Global asymptotics toward the rerefaction waves for a parabolic-elliptic system related to the Camassa-Holm shallow water equation

Abstract

This article is concerned with the global existence and large time behavior of solutions to the Cauchy problem for a parabolic-elliptic system related to the Camassa-Holm shallow water equation (E) { u t + ( u 2 2 ) x + P x = ɛ u xx , t > 0 , x ∈ R , - α P xx + P = f ( u ) + α 2 u x 2 - 1 2 u 2 , t > 0 , x ∈ R , with the initial data (I) u ( 0 , x ) = u 0 ( x ) → u ± , as x → ± ∞ . Here, u − < u + are two constants and f(u) is a sufficiently smooth function satisfying f″(u) > 0 for all u under consideration. Main aim of this article is to study the relation between solutions to the above Cauchy problem and those to the Riemann problem of the following nonlinear conservation law (R) { u t + f ( u ) x = 0 , u ( 0 , x ) = { u - , x < 0 , u + x > 0. It is well known that if u − < u + , the above Riemann problem admits a unique global entropy solution u R(x/t) u R ( x / t ) = { u - x ≤ f ′ ( u - ) t , ( f ′ ) - 1 ( x / t ) , f ′ ( u - ) t ≤ x ≤ f ′ ( u + ) t , u + , x ≥ f ′ ( u + ) t . Let U(t, x) be the smooth approximation of the rarefaction wave profile constructed similar to that of [21, 22, 23], we show that if u o(x) − U(0,x) ∈ H 1 ( R) and u − < u +, the above Cauchy problem (E) and (I) admits a unique global classical solution u(t, x) which tends to the rarefaction wave u R (x/t) as t → +∞ in the maximum norm. The proof is given by an elementary energy method.