LOCAL WELL-POSEDNESS FOR THE GREAT LAKE EQUATION WITH TRANSPORT NOISE.

This work is a continuation of the authors' work in [11]. In [11] the equation satisfied by an incompressible uid 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 uid in a basin with a free upper surface and a spatially varying bottom topography (see [23]). 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 [11], 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. [ABSTRACT FROM AUTHOR]