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We study the Cauchy problem for the advection-diffusion equation $\partial_tu + \mathrm{div} (u b ) = \Delta u$ associated with a merely integrabledivergence-free vector field $b$ 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 theinitial datum. We offer an up-to-date picture of the available resultsscattered in the literature, and we include some original proofs. We alsopropose some open problems, motivated by very recent results which showill-posedness of the equation in certain regimes of integrability via convexintegration schemes.

Weak and parabolic solutions of advection-diffusion equations with rough velocity field