It is known, after Jabin (J Differ Equ 260(5):4739–4757, 2016) and Alberti et al. (Ann PDE 5(1):9, 2019), that ODE flows and solutions of the transport equation associated to Sobolev vector fields do not propagate Sobolev regularity, even of fractional order. In this paper, we improve the result at Clop and Jylha (J Differ Equ 266(8):4544–4567, 2019) and show that some kind of propagation of Sobolev regularity happens as soon as the gradient of the drift is exponentially integrable. We provide sharp Sobolev estimates and new examples. As an application of our main theorem, we generalize a regularity result for the 2D Euler equation obtained by Bahouri and Chemin in Bahouri and Chemin (Arch Ration Mech Anal 127(2):159–181, 1994).

中文翻译：

---

与非利普希茨漂移相关的输运方程解和 ODE 流的 Sobolev 估计

众所周知，在 Jabin (J Differ Equ 260(5):4739–4757, 2016) 和 Alberti 等人之后。（Ann PDE 5(1):9, 2019），ODE 流和与 Sobolev 向量场相关的传输方程的解不会传播 Sobolev 正则性，即使是分数阶。在本文中，我们改进了 Clop 和 Jylha 的结果（J Differ Equ 266(8):4544–4567, 2019），并表明只要漂移的梯度是指数可积的，Sobolev 正则性的某种传播就会发生。我们提供了敏锐的 Sobolev 估计和新的例子。作为我们主要定理的应用，我们概括了 Bahouri 和 Chemin 在 Bahouri 和 Chemin 中获得的二维欧拉方程的正则性结果（Arch Ration Mech Anal 127(2):159-181, 1994）。