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Well-Posedness Properties for a Stochastic Rotating Shallow Water Model.

Dan Crisan1, Oana Lang1

  • 1Department of Mathematics, Imperial College London, London, UK.

Journal of Dynamics and Differential Equations
|November 18, 2024
PubMed
Summary
This summary is machine-generated.

This study investigates the stochastic rotating shallow water system, demonstrating a unique maximal solution exists and is continuous with initial conditions. The research confirms a strictly positive interval of existence and global solutions with positive probability.

Keywords:
SALT noiseStochastic rotating shallow water models

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Area of Science:

  • Fluid Dynamics
  • Stochastic Partial Differential Equations
  • Mathematical Physics

Background:

  • The stochastic rotating shallow water system is a crucial model in geophysical fluid dynamics.
  • Previous work established an inviscid version of this model.
  • The incorporation of noise is based on the Stochastic Advection by Lie Transport theory.

Purpose of the Study:

  • To analyze the well-posedness properties of the stochastic rotating shallow water system.
  • To investigate the existence and uniqueness of solutions under specific noise conditions.
  • To determine the nature of the solution's existence interval and global behavior.

Main Methods:

  • The study employs mathematical analysis to examine the system's behavior.
  • Well-posedness is investigated for a system perturbed by noise modulated by a non-Lipschitz function.
  • Techniques are used to establish the existence and continuity of solutions.

Main Results:

  • A unique maximal solution is shown to exist for the stochastic rotating shallow water system.
  • The solution demonstrates continuous dependence on the initial condition.
  • The interval of existence is proven to be strictly positive, with global solutions occurring with positive probability.

Conclusions:

  • The stochastic rotating shallow water system exhibits well-posedness properties.
  • The findings confirm the existence of a unique, continuous, and globally existing solution under specific stochastic perturbations.
  • This research contributes to the understanding of complex fluid dynamics models with stochastic influences.