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Stability of Relaxation Rarefaction Profiles

By Hailiang Liu


We consider the large time behavior of solutions for a hyperbolic relaxation system. For a certain class of initial data the solution is shown to converge to relaxation rarefaction profiles at a determined asymptotic rate. The result is established without the smallness conditions of the wave strength and the initial disturbances. Key words: Decay rate, rarefaction wave, relaxation. AMS Classification: 35L45, 35L65, 35B40. 1 Introduction Our main point of interest is the large time behavior of solutions developed by relaxation dynamics starting with a certain class of initial data. We consider a hyperbolic relaxation system of the form u t + v x = 0; v t + au x = f(u) \Gamma v; (1.1) where u; v are scalars, the constant a ? 0 is assumed to be large enough, so that it dominates a priori the velocities f 0 (u). We assume that f is convex, i.e., f 00 (u) ff; for u 2 IR: Now, let us define a set of the equilibrium states of (1.1) as \Gamma(u) := f(u; v); v = f(u)g: Manuscript ..

Topics: Key words, Decay rate, rarefaction wave, relaxation
Year: 2007
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