Nonuniform dependence of the R‐b‐family system in Besov spaces

Abstract

In this paper, we consider a generalized rotation b‐family system (R‐b‐family system) which models the evolution of equatorial water waves. Based on local well‐posedness results and lifespan estimates, we establish sharpness of continuity on the data‐to‐solution map by showing that it is not uniformly continuous from Bp,rs×Bp,rs−1 to C(false[0,Tfalse];Bp,rs×Bp,rs−1). The proof of nonuniform dependence is based upon approximate solutions and Littlewood‐Paley decomposition theory.

Nonuniform dependence on initial data for the 2D viscous shallow water equations

“…One of the first results of this type was proved by Kato [24] who showed that the solution operator for the (inviscid) Burgers equation is not Hölder continuous in the 𝐻 𝑠 (𝕋)-norm (𝑠 > 3∕2) for any Hölder exponent. After the phenomenon of nonuniform continuity for some dispersive equations was studied by Kenig et al [27], many results with regard to the nonuniform dependence on the initial data have been obtained for other nonlinear PDEs including the Euler equations [18,37], the Camassa-Holm equation [19,20,31,32], the Benjamin-Ono equation [28], the compressible gas dynamics [21,26], the Hunter-Saxton equation [22], the R-b-family system [23], and so on. Nevertheless, we notice that almost the above system mentioned is hyperbolic.…”