Non-uniform Dependence for the Novikov Equation in Besov Spaces
- PDF / 361,153 Bytes
- 10 Pages / 547.087 x 737.008 pts Page_size
- 36 Downloads / 204 Views
Journal of Mathematical Fluid Mechanics
Non-uniform Dependence for the Novikov Equation in Besov Spaces Jinlu Li, Min Li and Weipeng Zhu Communicated by A. Constantin
Abstract. In this paper, we investigate the dependence on initial data of solutions to the Novikov equation. We show that s (R), s > max{1 + 1 , 3 }. the solution map is not uniformly continuous dependence on the initial data in Besov spaces Bp,r p 2 Mathematics Subject Classification. 35Q35. Keywords. Novikov equation, Non-uniform dependence.
1. Introduction and Main Result In this paper, we consider the Cauchy problem for the Novikov equation (1 − ∂x2 )ut = 3uux uxx + u2 uxxx − 4u2 ux , t > 0, u(x, 0) = u0 (x).
(1.1)
What we are most concerned about is the issue of non-uniform dependence on the initial data. This equation was discovered very recently by Novikov in a symmetry classification of nonlocal PDEs with cubic nonlinearity. He showed that Eq. (1.1) is integrable by using a definition of the existence of an infinite hierarchy of quasi-local higher symmetries [38]. It has a bi-Hamiltonian structure and admits √ exact peakon solutions u(t, x) = ± ce|x−ct| with c > 0 [28]. The Novikov equation had been studied by many authors. Indeed, it is locally well-posed in certain Sobolev spaces and Besov spaces [43,44,46,47]. Moreover, it has global strong solutions [43], finite-time blow up solutions [47] and global weak solutions [30,42]. The Novikov equation can be thought as a generalization of the well-known Camassa-*Holm (CH) equation (1 − ∂x2 )ut = 3uux − 2ux uxx − uuxxx . This equation is known as the shallow water wave equation [2,13]. It is completely integrable, which has been studied extensively by many authors [2,6,14]. The CH equation also has a Hamiltonian structure [4,22], and admits exact peaked solitons of the form ce|x−ct| with c > 0 which are orbitally stable [15]. These peaked solutions also mimic the pattern specific to the waves of greatest height [7,11,40]. The local well-posedness for the Cauchy problem of CH equation in Sobolev spaces and Besov spaces was established in [9,10,16,39]. Moreover, the CH equation has global strong solutions [5,9,10], finitetime blow-up strong solutions [5,8–10], unique global weak solution [45], and it is continuous dependence on initial data [32]. The CH equation was the only known integrable equation having peakon solutions until 2002 when another such equation was discovered by Degasperis and Procesi [18] (1 − ∂x2 )ut = 4uux − 3ux uxx − uuxxx . 0123456789().: V,-vol
Page 2 of 10
50
J. Li et al.
JMFM
The DP equation can be regarded as a model for nonlinear shallow water dynamics and its asymptotic accuracy is the same as for the CH shallow water equation [19], also, it’s integrable with a bi-Hamiltonian structure [12,17]. Similar to the CH equation, the DP equation has travelling wave solutions [31,41]. The Cauchy problem of the DP equation is locally well-posed in certain Sobolev spaces and Besov spaces [23,24,48]. In addition, it has global strong solutions [36,48,50], the finite-time blow-u
Data Loading...
