On the advection-diffusion equation with rough coefficients: Weak solutions and vanishing viscosity
On the advection-diffusion equation with rough coefficients: Weak solutions and vanishing viscosity
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关于具有粗糙系数的对流扩散方程：弱解和零粘性
DOI：
10.1016/j.matpur.2022.09.005
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发表时间：
2022-10-14
影响因子：
2.3
通讯作者：
Crippa, Gianluca
中科院分区：
文献类型：
--
作者：
Bonicatto, Paolo;Ciampa, Gennaro;Crippa, Gianluca
本文讨论了输运/连续性方程的粘性消失格式，其中偏微分方程tu +div（u B）= 0被无发散向量场B漂移.在关于B的一般Sobolev假设下，我们证明了该格式收敛于输运方程的唯一拉格朗日解.我们的证明基于随机流的使用，并产生定量的收敛率。这为输运方程提供了一个完全通用的选择标准（甚至超出了分布区域），它补偿了凸积分结构引起的低可积性解的野生非唯一性现象，如最近的工作[8，28 -30]所示，并排除了异常耗散的可能性。(c)2022作者（S）出版社：Elsevier Masson SAS这是一个在CC BY许可证下的开放获取文章（http：//creativecommons.org/licenses/by/4.0/）。
We deal with the vanishing viscosity scheme for the transport/continuity equation partial differential tu +div(ub) = 0 drifted by a divergence-free vector field b. Under general Sobolev assumptions on b, we show the convergence of such scheme to the unique Lagrangian solution of the transport equation. Our proof is based on the use of stochastic flows and yields quantitative rates of convergence. This offers a completely general selection criterion for the transport equation (even beyond the distributional regime) which compensates the wild non-uniqueness phenomenon for solutions with low integrability arising from convex integration constructions, as shown in recent works [8,28-30], and rules out the possibility of anomalous dissipation.(c) 2022 The Author(s). Published by Elsevier Masson SAS. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).