Nonuniform Dependence of a Two-Component NOVIKOV System in Besov Spaces

Abstract

Considered herein is the Cauchy problem of the two-component Novikov system. In the periodic case, we first constructed an approximate solution sequence that possesses the nonuniform dependence property; then, by applying the energy methods, we managed to prove that the difference between the approximate and actual solution is negligible, thus succeeding in proving the nonuniform dependence result in both supercritical Besov spaces Bp,rs(T)×Bp,rs(T) with s>max{32,1+1p},1≤p≤∞,1≤r<∞ and critical Besov space B2,132(T)×B2,132(T). In the non-periodic case, we constructed two sequences of initial data with high and low-frequency terms by analyzing the inner structure of the system under investigation in detail, and we proved that the distance between the two corresponding solution sequences is lower-bounded by time t, but converges to zero at initial time. This implies that the solution map is not uniformly continuous both in supercritical Besov spaces Bp,rs(R)×Bp,rs(R) with s>max{32,1+1p},1≤p≤∞,1≤r<∞ and critical Besov spaces Bp,11+1p(R)×Bp,11+1p(R) with 1≤p≤2. The proof of nonuniform dependence is based on approximate solutions and Littlewood–Paley decomposition theory. These approaches are widely applicable in the study of continuous properties for shallow water equations.