Preprint 2004-018

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

Abstract: We give examples of divergence free vector fields

a(x,y) ∈ ∩_1≤p<∞W^1,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^∞(R^d). 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^∞(R^d) 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 W^1,1⊂BV so that for no t>0 is it ture that u(t,.) is of bounded variation.

Paper:

Available as PDF (192 Kbytes).

Author(s):

Ferruccio Colombini, <colombini@dm.unipi.it>

Tao Luo, <tl48@georgetown.edu>

Jeffrey Rauch, <rauch@umich.edu>

Publishing information:

Comments:

Submitted by:

<tl48@georgetown.edu> April 21 2004.