On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity

In the first part of the paper, we study the Cauchy problem for the advection-diffusion equation $\partial_t v + \text{div }(v\boldsymbol{b} ) = \Delta v$ associated with a merely integrable, divergence-free vector field $\boldsymbol{b}$ defined on the torus. We first introduce two notions of solutions (distributional and parabolic), recalling the corresponding available results of existence and uniqueness. Then, we establish a regularity criterion, which in turn provides uniqueness for distributional solutions. This is motivated by the recent results in [31] where the authors showed non-uniqueness of distributional solutions to the advection-diffusion equation despite the parabolic one is unique. In the second part of the paper, we precisely describe the vanishing viscosity scheme for the transport/continuity equation drifted by $\boldsymbol{b}$, i.e. $\partial_t u + \text{div }(u\boldsymbol{b} ) = 0$. Under Sobolev assumptions on $\boldsymbol{b} $, we give two independent proofs of the convergence of such scheme to the Lagrangian solution of the transport equation. The first proof slightly generalizes the original one of [21]. The other one is quantitative and yields rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration schemes, as shown in recent works [10, 31, 32, 33], and rules out the possibility of anomalous dissipation.