Abstract:We give examples of divergence free vector fieldsFor such fields the Cauchy problem for the linear transport equationa(x,y) ∈ ∩_{1≤p<∞}W^{1,p}(R^{2}).has unique bounded solutions for u∂_{t}u + a_{1}(x,y)∂_{x}u + + a_{2}(x,y)∂_{y}u = 0, div a := ∂_{x}a_{1}+ ∂_{y}a_{2}= 0, (1)_{0}∈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 u_{0}∈C_{0}^{∞}(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.

[ 1996 | 1997 | 1998 | 1999 | 2000 | 2001 | 2002 | 2003 | 2004 | All Preprints | Preprint Server Homepage ]

Conservation Laws Preprint Server <conservation@math.ntnu.no> Last modified: Thu Apr 22 09:55:06 MEST 2004