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Global existence of dissipative solutions to the Camassa--Holm equation with transport noise

Luca Galimberti, Helge Holden, Kenneth H. Karlsen, Peter H. C. Pang

Abstract

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term.

Global existence of dissipative solutions to the Camassa--Holm equation with transport noise

Abstract

We consider a nonlinear stochastic partial differential equation (SPDE) that takes the form of the Camassa--Holm equation perturbed by a convective, position-dependent, noise term. We establish the first global-in-time existence result for dissipative weak martingale solutions to this SPDE, with general finite-energy initial data. The solution is obtained as the limit of classical solutions to parabolic SPDEs. The proof combines model-specific statistical estimates with stochastic propagation of compactness techniques, along with the systematic use of tightness and a.s. representations of random variables on specific quasi-Polish spaces. The spatial dependence of the noise function makes more difficult the analysis of a priori estimates and various renormalisations, giving rise to nonlinear terms induced by the martingale part of the equation and the second-order Stratonovich--Itô correction term.
Paper Structure (25 sections, 43 theorems, 377 equations)

This paper contains 25 sections, 43 theorems, 377 equations.

Table of Contents

  1. Introduction
  2. Background and main result
  3. Outline of main ideas
  4. Preliminaries and solution concepts
  5. Some a-priori estimates
  6. Tightness and a.s. representations
  7. Renormalisations
  8. Random mappings and path spaces
  9. Compactness and tightness criteria
  10. Tightness and a.s. representations
  11. Properties of a.s. representations
  12. Existence of martingale solutions
  13. Identification of a weak limit
  14. Energy inequalities and a right-continuity property
  15. Equation for the weak limits $\overline{S(q)}$
  16. ...and 10 more sections

Key Result

Theorem 1.1

Let $\sigma \in W^{2, \infty}({\mathbb{S}^1})$, and fix some $p_0>4$. For any initial probability distribution $\Lambda$ supported on $H^1({\mathbb{S}^1})$, satisfying there exists a dissipative weak martingale solution $\bigl(\tilde{\mathcal{S}},\tilde{u},\tilde{W}\bigr)$ to the stochastic CH equation eq:u_ch with random initial data $\tilde{u}_0$ distributed according to $\Lambda$ (${\tilde{u}_

Theorems & Definitions (100)

  • Theorem 1.1: existence of dissipative solution
  • Definition 2.1: $H^m$ martingale solution of viscous SPDE
  • Definition 2.2: strong $H^m$ solution of viscous equation
  • Theorem 2.3: strong well-posedness of viscous SPDE
  • Definition 2.4: dissipative weak martingale solution
  • Lemma 3.1: basic estimates
  • proof
  • Proposition 3.2: higher integrability
  • proof
  • Proposition 3.3: temporal translation estimate
  • ...and 90 more