Preprint 2004-017

Uniqueness and Nonuniquess for Nonsmooth Divergence Free Transport

Abstract: We present an example of a uniformly bounded divergence free vector field a(x).∂_x on R^d which has the property that the linear transport equation

∂_t u + ∑^d_i=1 a_j(t,x) ∂_xj u = 0,          div a = ∑^d_i=1 ∂_xj a_j = 0         (1)

has a nontrivial bounded solution with vanishing Cauchy data. The coefficients have the property that x₃∇a is a bounded measure.

For the same equation we prove uniqueness in the Cauchy problem when the coefficients a and u belong to (H^1/2∩L^∞)([0,T]×R^d).

Paper:

Available as PDF (176 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.