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

Abstract

We study the Cauchy problem for the advection-diffusion equation ${\partial }_{t}u+\mathrm{div}(ub)=\Delta u$ associated with a merely integrable divergence-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 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.

arXiv commentPart of these notes was originally contained in an preliminary version of arXiv:2107.03659

References in corpus (4)

• Convex integration solutions to the transport equation with full dimensional concentration
• Advection diffusion equations with Sobolev velocity field
• On the advection-diffusion equation with rough coefficients: weak solutions and vanishing viscosity
• Quantitative stability estimates for Fokker-Planck equations

Cited by in corpus (3)

• On the conservation of helicity by weak solutions of the 3D Euler and inviscid MHD equations
• Enhanced Dissipation via time-modulated velocity fields
• A regularity result for the Fokker-Planck equation with non-smooth drift and diffusion