Global regularity of non-diffusive temperature fronts for the 2D viscous Boussinesq system

Abstract

In this paper we address the temperature patch problem of the 2D viscous Boussinesq system without heat diffusion term. The temperature satisfies the transport equation and the initial data of temperature is given in the form of non-constant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that our partially viscous Boussinesq system admits a unique global regular solution and the initial C k,γ and W 2,∞ regularity of the temperature front boundary with k ∈ Z + = {1, 2, • • • } and γ ∈ (0, 1) will be preserved for all the time. In particular, this naturally extends the previous work by Danchin & Zhang (2017) and Gancedo & García-Juárez (2017). In the proof of the persistence result of higher boundary regularity, we introduce the striated type Besov space B s,ℓ p,r,W (R d ) and establish a series of refined striated estimates in such a function space, which may have its own interest.

Global derivation of a Boussinesq-Navier-Stokes type system from fluid-kinetic equations

“…The existence theory (in the less-diffusivity case) has been for instance developed in [55,15,1,50,20,51], while stability of hydrodynamic equilibria is studied in [25,74,65,26]. We also refer to the so-called temperature patch (or front) problem adressed in [21,33,16]. Note that there is no diffusivity in the transport equation on the density, which makes the mathematical analysis much more challenging than the thermal diffusion case.…”