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Optimality of integrability estimates for advection-diffusion equations. (English) Zbl 1373.35155

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.

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
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[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
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