Global existence and optimal temporal decay estimates for systems of parabolic conservation laws i: the one-dimensional case

Abstract

This paper is closely related to the work of D. Hoff and J. A. Smoller [8], M. Schonbek [l8] and is concerned with the global existence and the optimal temporal decay estimates for the following one-dimensional parabolic conservation laws

Where is the unknown vector is an arbitrary n × 1 smooth vector—valued function defined in a ball of radius r centered at a fixed vector is a constant, diagonalizable martrix with positive eigenvalues.

Our results shows that if the corresponding inviscid system (i.e (1) with D = 0) is hyperbolic at some fixed point there exists a nonsigular matrix such that and then for with sufficiently small, the Cauchy problem (1) admits a unique globally smooth solution and satisfies the following temporal decay estimates: For each k = 0,1,2,…

The above decay estimates are optimal in the sense that they coincide with the corresponding decay estimates for the solution to the linear part of the corresponding Cauchy problem.

Keywords:

• Nonlinear parabolic conservation laws
• global existence
• optimal decay estimates
• Sobolev imbedding theorems
• entropy—entropy flux pairs