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

Helge Holden; Kenneth H. Karlsen; Peter H. C. Pang

doi:10.3934/dcds.2022163

Article Contents

2023, Volume 43, Issue 2: 568-618. Doi: 10.3934/dcds.2022163 open access

This issue Previous Article Linear stability of the elliptic relative equilibrium with $ (1 + 7) $-gon central configuration in planar $ N $-body problem Next Article A 3D-Schrödinger operator under magnetic steps with semiclassical applications

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

1.
Department of Mathematical Sciences, NTNU Norwegian University of Science and Technology, NO-7491 Trondheim, Norway
2.
Department of Mathematics, University of Oslo, NO-0316 Oslo, Norway

^*Corresponding author: Helge Holden (helge.holden@ntnu.no)
^*Corresponding author: Helge Holden (helge.holden@ntnu.no)

Received: September 2022

Revised: October 2022

Early access: November 2022

Published: February 2023

This research was partially supported by the Research Council of Norway Toppforsk project Waves and Nonlinear Phenomena (250070). The final author is also partially supported by the Research Council of Norway project INICE (301538)

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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\in\mathbb{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.

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Mathematics Subject Classification: Primary: 35R60, 35G25, 60H15; Secondary: 35A01, 35A02.

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