On the Well-Posedness of Stochastic Boussinesq Equations with Transport Noise

Abstract

The Boussinesq equations are fundamental in meteorology. Among other aspects, they aim to model the process of front formation. We use the approach presented in [Hol15] to introduce stochasticity into the incompressible Boussinesq equations. This is, we introduce cylindrical transport noise in a way that the geometric properties in the Euler-Poincaré formulation are preserved. One of our main results establishes the local well-posedness of regular solutions for these new stochastic Boussinesq equations. We also construct a blow-up criterion and derive some general estimates, which are crucial for showing well-posedness of a wide range of similar SPDEs.

The Burgers’ equation with stochastic transport: shock formation, local and global existence of smooth solutions

“…ThereNow, consider the space W T := L 2 (Ξ; C([0, T ]; H s (T, R))). One can show that Υ is a contraction on W T by following the same arguments in [CFH17,AOBdL18]. Therefore, by applying Picard's iteration argument, one can construct a local solution.…”

“…where the coefficients a(x), b(x) are smooth and bounded. We state the main result of this section: We will provide a sketch of the proof, which follows closely the approach developed in [AOBdL18,CFH17]. For clarity of exposition, let us divide the argument into several steps.…”

“…[Gol85]), one can prove that for positive times T > δ > 0, each term in the mild equation (4.4) enjoys higher regularity, namely, u ∈ L 2 (Ξ; C([δ, T ]; H s+2 )) for every T > δ > 0. All the computations can be carried out easily by mimicking the same ideas as in [CFH17,AOBdL18].…”

“…For instance, when deriving a priori estimates, certain high order terms appear, which need to be treated carefully. Recently, the same type of multiplicative noise has been treated for the Euler equation [CFH17, FL18] and the Boussinesq system [AOBdL18], whose techniques we follow closely in our proof of local well-posedness. There is also the recent work [HNS18] showing the local well-posedness of weak solutions in the viscous Burgers' equation (ν > 0) driven by rough paths in the transport velocity.…”

“…Proof. For this, we refer the reader to [AOBdL18,CFH17]. It suffices to defineū = u 1 − u 2 , and perform standard estimates for the evolution of the L 2 norm ofū.…”