Nearly Lipschitzean Divergence Free Transport Propogates neither Continuity nor BV Regularity

We give examples of divergence free vector fields a(x, y) ∈ ∩ 1≤p<∞ W 1,p (R 2 ) .For such fields the Cauchy problem for the linear transport equation ∂u ∂t + a 1 (x, y) ∂u ∂x + a 2 (x, y) ∂u ∂y = 0, div a := ∂a 1 ∂x + ∂a 2 ∂y = 0 , (0.1) has unique bounded solutions for u 0 ∈ L ∞ (R 2 ).The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics.In addition, there are smooth initial data u 0 ∈ C ∞ 0 (R 2 ) 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 initial data in W 1,1 ⊂ BV so that for no t > 0 is it true that u(t, •) is of bounded variation.