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Well-posedness and existence of an invariant measure for the linearly-damped KdV equations driven by a jump noise

Krutika Tawri, Roger Temam, Xinwu Yang

Abstract

In this paper, we investigate the linearly damped KdV equation on the one-dimensional torus $\mathbb{T}$, perturbed by a multiplicative Lévy noise. For any damping coefficient $γ> 0$, we establish the existence and uniqueness of a pathwise weak solution with values in $H^2(\mathbb{T})$. In the second part of the paper, we analyze the long-time behavior of these solutions. This study is particularly subtle as the presence of jumps in time can significantly influence the asymptotics. We show, using the techniques of Maslowski and Seidler, that, provided the frictional damping coefficient $γ> 0$ is sufficiently large, the system influenced by square-integrable jumps admits an invariant measure in $H^2(\mathbb{T})$.

Well-posedness and existence of an invariant measure for the linearly-damped KdV equations driven by a jump noise

Abstract

In this paper, we investigate the linearly damped KdV equation on the one-dimensional torus , perturbed by a multiplicative Lévy noise. For any damping coefficient , we establish the existence and uniqueness of a pathwise weak solution with values in . In the second part of the paper, we analyze the long-time behavior of these solutions. This study is particularly subtle as the presence of jumps in time can significantly influence the asymptotics. We show, using the techniques of Maslowski and Seidler, that, provided the frictional damping coefficient is sufficiently large, the system influenced by square-integrable jumps admits an invariant measure in .
Paper Structure (11 sections, 27 theorems, 223 equations)

This paper contains 11 sections, 27 theorems, 223 equations.

Table of Contents

  1. Introduction
  2. Mathematical Framework
  3. Main Results
  4. Existence of pathwise solution
  5. Galerkin Scheme
  6. Moment estimates
  7. Tightness
  8. Passage to the limit $N \to \infty$
  9. Global pathwise solution
  10. Existence of invariant measure
  11. Proof of Convergence Lemma \ref{['cor:L^1 convergence of K']}

Key Result

Theorem 3.1

Assume that we are given an $\mathcal{F}_0$-measurable random variable $u_0: \Omega \to H^2(\mathbb{T})$ such that $u_0 \in L^8(\Omega, \mathcal{F}_0, \mathbb{P}; L^2(\mathbb{T})) \cap L^2(\Omega, \mathcal{F}_0, \mathbb{P}; H^2(\mathbb{T}))$. We also assume that $G$ and $K$ satisfy the assumptions a

Theorems & Definitions (49)

  • Definition 1
  • Remark 2.1
  • Definition 2: Martingale solutions to \ref{['eqn: skdv']}
  • Definition 3: Pathwise solutions to \ref{['eqn: skdv']}
  • Theorem 3.1
  • Theorem 3.2
  • Theorem 4.1: $L^8_\mathbb{P} L^\infty_t L^2_x$ regularity for $u_\epsilon^N$
  • proof
  • Theorem 4.2: $L^2_\mathbb{P} L^\infty_t H^2_x$ estimates for $u_\epsilon^N$
  • proof
  • ...and 39 more