Preprint 2004-048

Asymptotic Behavior of Spatially Inhomogeneous Balance Laws

Abstract: We study the longtime behavior of spatially inhomogeneous scalar balance laws

u_t+f(u)_x=g(x,u)

with periodic initial data and a convex flux f.

Our main result states that for a large class of initial data the entropy solution will either converge uniformly to some steady state or to a discontinuous time-periodic solution.

This extends results of Lyberopoulos, Sinestrari and Fan & Hale obtained in the spatially homogeneous case.

The proof is based on the method of generalized characteristics together with ideas from dynamical systems theory.

A major difficulty consists of finding the periodic solutions which determine the asymptotic behavior. To this end we introduce a new tool, the Rankine–Hugoniot vector field, which describes the motion of a (hypothetical) shock with certain prescribed left and right states. We then show the existence of periodic solutions of the Rankine–Hugoniot vector field and prove that the actual shock curves converge to these periodic solutions.

Paper:

Available as PDF (392 Kbytes; not suited for on-screen viewing, but prints OK), Postscript (456 Kbytes) or gzipped PostScript (184 Kbytes; uncompress using gunzip).

Author(s):

Julia Ehrt

Jörg Härterich, <haerter@mi.fu-berlin.de>

Publishing information:

Comments:

Submitted by:

<haerter@mi.fu-berlin.de> October 8 2004.