A note on the Lagrangian flow associated to a partially regular vector field

Abstract

In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$b(t,x_1,x_2)=(b_1(t,x_1),b_2(t,x_1,x_2))\in {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2},(x_1,x_2)\in {\mathbb{R}}^{n_1}\times {\mathbb{R}}^{n_2}.$$ We assume that the first component $b_1$ does not depend on the second variable $x_2$, and has Sobolev $W^{1,p}$ regularity in the variable $x_1$, for some $p>1$. On the other hand, the second component $b_2$ has Sobolev $W^{1,p}$ regularity in the variable $x_2$, but only fractional Sobolev $W^{\alpha ,1}$ regularity in the variable $x_1$, for some $\alpha >1/2$. These estimates imply well-posedness, compactness, and quantitative stability for the Lagrangian flow associated to such a vector field.