The stochastic primitive equations with transport noise and turbulent pressure

Abstract

In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L 2regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well. Contents 1. Introduction 1 2. Preliminaries 6 3. Local and global existence in the strong-weak setting 8 4. L 2 -estimates for the linearized problem 14 5. Proof of the main results in the strong-weak setting 21 6. Local and global existence in the strong-strong setting 39 7. Inhomogeneous viscosity and conductivity 49 8. Transport noise of Stratonovich type 55 Appendix A. Kadlec's formulas 59 References 62

Nonlinear parabolic stochastic evolution equations in critical spaces part II

“…where in the case p = 2, κ = 0, θ ∈ (0, 1 2 ), we replace the range space by L 2 ( ; C([σ, T ]; X 1/2 )) (which is constant in θ ∈ (0, 1 2 )). Moreover, we set…”

“…According to Theorem 1.2 (2) this can only happen if P(σ < ∞) = 0, and thus σ = ∞ a.s. Similar considerations hold for Theorem 1.2 (1) and (3). Of course to obtain an a priori bound or energy estimate we need to use structural properties of a given SPDE.…”

“…where u : [0, ∞) × × T → R is the unknown process and w c t is a colored noise on T, i.e., an H λ (T)-cylindrical Brownian motion with λ ∈ ( 1 2 , 1), and h ∈ [1,3). The following is a special case of Theorems 7.2 and 7.3, where u 3 and |u| h are replaced by more general nonlinearities.…”

“…The estimate for v 2 follows by combining (3.9) for θ = 0 and θ ∈ ((1 + κ)/ p, 1/2) with Proposition 2.5. To prove (2), one can argue similarly using Proposition 2.5(2) instead of Proposition 2.5 (1).…”

“…In particular, for d = 2 new global well-posedness and regularity properties have been obtained using our new temporal weighted methods, and we do not know how to derive these results without our new methods. In [1] global well-posedness for the stochastic primitive equations with transport noise in 3d is obtained by applying our local existence theory from [3] and blow-up criteria Theorem 4.11. In [7,8] we will use our theory to build an L p -theory for stochastic reaction-diffusion equations with transport noise.…”