线性Sobolev输运中的极端时间不稳定性：几乎光滑非唯一解
Extreme temporal intermittency in the linear Sobolev transport: almost smooth nonunique solutions
摘要：
在本文中，我们重新审视的概念的时间不一致性得到尖锐的非唯一性结果的线性迁移方程。我们构造了空间维$d\geq 2$中所有$p<\infty$的具有尖锐Sobolev正则性$L^1_t W^{1,p}$的无发散向量场，其输运方程允许所有$p<\infty$和$k\in \mathbb{N}$的属于$L^p_tC^k$的非唯一弱解.特别地，我们的结果表明DiPerna-Lions理论的唯一性中的时间可积性假设是尖锐的。同样的结果也适用于任意维数$d \geq 2$中具有任意大阶扩散算子的输运扩散方程.
英文摘要：
In this paper, we revisit the notion of temporal intermittency to obtain sharp nonuniqueness results for linear transport equations. We construct divergence-free vector fields with sharp Sobolev regularity $L^1_t W^{1,p}$ for all $p<\infty$ in space dimensions $d\geq 2$ whose transport equations admit nonunique weak solutions belonging to $L^p_tC^k$ for all $p<\infty$ and $k\in \mathbb{N}$. In particular, our result shows that the time-integrability assumption in the uniqueness of the DiPerna-Lions theory is sharp. The same result also holds for transport-diffusion equations with diffusion operators of arbitrarily large order in any dimensions $d \geq 2$.
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