The stochastic primitive equations with non-isothermal turbulent pressure

Abstract

In this paper we introduce and study the primitive equations with non-isothermal turbulent pressure and transport noise. They are derived from the Navier-Stokes equations by employing stochastic versions of the Boussinesq and the hydrostatic approximations. The temperature dependence of the turbulent pressure can be seen as a consequence of an additive noise acting on the small vertical dynamics. For such model we prove global well-posedness in H 1 where the noise is considered in both the Itô and Stratonovich formulations. Compared to previous variants of the primitive equations, the one considered here present a more intricate coupling between the velocity field and the temperature. The corresponding analysis is seriously more involved than in the deterministic setting. Finally, the continuous dependence on the initial data and the energy estimates proven here are new, even in the case of isothermal turbulent pressure.

Global existence for the stochastic Boussinesq equations with transport noise and small rough data

“…where U = (u, ρ) := (U 1 , U 2 , U 3 , U 4 ), AU = (∆u, ∆ρ), B(U ) := −P(u • ∇U ), G(U ) = P(ρe 3 , 0) = P(U 4 e 3 , 0), σ(U ) = (Pσ (1) (U ), σ (2) (U )). Notice that when U 0 dx = 0, ∇ • b = 0, and σ satisfies condition (2.11) below, one can integrate system (2.4) in T 3 to obtain that U dx = 0 for all t ≥ 0.…”

“…We are interested in the stochastic Boussinesq equations on a 3D torus T 3 du = (∆u − P(u • ∇u − ρe 3 ))dt + P(b • ∇u + σ (1) (u, ρ))dW t , ∇ • u = 0, dρ = (∆ρ − u • ∇ρ)dt + (b • ∇ρ + σ (2) (u, ρ))dW t .…”

“…Here u = u(x, t) is the velocity vector field, ρ = ρ(x, t) is the scalar temperature or density of fluid, P is the Leray projection (see (2.3)), and W t is a cylindrical Wiener process valued in some separable Hilbert space and the corresponding stochastic integral is in the Itô sense. The term (P(b • ∇u), b • ∇ρ)dW t represents the transport noise and (P(σ (1) (u, ρ)), σ (2) (u, ρ))dW t is the multiplicative noise. The main result of the paper is the global well-posedness of the system (1.1) in L p (Ω; L p (T 3 )) with p > 5 and small initial data.…”

“…The transport noise represent a wider and more general class of noise than the multiplicative noise. In the study of turbulent flows, the transport noise has been introduced in [26,27] and widely studied in many different SPDEs, see for example [2,3,5,10,19,35,36] and refereces therein. It is worthwhile to mention that with transport noise we are able to prove the existence and uniqueness of solutions when (u 0 , ρ 0 ) ∈ L p (Ω; L p (T 3 )) with p > 5.…”

Pathwise Solutions for Stochastic Hydrostatic Euler Equations and Hydrostatic Navier-Stokes Equations Under the Local Rayleigh Condition

“…It can also offer predictions of not only a realistic trajectory but also the associated uncertainties. Along this line of research, the stochastic PEs with full viscosity was investigated in 2D [25,26] and in 3D [2,3,9,19,20]. With only horizontal viscosity, the global existence and uniqueness of strong solutions have been established in [50].…”