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The viscous variational wave equation with transport noise

Peter H. C. Pang

TL;DR

This work analyzes the viscous variational wave equation with transport-type noise on the torus and proves pathwise global existence, uniqueness, and temporal continuity of solutions in $L^2(T)$. A two-level Galerkin framework with a nonlinear cutoff yields approximate equations for $(R,S)$, from which martingale solutions are obtained via Skorokhod--Jakubowski with Dudley maps; pathwise uniqueness is then established through renormalization and double commutator estimates, enabling a Gyöngy–Krylov/Yamada–Watanabe upgrade to strong solutions. The analysis blends deterministic variational-wave techniques with stochastic compactness and refined commutator controls for the transport noise, culminating in continuous-in-time $L^2$-paths for the limit. The results provide a rigorous foundation for well-posedness of stochastic viscoelastic-type wave dynamics under transport noise and set the stage for vanishing-viscosity limits and dissipative/conservative behavior studies in higher complexity models.

Abstract

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal continuity of solutions to this system in $L^2_x$. Martingale solutions are extracted from a two-level Galerkin approximation via the Skorokhod--Jakubowski theorem. We use the apparatus of Dudley maps to streamline this stochastic compactness method, bypassing the usual martingale identification argument. Pathwise uniqueness for the system is established through a renormalisation procedure that involves double commutator estimates and a delicate handling of noise and nonlinear terms. New model-specific commutator estimates are proven.

The viscous variational wave equation with transport noise

TL;DR

This work analyzes the viscous variational wave equation with transport-type noise on the torus and proves pathwise global existence, uniqueness, and temporal continuity of solutions in . A two-level Galerkin framework with a nonlinear cutoff yields approximate equations for , from which martingale solutions are obtained via Skorokhod--Jakubowski with Dudley maps; pathwise uniqueness is then established through renormalization and double commutator estimates, enabling a Gyöngy–Krylov/Yamada–Watanabe upgrade to strong solutions. The analysis blends deterministic variational-wave techniques with stochastic compactness and refined commutator controls for the transport noise, culminating in continuous-in-time -paths for the limit. The results provide a rigorous foundation for well-posedness of stochastic viscoelastic-type wave dynamics under transport noise and set the stage for vanishing-viscosity limits and dissipative/conservative behavior studies in higher complexity models.

Abstract

This article considers the variational wave equation with viscosity and transport noise as a system of three coupled nonlinear stochastic partial differential equations. We prove pathwise global existence, uniqueness, and temporal continuity of solutions to this system in . Martingale solutions are extracted from a two-level Galerkin approximation via the Skorokhod--Jakubowski theorem. We use the apparatus of Dudley maps to streamline this stochastic compactness method, bypassing the usual martingale identification argument. Pathwise uniqueness for the system is established through a renormalisation procedure that involves double commutator estimates and a delicate handling of noise and nonlinear terms. New model-specific commutator estimates are proven.
Paper Structure (18 sections, 25 theorems, 169 equations)

This paper contains 18 sections, 25 theorems, 169 equations.

Table of Contents

  1. Introduction
  2. Background
  3. Viscosity
  4. Strategy and outline of paper
  5. Definitions of solutions
  6. Galerkin approximations
  7. The Galerkin scheme
  8. A priori estimates and tightness
  9. Tightness of laws for $\{u_N\}$
  10. The Skorokhod argument
  11. Skorokhod representatives
  12. Consequences of the joint equality of laws
  13. Martingale solutions to the viscous variational wave equations
  14. Existence and uniqueness of pathwise solutions
  15. Commutator estimates for $c(\tilde{u})$
  16. ...and 3 more sections

Key Result

Theorem 1.2

Let $(R^0,S^0) \in \left( L^2(\mathbb{T}) \right)^2$ have finite $2p_0 > 4$ moments, and be such that $\int_\mathbb{T} R^0 - S^0 \,\mathrm{d} x = 0$. On Assumption sum:c_sigma, the viscous variational wave equation with transport noise eq:vvw -- eq:constitutive, with initial condition $(R^0,S^0)$ h

Theorems & Definitions (50)

  • Theorem 1.2
  • Definition 1.3: Martingale solutions
  • Definition 1.4: Pathwise solutions
  • Lemma 2.1
  • Remark 2.2
  • proof
  • Proposition 2.3: Well-posedness of the $k = \infty$ Galerkin scheme
  • proof
  • Lemma 2.4: Aubin--Lions lemma Simon:1987vn
  • Lemma 2.5
  • ...and 40 more