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Structure preservation and emergent dissipation in stochastic wave equations with transport noise

Chang Liu, Dejun Luo

TL;DR

The paper analyzes nonlinear wave equations perturbed by transport-type noise acting on either displacement or velocity, showing that small-scale stochastic advection can yield either a structure-preserving deterministic limit or an emergent Laplacian damping resembling Westervelt-type acoustics. It establishes well-posedness for the displacement-noise model via a direct Galerkin approach with pathwise uniqueness, while proving weak existence for the velocity-noise model through compactness, leaving pathwise uniqueness open. Under a covariance-scaling regime, the authors prove that displacement-noise converges to the deterministic wave equation, whereas velocity-noise introduces a damped term $\kappa\Delta\partial_t u$ in the limit; they also provide a quantitative convergence rate in $2$ and $3$ dimensions for the velocity-noise case. The results highlight how stochastic fluctuations at small scales can induce large-scale dissipation and connect stochastic perturbations to classical dissipative models in nonlinear acoustics. The techniques combine energy estimates, Galerkin approximations, compactness arguments, and careful analysis of stochastic convolutions with scaled covariances, yielding both qualitative and, under extra regularity, quantitative convergence results.

Abstract

We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.

Structure preservation and emergent dissipation in stochastic wave equations with transport noise

TL;DR

The paper analyzes nonlinear wave equations perturbed by transport-type noise acting on either displacement or velocity, showing that small-scale stochastic advection can yield either a structure-preserving deterministic limit or an emergent Laplacian damping resembling Westervelt-type acoustics. It establishes well-posedness for the displacement-noise model via a direct Galerkin approach with pathwise uniqueness, while proving weak existence for the velocity-noise model through compactness, leaving pathwise uniqueness open. Under a covariance-scaling regime, the authors prove that displacement-noise converges to the deterministic wave equation, whereas velocity-noise introduces a damped term in the limit; they also provide a quantitative convergence rate in and dimensions for the velocity-noise case. The results highlight how stochastic fluctuations at small scales can induce large-scale dissipation and connect stochastic perturbations to classical dissipative models in nonlinear acoustics. The techniques combine energy estimates, Galerkin approximations, compactness arguments, and careful analysis of stochastic convolutions with scaled covariances, yielding both qualitative and, under extra regularity, quantitative convergence results.

Abstract

We study nonlinear wave equations perturbed by transport noise acting either on the displacement or on the velocity. Such noise models random advection and, under suitable scaling of space covariance, may generate an effective dissipative term. We establish well-posedness in both cases and analyse the associated scaling limits. When the noise acts on the displacement, the system preserves its original structure and converges to the deterministic nonlinear wave equation, whereas if it acts on the velocity, the rescaled dynamics produce an additional Laplacian damping term, leading to a stochastic derivation of a Westervelt-type acoustic model.
Paper Structure (11 sections, 17 theorems, 164 equations)

This paper contains 11 sections, 17 theorems, 164 equations.

Table of Contents

  1. Introduction
  2. Well-posedness
  3. Scaling limits
  4. Organization of the paper
  5. Preliminaries
  6. Well-posedness for \ref{['eq-SWE']}
  7. Weak existence for \ref{['eq-SWE2']}
  8. Scaling limit for \ref{['eq-SWE']}
  9. Scaling limit for \ref{['eq-SWE2']}
  10. Weak convergence
  11. Quantitative estimate

Key Result

Theorem 1.2

Given any $(u_0,v_0) \in H^1 \times L^2$ and $T>0$, equation eq-SWE admits a pathwise unique weak solution $(u,v)$ on the interval $[0,T]$. More precisely, given a filtered probability space $(\Omega,\mathcal{F},(\mathcal{F}_t),\mathbb{P})$ and a family of independent $(\mathcal{F}_t)$-Brownian moti Furthermore, the weak solution $(u,v)$ satisfies the following uniform bound:

Theorems & Definitions (37)

  • Theorem 1.2
  • Definition 1.3
  • Theorem 1.4
  • Remark 1.5
  • Theorem 1.7
  • Theorem 1.8
  • Remark 1.9
  • Example 2.1
  • Example 2.2
  • Lemma 2.3
  • ...and 27 more