

Abstract

We prove well-posedness and rough path stability of a class of linear and semi-linear rough PDE's on R d using the variational approach. This includes well-posedness of (possibly degenerate) linear rough PDE's in L p (R d ), and then -based on a new method -energy estimates for non-degenerate linear rough PDE's. We accomplish this by controlling the energy in a properly chosen weighted L 2 -space, where the weight is given as a solution of an associated backward equation. These estimates then allow us to extend well-posedness for linear rough PDE's to semi-linear perturbations.1991 Mathematics Subject Classification. 60H15.

Generalized Burgers equation with rough transport noise

“…The main tool that we will use is the Feynman-Kac formula extended to the rough path setting as demonstrated in [6] and later explored in the L 2 (R d ) setting in [9]. Let us briefly explain the idea.…”

“…In [6,9] it is shown that the expressions, (3.6), (3.7) and (3.8) extend to case where Z is a geometric rough path and the relationship between these equations is still preserved in the limit. The solution X to (3.6) has however no meaning a priori, even in the rough path sense.…”

“…Based on this we will introduce two equations and their corresponding solutions that will be helpful for obtaining energy estimates. For a proof of the following result we refer to [9].…”

“…Proof. In [9] it is shown that M is a bounded solution in H 3 (R d × R d ) with the appropriate bounds on M ♮ and M ♭ under the weaker assumption that…”

Global Strong Well-Posedness of the stochastic bidomain equations with FitzHugh-Nagumo transport

“…Abstract stochastic semi-and quasilinear evolution equation of the form du + A(u)u dt = F (u) dt +H(u) dW, (1.7) have been considered before by many authors, see e.g., [4,7,9,14,17,22]. In fact, strong well-posedness results for (1.7) were shown by van Neerven, Veraar and Weis [22] as well as by Hornung [14] under Lipschitz conditions on F and H. Their results imply local existence results for (1.4).…”