GLOBAL REGULARITY OF NONDIFFUSIVE TEMPERATURE FRONTS FOR THE TWO-DIMENSIONAL VISCOUS BOUSSINESQ SYSTEM

In this paper we address the temperature patch problem of the two-dimensional 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 nonconstant patch, usually called the temperature front initial data. Introducing a good unknown and applying the method of striated estimates, we prove that the partially viscous Boussinesq system admits a unique global regular solution and the initial C^k,\gamma and W^2,∞ regularity of the temperature front boundary with k \in \BbbZ ⁺ = \{ 1, 2, \cdot \cdot \cdot \} and \gamma \in (0, 1) will be preserved for all the time. In particular, this naturally extends the previous work by Danchin and Zhang [Comm. Partial Differential Equations, 42 (2017), pp. 68-99] and Gancedo and Garc\'{\i}a-Ju\'arez [Ann. PDE, 3 (2017), 14]. In the proof of the persistence result of higher boundary regularity, we introduce the striated type Besov space \scrB _p,r^s,\ell_\scrW (\BbbR ^d) and establish a series of refined striated estimates in such a function space, which may have its own interest.