Preprint 2005-039

Optimal Tracing of Viscous Shocks in Solutions of Viscous Conservation Laws

Abstract: This paper contains a qualitative study of a scalar conservation law with viscosity:

u_t+f(u)_x=u_xx. We consider the problem of identifying the location of viscous shocks, thus obtaining an optimal finite dimensional description of solutions to the viscous conservation law. We introduce a nonlinear functional whose minimizers yield the viscous travelling profiles which ``optimally fit'' the given solution. We prove that, outside an initial time interval and away from times of shock interactions, our functional remains very small, i.e.~the solution can be accurately represented by a finite number of viscous travelling waves.

Paper:

Available as PDF (216 Kbytes),

Author(s):

W. Shen, <shen_w@math.psu.edu>

M. Park, <park_m@math.psu.edu>

Publishing information:

Submitted to SIAM J. Math. Anal. (2005)

Comments:

Revised May 25 2006

Submitted by:

<shen_w@math.psu.edu> October 17 2005 and May 26 2006.