We address a question concerning the issue of existence to a Boussinesq type system with a heat source. The problem is studied in the whole two dimensional plane and the heat source is a measure transported by the flow. For arbitrary initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructing solutions and make us work in a non-standard framework of inhomogeneous Besov spaces of the $L^\infty(0,T;B^s_{p,\infty})$-type. Application of the Lagrangian coordinates yields uniqueness omitting difficulties with comparison of measures.

中文翻译：

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带测度强迫的 Boussinesq 系统

我们解决了一个关于具有热源的 Boussinesq 型系统的存在问题的问题。该问题是在整个二维平面上研究的，热源是由流动传输的量度。对于任意初始数据，我们证明了唯一正则解在时间上的全局存在性。作为热源的度量限制了构建解决方案的规律性，并使我们在 $L^\infty(0,T;B^s_{p,\infty})$ 类型的非齐次 Besov 空间的非标准框架中工作。拉格朗日坐标的应用产生了唯一性，忽略了度量比较的困难。