Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The primitive equations with stochastic wind driven boundary conditions

Tim Binz, Matthias Hieber, Amru Hussein, Martin Saal

TL;DR

The paper addresses the well-posedness of the three-dimensional stochastic primitive equations with stochastic wind-driven boundary data on a cylindrical domain. It combines a Da Prato–Zabczyk-type treatment of stochastic boundary conditions with anisotropic maximal regularity for the hydrostatic Stokes operator, using a hydrostatic Neumann map to convert boundary noise into interior forcing and a decomposition V=V_b+Z_b+Z_f. The main result is a unique local pathwise solution in the critical $L^q_t$-$H^{-1,p}_zL^p_{xy}$ setting, with detailed nonlinear estimates and regularity properties for the stochastic and deterministic components. This provides a rigorous foundation for stochastic boundary-driven geophysical flows and offers a framework that could extend to other hydrostatic SPDEs with stochastic boundaries.

Abstract

The primitive equations for geophysical flows are studied under the influence of {\em stochastic wind driven boundary conditions} modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic $L^q_t$-$H^{-1,p}_zL^p_{xy}$-setting. This solution is constructed in critical spaces.

The primitive equations with stochastic wind driven boundary conditions

TL;DR

The paper addresses the well-posedness of the three-dimensional stochastic primitive equations with stochastic wind-driven boundary data on a cylindrical domain. It combines a Da Prato–Zabczyk-type treatment of stochastic boundary conditions with anisotropic maximal regularity for the hydrostatic Stokes operator, using a hydrostatic Neumann map to convert boundary noise into interior forcing and a decomposition V=V_b+Z_b+Z_f. The main result is a unique local pathwise solution in the critical - setting, with detailed nonlinear estimates and regularity properties for the stochastic and deterministic components. This provides a rigorous foundation for stochastic boundary-driven geophysical flows and offers a framework that could extend to other hydrostatic SPDEs with stochastic boundaries.

Abstract

The primitive equations for geophysical flows are studied under the influence of {\em stochastic wind driven boundary conditions} modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic --setting. This solution is constructed in critical spaces.

Paper Structure

This paper contains 15 sections, 8 theorems, 157 equations.

Table of Contents

  1. Introduction
  2. First formulation of the stochastic primitive equations
  3. Linear deterministic theory
  4. Anisotropic $H_{z}^{s,p}H^{m,p}_{xy}$-spaces
  5. Solenoidal spaces and the hydrostatic Helmholtz projection
  6. The hydrostatic Stokes operator
  7. Maximal $L^q_t$-regularity and interpolations spaces
  8. The hydrostatic Neumann map
  9. Linear stochastic theory
  10. Cylindrical Wiener processes and stochastic convolutions
  11. Rewriting the stochastic boundary condition as a forcing term
  12. Local pathwise well-posedness
  13. Notion of solution and main result
  14. Non-linear estimates
  15. Local well-posedness

Key Result

Proposition 3.4

For $s,m\in \mathbb{R}$ the operator $A_{p,\mathcal{H}}^{(s,m)}+\nu$ admits for any $\nu >0$ a bounded $H^\infty$-calculus of angle zero. In particular, $A_{p,\mathcal{H}}^{(s,m)}$ has the property of deterministic maximal $L^q_{\mu}$-regularity on any finite time interval $(0,T)$ for $T\in (0,\inft is an isomorphism Moreover, for $\theta\in (0,1)$ and $q\in (1,\infty)$

Theorems & Definitions (21)

  • Remark 3.1: Geometry of the spaces
  • Remark 3.2: Extension of operators from scalar to $\mathcal{H}$-valued spaces
  • Remark 3.3: Properties of the hydrostatic Stokes operator
  • Proposition 3.4: Bounded $H^\infty$-calculus
  • proof
  • Proposition 3.5: Hydrostatic Neumann map
  • proof : Proof of Proposition \ref{['prop:NeumannMap']}
  • Lemma 3.6
  • proof
  • Proposition 4.1: Stochastic maximal regularity for the hydrostatic Stokes equations
  • ...and 11 more