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Damping-Diffusion-Noise Interactions in the Stochastic Camassa--Holm Equation

Diego Alonso-Orán, Peter H. C. Pang, Hao Tang

Abstract

We investigate the effects of the interaction between time-inhomogeneous damping, non-local diffusion, and noise on classical solutions to the Camassa--Holm equations that incorporate these additional factors. Initially, a local-in-time theory is developed, covering the existence, uniqueness, and a blow-up criterion under relatively general conditions for these interacting factors. Subsequently, we identify different conditions on the interactions between damping, diffusion, and noise that enable the global regularity and long-time behaviour of classical solutions. Notably, we demonstrate the existence of an evolution system of measures, generalising the concept of invariant measures to the time-inhomogeneous system.

Damping-Diffusion-Noise Interactions in the Stochastic Camassa--Holm Equation

Abstract

We investigate the effects of the interaction between time-inhomogeneous damping, non-local diffusion, and noise on classical solutions to the Camassa--Holm equations that incorporate these additional factors. Initially, a local-in-time theory is developed, covering the existence, uniqueness, and a blow-up criterion under relatively general conditions for these interacting factors. Subsequently, we identify different conditions on the interactions between damping, diffusion, and noise that enable the global regularity and long-time behaviour of classical solutions. Notably, we demonstrate the existence of an evolution system of measures, generalising the concept of invariant measures to the time-inhomogeneous system.

Paper Structure

This paper contains 21 sections, 13 theorems, 173 equations.

Table of Contents

  1. Introduction
  2. Motivations and main goal
  3. Summary of the main results
  4. Background, main results and remarks
  5. Notations
  6. Definitions
  7. Hypotheses and examples
  8. Statements of main results
  9. Remarks on difficulties and novelties
  10. Auxiliary results
  11. Local pathwise classical solution
  12. Step 1: the approximation scheme
  13. Step 2: convergence of the approximation solutions
  14. Step 3: concluding the proof for Theorem \ref{['Thm:local-theory']}
  15. Diffusion-damping-noise interaction: Case 1
  16. ...and 6 more sections

Key Result

Theorem 2.1

Let Hypotheses H-Q, H-h and H-parameters hold. If $u_0$ is an $H^s$-valued $\mathcal{F}_0$-measurable random variable with $s>3/2+\max\{2\gamma_0,1,2\theta{\bf 1}_{\{\varepsilon>0\}}\}$, where $\gamma_0$ is given in gamma0, then

Theorems & Definitions (33)

  • Definition 2.1
  • Remark 2.1
  • Definition 2.2
  • Definition 2.3
  • Definition 2.4
  • Example 2.1: Examples of classes \ref{['Ak:skew adjoint']} and \ref{['Bk:skew adjoint']}
  • Example 2.2: Examples of $h_k$ satisfying Hypothesis \ref{['H-h']}
  • Example 2.3: Examples of $h_k$ satisfying Hypotheses \ref{['H-h']} and \ref{['H-h-large']}
  • Theorem 2.1
  • Theorem 2.2
  • ...and 23 more