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

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 $C^{k,\gamma}$ and $W^{2,\infty}$ regularity of the temperature front boundary with $k\in \mathbb{Z}^+ = \{1,2,\cdots\}$ and $\gamma\in (0,1)$ will be preserved for all the time. In particular, this naturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo $\&$ Garc\'ia-Ju\'arez (2017). In the proof of the persistence result of higher boundary regularity, we introduce the Besov-type space $B^{s,n}_{p,r}(\mathbb{R}^d)$ and its striated type space $\mathcal{B}^{s,\ell,n}_{p,r,W}(\mathbb{R}^d)$ and establish a series of refined striated estimates in these function spaces, which may have its own interest.