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• W3185023325 abstract "We present sharp conditions on divergence-free drifts in Lebesgue spaces for the passive scalar advection–diffusion equation $$begin{aligned} partial _t theta - Delta theta + b cdot nabla theta = 0, end{aligned}$$ to satisfy local boundedness, a single-scale Harnack inequality, and upper bounds on fundamental solutions. We demonstrate these properties for drifts b belonging to $$L^q_t L^p_x$$ , where $$frac{2}{q} + frac{n}{p} < 2$$ , or $$L^p_x L^q_t$$ , where $$frac{3}{q} + frac{n-1}{p} < 2$$ . For steady drifts, the condition reduces to $$b in L^{frac{n-1}{2}+}$$ . The space $$L^1_t L^infty _x$$ of drifts with ‘bounded total speed’ is a borderline case and plays a special role in the theory. To demonstrate sharpness, we construct counterexamples whose goal is to transport anomalous singularities into the domain ‘before’ they can be dissipated." @default.
• W3185023325 created "2021-08-02" @default.
• W3185023325 creator A5014900083 @default.
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• W3185023325 date "2023-08-03" @default.
• W3185023325 modified "2023-10-11" @default.
• W3185023325 title "Regularity Properties of Passive Scalars with Rough Divergence-Free Drifts" @default.
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• W3185023325 doi "https://doi.org/10.1007/s00205-023-01896-3" @default.
• W3185023325 hasPublicationYear "2023" @default.
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