Differentiability properties of the flow of 2d autonomous vector fields

[摘要] We investigate under which assumptions the flow associated to autonomous planar vector fields inherits the Sobolev or BV regularity of the vector field. We consider nearly incompressible and divergence-free vector fields, taking advantage in both cases of the underlying Hamiltonian structure. Finally, we provide an example of an autonomous planar Sobolev divergence-free vector field, such that the corresponding regular Lagrangian flow has no bounded variation. (c) 2021 The Author(s). Published by Elsevier Inc. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/4.0/).

[发布日期] 2021-11-15

[关键词] Transport equation;Regular Lagrangian flow;Sobolev regularity;Hamiltonian structure;Nearly incompressible vector fields