Preprint 2004-018

Nearly Lipshitzean Divergence Free Transport Does Not Propagate Continuity And BV Regularity

Ferruccio Colombini, Tao Luo and Jeffrey Rauch

Abstract: We give examples of divergence free vector fields

a(x,y) ∈ ∩1≤p<∞W1,p(R2).
For such fields the Cauchy problem for the linear transport equation
tu + a1(x,y)∂xu + + a2(x,y)∂yu = 0,          div a := ∂xa1 + ∂ya2 = 0,          (1)
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.

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.

Available as PDF (192 Kbytes).
Ferruccio Colombini, <>
Tao Luo, <>
Jeffrey Rauch, <>
Publishing information:
Submitted by:
<> April 21 2004.

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