We study the Cauchy problem for the advection–diffusion equation \(\partial _t u + {{\,\mathrm{\textrm{div}}\,}}(u\varvec{b}) = \Delta u\) associated with a merely integrable divergence-free vector field \(\varvec{b}\) defined on the torus. We discuss existence, regularity and uniqueness results for distributional and parabolic solutions, in different regimes of integrability both for the vector field and for the initial datum. We offer an up-to-date picture of the available results scattered in the literature, and we include some original proofs. We also propose some open problems, motivated by very recent results which show ill-posedness of the equation in certain regimes of integrability via convex integration schemes.

我们研究平流扩散方程的柯西问题\(\partial _t u + {{\,\mathrm{\textrm{div}}\,}}( u\varvec{b}) = \Delta u\) 与仅可积的无散度向量场 \(\varvec{b}\) 相关联 定义在圆环上。我们讨论向量场和初始数据的不同可积机制中分布解和抛物线解的存在性、规律性和唯一性结果。我们提供了散布在文献中的可用结果的最新图片，并且包括一些原始证明。我们还提出了一些开放性问题，这些问题的动机是最近的结果表明，通过凸积分方案，方程在某些可积范围内是不适定的。