Abstract
We discuss \(L^p\) integrability estimates for the solution u of the advection–diffusion equation \(\partial _t u + \mathrm{div}\,(bu) = \Delta u\), where the velocity field \(b \in L^r_t L^q_x\). We first summarize some classical results proving such estimates for certain ranges of the exponents r and q. Afterwards we prove the optimality of such ranges by means of new original examples.
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Dedicated to Alberto Bressan on the occasion of his 60th birthday.
This article is part of the topical collection “Hyperbolic PDEs, Fluids, Transport and Applications: Dedicated to Alberto Bressan for his 60th birthday” guest edited by Fabio Ancona, Stefano Bianchini, Pierangelo Marcati, Andrea Marson.
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Bianchini, S., Colombo, M., Crippa, G. et al. Optimality of integrability estimates for advection–diffusion equations. Nonlinear Differ. Equ. Appl. 24, 33 (2017). https://doi.org/10.1007/s00030-017-0455-9
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DOI: https://doi.org/10.1007/s00030-017-0455-9
Keywords
- Advection–diffusion equations
- Parabolic equations
- Integrability estimates and their optimality
- Duhamel formula
- Self-similar solutions