© 2016, Springer-Verlag Berlin Heidelberg. We face the well-posedness of linear transport Cauchy problems (Formula presented.) under borderline integrability assumptions on the divergence of the velocity field b. For (Formula presented.) vector fields b satisfying (Formula presented.) and (Formula presented.), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every (Formula presented.) , we construct an example of a bounded autonomous velocity field b with (Formula presented.) for which the associate Cauchy problem for the transport equation admits infinitely many solutions. Stability questions and further extensions to the BV setting are also addressed.
|Journal||Calculus of Variations and Partial Differential Equations|
|Publication status||Published - 1 Feb 2016|
- Primary 35F05
- Secondary 35F10