Bonicatto, Paolo and Ciampa, Gennaro and Crippa, Gianluca. (2021) On the advectiondiffusion equation with rough coefficients: weak solutions and vanishing viscosity. Preprints Fachbereich Mathematik, 2021 (19).

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Abstract
In the first part of the paper, we study the Cauchy problem for the advectiondiffusion equation $\partial_t v + \text{div }(v\boldsymbol{b} ) = \Delta v$ associated with a merely integrable, divergencefree 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 nonuniqueness of distributional solutions to the advectiondiffusion 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 nonuniqueness 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.
Faculties and Departments:  05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa) 12 Special Collections > Preprints Fachbereich Mathematik 

UniBasel Contributors:  Crippa, Gianluca 
Item Type:  Preprint 
Publisher:  Universität Basel 
Language:  English 
edoc DOI:  
Last Modified:  03 Nov 2021 07:56 
Deposited On:  03 Nov 2021 07:56 
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