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In this paper we address the temperature patch problem of the 2D viscousBoussinesq system without heat diffusion term. The temperature satisfies thetransport equation and the initial data of temperature is given in the form ofnon-constant patch, usually called the temperature front initial data.Introducing a good unknown and applying the method of striated estimates, weprove that our partially viscous Boussinesq system admits a unique globalregular solution and the initial $C^{k,\gamma}$ and $W^{2,\infty}$ regularityof 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, thisnaturally extends the previous work by Danchin $\&$ Zhang (2017) and Gancedo$\&$ Garc\'ia-Ju\'arez (2017). In the proof of the persistence result of higherboundary regularity, we introduce the striated type Besov space$\mathcal{B}^{s,\ell}_{p,r,W}(\mathbb{R}^d)$ and establish a series of refinedstriated estimates in such a function space, which may have its own interest.

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