Bianchini, Stefano; Colombo, Maria; Crippa, Gianluca; Spinolo, Laura V. Optimality of integrability estimates for advection-diffusion equations. (English) Zbl 1373.35155 NoDEA, Nonlinear Differ. Equ. Appl. 24, No. 4, Paper No. 33, 19 p. (2017). Summary: We discuss \(L^p\) integrability estimates for the solution \(u\) of the advection-diffusion equation \(\partial_tu+\mathrm{div }(bu)=\Delta u\), where the velocity field \(b\in L^r_tL^q_x\). 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. Cited in 10 Documents MSC: 35K10 Second-order parabolic equations 35B45 A priori estimates in context of PDEs 35R05 PDEs with low regular coefficients and/or low regular data 58J35 Heat and other parabolic equation methods for PDEs on manifolds Keywords:advection-diffusion equations; parabolic equations; integrability estimates and their optimality; Duhamel formula; self-similar solutions PDFBibTeX XMLCite \textit{S. Bianchini} et al., NoDEA, Nonlinear Differ. Equ. Appl. 24, No. 4, Paper No. 33, 19 p. (2017; Zbl 1373.35155) Full Text: DOI arXiv References: [1] Evans, L.C.: Partial differential equations. In: Graduate Studies in Mathematics, vol. 19, 2nd edn. American Mathematical Society, Providence (2010) · Zbl 1194.35001 [2] Ladyženskaja, O.A., Ural’ceva, N.N.: Linear and Quasilinear Elliptic Equations. Academic Press, New York (1968) [3] Ladyženskaja, O.A., Solonnikov, V.A., Ural’ceva, N.N.: Linear and quasilinear equations of parabolic type. In: Translations of Mathematical Monographs, vol. 23. American Mathematical Society, Providence (1968) · Zbl 1366.35066 [4] Mooney, C.: Finite time blowup for parabolic systems in two dimensions. Arch. Ration. Mech. Anal. 223(3), 1039-1055 (2017) · Zbl 1366.35066 · doi:10.1007/s00205-016-1052-5 [5] Serrin, J.: On the interior regularity of weak solutions of the Navier-Stokes equations. Arch. Ration. Mech. Anal. 9(1), 187-195 (1962) · Zbl 0106.18302 · doi:10.1007/BF00253344 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.