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

Crippa, Gianluca and Ligabue, Silvia. (2021) A note on the Lagrangian flow associated to a partially regular vector field. Preprints Fachbereich Mathematik, 2021 (11).

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Official URL: https://edoc.unibas.ch/82582/

## Abstract

In this paper we derive quantitative estimates for the Lagrangian flow associated to a partially regular vector field of the form $$b(t,x_1,x_2) = (b_1(t,x_1),b_2(t,x_1,x_2)) \in {\mathbb R}^{n_1}\times{\mathbb R}^{n_2} \,, \qquad (x_1,x_2)\in{\mathbb R}^{n_1}\times{\mathbb R}^{n_2}\,.$$ 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.
Faculties and Departments: 05 Faculty of Science > Departement Mathematik und Informatik > Mathematik > Analysis (Crippa)12 Special Collections > Preprints Fachbereich Mathematik Crippa, Gianluca and Ligabue, Silvia Preprint Universität Basel English 10.5451/unibas-ep82582 08 Apr 2021 19:51 08 Apr 2021 19:51

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