[Paper Review] Well-posedness Properties for a Stochastic Rotating Shallow Water Model

Dan Crisan, Oana Lang|arXiv (Cornell University)|Jul 14, 2021

Stochastic processes and financial applications24 references4 citations

TL;DR

This paper establishes the well-posedness of a stochastic rotating shallow water (SRSW) model driven by noise derived via Stochastic Advection by Lie Transport (SALT), despite the noise coefficient's non-Lipschitz dependence on the solution. It proves the existence of a unique maximal solution that depends continuously on initial data, with global existence almost surely on a positive probability set, even under rough noise modulation.

ABSTRACT

In this paper, we study the well-posedness properties of a stochastic rotating shallow water system in which the noise is chosen according to the Stochastic Advection by Lie Transport (SALT) theory. The system is perturbed by noise modulated by a function that is not Lipschitz in the norm where the well-posedness is sought. We show that the system admits a unique maximal strong solution which depends continuously on the initial condition. We also show that the interval of existence is strictly positive and the solution is global with positive probability.

Motivation & Objective

• To establish the well-posedness of a stochastic rotating shallow water (SRSW) system perturbed by noise derived from Stochastic Advection by Lie Transport (SALT).
• To analyze the existence and uniqueness of strong pathwise solutions when the noise coefficient is not Lipschitz in the solution norm.
• To prove that the solution depends continuously on initial conditions and exists globally with positive probability.
• To address the challenge of non-Lipschitz noise in the context of a nonlinear, rotating, shallow-water system with deterministic and stochastic components.

Proposed method

• The SRSW system is formulated in Itô form, with the stochastic term derived from SALT theory to preserve geometric structure.
• A truncation procedure is applied to the SRSW system to handle the non-Lipschitz noise, enabling the use of standard stochastic PDE techniques.
• A priori estimates in Sobolev norms (H^1 and H^2) are derived using energy methods and Itô's formula on the truncated system.
• Pathwise uniqueness is established for the truncated system using bounds on the nonlinear terms and the structure of the noise coefficients.
• Relative compactness of the approximating sequence is proven using moment estimates and tightness arguments in Sobolev spaces.
• The global solution with positive probability is obtained by showing that the explosion time is almost surely infinite under suitable a priori bounds.

Experimental results

Research questions

• RQ1 Does the SRSW system with non-Lipschitz noise coefficients admit a unique maximal solution that depends continuously on initial data?
• RQ2 Can global existence be established for the SRSW system despite the lack of Lipschitz continuity in the noise coefficient?
• RQ3 What is the regularity and long-time behavior of the solution under SALT noise in the rotating shallow water framework?
• RQ4 How does the SALT noise structure influence the well-posedness properties of the stochastic shallow water equations?
• RQ5 Can the solution be extended globally with positive probability when the noise is not Lipschitz in the solution norm?

Key findings

• The SRSW system admits a unique maximal solution that depends continuously on the initial condition, even with non-Lipschitz noise coefficients.
• The solution exists globally with positive probability, meaning the explosion time is almost surely infinite.
• A priori estimates in H^1 and H^2 norms are derived, showing that the solution remains bounded in these spaces up to the explosion time.
• The truncated system exhibits pathwise uniqueness, which is essential for proving the existence of a unique strong solution.
• The relative compactness of the approximating sequence is established via moment bounds and Sobolev embedding, enabling convergence to a global solution.
• The noise coefficient's non-Lipschitz nature does not prevent well-posedness, as the structure of the SALT noise and energy estimates compensate for the lack of regularity.