Abstract: We give examples of divergence free vector fieldsFor such fields the Cauchy problem for the linear transport equationa(x,y) ∈ ∩1≤p<∞W1,p(R2).has unique bounded solutions for u0∈L∞(Rd). The first example has nonuniqueness in the Cauchy problem for the ordinary differential equation defining characteristics. In addition, there are smooth initial data u0∈C0∞(Rd) so that the unique bounded solution is not continuous on any neighborhood of the origin.∂tu + a1(x,y)∂xu + + a2(x,y)∂yu = 0, div a := ∂xa1 + ∂ya2 = 0, (1)
The second example is a field of similar regularity and inital data in W1,1⊂BV so that for no t>0 is it ture that u(t,.) is of bounded variation.
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