Kiselev与Nazarov主题的变奏：沿非无散度BMO场的非局部输运-扩散的Hölder估计
Variation on a theme by Kiselev and Nazarov: H{ö}lder estimates for non-local transport-diffusion, along a non-divergence-free BMO field
AI总结：
本文利用对偶演化技术，证明了在扩散主导小尺度且散度负部足够小的条件下，带有分数阶扩散算子和一般BMO输运场的输运-扩散方程的一致Hölder正则性估计，放宽了输运场需有界的限制。
摘要：
我们证明了带有分数阶扩散算子和一般BMO中输运场的输运-扩散方程的一致Hölder正则性估计，只要扩散阶数在小尺度上主导输运项；我们唯一的要求是散度的负部在某个临界Lebesgue空间中足够小。与L.Silvestre（2012）的著名结果相比，我们的输运场不需要是有界的。如果输运场是Hölder连续的，在超临界情况下也能获得类似结果。我们的证明受A.Kiselev和F.Nazarov（2010）启发，并基于对偶演化技术。其思想是在与分数阶扩散算子耦合时，在对偶守恒律下传播原子性质（即在Lebesgue空间中的局部化和可积性）。
英文摘要：
We prove uniform Hölder regularity estimates for a transport-diffusion equation with a fractional diffusion operator,and a general advection field in BMO, as long as the order of the diffusion dominates the transport term at small scales;our only requirement is the smallness of the negative part of the divergence in some critical Lebesgue space. In comparison to a celebrated result by L.Silvestre (2012), our advection field does not need to be bounded. A similar result can be obtained in the super-critical case if the advection field is Hölder continuous. Our proof is inspired by A.Kiselev and F.Nazarov (2010) and is based on the dual evolution technique. The idea is to propagate an atom property (i.e. localization and integrability in Lebesgue spaces) under the dual conservation law, when it is coupled with the fractional diffusion operator.
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