Nearly Lipschitzean Divergence Free Transport Propogates neither Continuity nor BV Regularity

Abstract

We give examples of divergence free vector fields. For such fields the Cauchy problem for the linear transport equation has unique bounded solutions. The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics. In addition, there are smooth initial data so that the unique bounded solution is not continuous on any neighborhood of the origin. The second example is a field of similar regularity and intial data of bounded variation.