Well-posedness Properties for a Stochastic Rotating Shallow Water Model

Abstract

In this paper, we study the well-posedness properties of a stochastic rotating shallow water system. An inviscid version of this model has first been derived in [17] and the noise is chosen according to the Stochastic Advection by Lie Transport theory presented in [17]. The system is perturbed by noise modulated by a function that is not Lipschitz in the norm where the wellposedness is sought. We show that the system admits a unique maximal 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.

Bayesian Inference for Fluid Dynamics: A Case Study for the Stochastic Rotating Shallow Water Model

“…In our case X is given by a pathwise solution of the stochastic rotating shallow water system computed on a staggered grid. We have proven in [8] that such a solution exists. 1 The nonlinear filtering problem consists in finding the best approximation of the posterior distribution of the signal X t given the observations Z 1 , Z 2 , .…”

“…The rotating shallow water model (RSW) is a classical nonlinear fluid dynamics model which contains key aspects of the oceanic and atmospheric dynamics. A detailed analytical description of this model has been provided in [8]. From a numerical perspective, challenges are generated especially by the nonlinear advective terms.…”

“…Note that in[8] we work with infinite dimensional function spaces, while here the state space will be R d X 2. For a mathematical introduction on the subject see[1].…”

Existence and Uniqueness of Maximal Solutions to a 3D Navier-Stokes Equation with Stochastic Lie Transport

“…There has been limited progress in proving well-posedness for this class of equations: Crisan, Flandoli and Holm [5] have shown local existence and uniqueness for the 3D Euler Equation on the torus, whilst Crisan and Lang ( [6], [17], [18]) demonstrated the same result for the Euler, Rotating Shallow Water and Great Lake Equations on the torus once more. Whilst this represents a strong start in the theoretical analysis (alongside works for SPDEs with general transport noise e.g.…”

Existence and Uniqueness of Maximal Solutions to SPDEs with Applications to Viscous Fluid Equations

“…The significance of such equations in modelling, numerical schemes and data assimilation continues to be well documented, see ( [6], [7], [32], [31] [49], [37], [9], [15], [36], [5], [20], [21], [2]). In contrast there has been limited progress in proving well-posedness for this class of equations: Crisan, Flandoli and Holm [8] have shown the existence and uniqueness of maximal solutions for the 3D Euler Equation on the torus, whilst Crisan and Lang ([10], [11], [12]) extended the well-posedness theory for the Euler, Rotating Shallow Water and Great Lake Equations on the torus once more. Alonso-Orán and Bethencourt de León [1] show the same properties for the Boussinesq Equations again on the torus, whilst Brzeźniak and Slavík [4] demonstrate these properties on a bounded domain for the Primitive Equations but for a specific choice of transport noise which facilitates their analysis.…”