Weak and parabolic solutions of advection–diffusion equations with rough velocity field
Paolo Bonicatto, Gennaro Ciampa, Gianluca Crippa
Abstract
Abstract We study the Cauchy problem for the advection–diffusion equation $$\partial _t u + {{\,\mathrm{\textrm{div}}\,}}(u\varvec{b}) = \Delta u$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:msub> <mml:mi>∂</mml:mi> <mml:mi>t</mml:mi> </mml:msub> <mml:mi>u</mml:mi> <mml:mo>+</mml:mo> <mml:mrow> <mml:mspace/> <mml:mtext>div</mml:mtext> <mml:mspace/> </mml:mrow> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>u</mml:mi> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> <mml:mo>)</mml:mo> </mml:mrow> <mml:mo>=</mml:mo> <mml:mi>Δ</mml:mi> <mml:mi>u</mml:mi> </mml:mrow> </mml:math> associated with a merely integrable divergence-free vector field $$\varvec{b}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mrow> <mml:mi>b</mml:mi> </mml:mrow> </mml:math> 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.