Linear transport equations for vector fields with subexponentially integrable divergence

We face the well-posedness of linear transport Cauchy problems {partial derivative u/partial derivative t + b . del u + cu = f (0, T) x R-n u(0, .) = u(0) is an element of L-infinity R-n under borderline integrability assumptions on the divergence of the velocity field b. For W-loc(1,1) vector fields b satisfying vertical bar b(x,t)vertical bar/1+vertical bar x vertical bar is an element of L1 (0, T; L-1) + L-1(0, T; L-infinity) and div b is an element of L-1(0, T; L-infinity) + L-1 (0,T; Exp (L/log L)), we prove existence and uniqueness of weak solutions. Moreover, optimality is shown in the following way: for every gamma > 1, we construct an example of a bounded autonomous velocity field b with div (b) is an element of Exp (L/log(gamma) L) 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.