Local well-posedness for the great lake equation with transport noise

Abstract

This work is a continuation of the authors' work in [8]. In [8] the equation satisfied by an incompressible fluid with stochastic transport is analysed. Here we lift the incompressibility constraint. Instead we assume a weighted incompressibility condition. This condition is inspired by a physical model for a fluid in a basin with a free upper surface and a spatially varying bottom topography (see [20]). Moreover, we assume a different form of the vorticity to stream function operator that generalizes the standard Biot-Savart operator which appears in the Euler equation. These two properties are exhibited in the physical model called the great lake equation. For this reason we refer to the model analysed here as the stochastic great lake equation. Just as in [8], the deterministic model is perturbed with transport type noise. The new vorticity to stream function operator generalizes the curl operator and it is shown to have good regularity properties. We also show that the initial smoothness of the solution is preserved. The arguments are based on constructing a family of viscous solutions which is proved to be relatively compact and to converge to a truncated version of the original equation. Finally, we show that the truncation can be removed up to a positive stopping time.

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.…”

The Vlasov-Poisson and Vlasov-Poisson-Fokker-Planck systems in stochastic electromagnetic fields: local well-posedness

“…The 2D Euler equations in vorticity form subjected to transport noise was studied in, for example [9,16], where strong solutions with bounded vorticity were constructed (see also [15]). The aforementioned papers [15,16] belong to the socalled theory of "Stochastic Advection by Lie Transport" (SALT), see the foundational paper [34], as well as [1,17]. The work [11] studies the 3D primitive equations with transport noise.…”