Global well-posedness of the viscous Camassa–Holm equation with gradient noise

Abstract

We analyse a nonlinear stochastic partial differential equation that corresponds to a viscous shallow water equation (of the Camassa-Holm type) perturbed by a convective, position-dependent noise term. We establish the existence of weak solutions in H m (m ∈ N) using Galerkin approximations and the stochastic compactness method. We derive a series of a priori estimates that combine a model-specific energy law with non-standard regularity estimates. We make systematic use of a stochastic Gronwall inequality and also stopping time techniques. The proof of convergence to a solution argues via tightness of the laws of the Galerkin solutions, and Skorokhod-Jakubowski a.s. representations of random variables in quasi-Polish spaces. The spatially dependent noise function constitutes a complication throughout the analysis, repeatedly giving rise to nonlinear terms that "balance" the martingale part of the equation against the second-order Stratonovich-to-Itô correction term. Finally, via pathwise uniqueness, we conclude that the constructed solutions are probabilistically strong. The uniqueness proof is based on a finite-dimensional Itô formula and a DiPerna-Lions type regularisation procedure, where the regularisation errors are controlled by first and second order commutators.

Second order commutator estimates in renormalisation theory for SPDEs with gradient-type noise

“…Since mollification and multiplication (by σ, say) do not commute, there are commutator brackets that must be shown to vanish in appropriate topologies as the mollifier tends to a Dirac mass. These commutators are the focus of this note, and their vanishing are the results we prove in Lemmas 2.1 and 2.2, and Theorem 2.3 below, following [15,Proposition 3.4], [11,Section 7] quite closely.…”

“…Most of the computations are directly inspired by [15, pp 654 -657]. The d = m = 1 analogue on T (obviously without the divergence-free condition) was worked out in [8,10,11].…”

“…Given the last three (δ-independent) bounds, it is sufficient to establish convergence of (11) under the assumption that σ, u are smooth (in x). The general case follows by density using the established bounds.…”

Weak convergence of stochastic integrals

“…This section presents a slight extension of Theorem 2.1 to a non-reflexive Banach space X. This extension is useful in the analysis of the stochastic Camassa-Holm equation (see [6,8]). While the context of this application is not necessary to understand the proof and result, a brief overview will be provided first.…”

“…This is a nonlinear SPDE modelling the evolution of certain types of waves or currents under the influence of random fluctuations. In this application, the unknown h n = h n (ω, t, x) represents a part of the total energy of a viscous perturbation [8]. The source term G n in (1.2) includes the accumulated viscous energy dissipation, which in the limit n → ∞ becomes a singular measure.…”