Global Well-posedness and Global Attractor for Two-dimensional Zakharov-Kuznetsov Equation
- PDF / 406,125 Bytes
- 32 Pages / 510.236 x 737.008 pts Page_size
- 40 Downloads / 194 Views
Acta Mathematica Sinica, English Series Springer-Verlag GmbH Germany & The Editorial Office of AMS 2020
Global Well-posedness and Global Attractor for Two-dimensional Zakharov–Kuznetsov Equation Min Jie SHAN Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, P. R. China E-mail : [email protected] Abstract The initial value problem for two-dimensional Zakharov–Kuznetsov equation is shown to be globally well-posed in H s (R2 ) for all 57 < s < 1 via using I-method in the context of atomic spaces. By means of the increment of modified energy, the existence of global attractor for the weakly damped, < s < 1. forced Zakharov–Kuznetsov equation is also established in H s (R2 ) for 10 11 Keywords spaces
Zakharov–Kuznetsov equation, global well-posedness, global attractor, I-method, atomic
MR(2010) Subject Classification
1
35Q53, 35B41, 35A01, 35Q35, 35B45
Introduction and Main Results
We consider the Cauchy problem for the symmetrized two-dimensional Zakharov–Kuznetsov (ZK) equation ut + (∂x3 + ∂y3 )u + (∂x + ∂y )u2 = 0, (x, y) ∈ R2 , t ≥ 0, (1.1) u(x, y, 0) = u0 (x, y) ∈ H s (R2 ), where u = u(x, y, t) is a real-valued function. The ZK equation was initially deduced as a model of nonlinear unidirectional ion-acoustic wave propagation in a magnetized plasma by Zakharov and Kuznetsov [19]. It may be treated as a higher dimensional generalization of the Korteweg–de Vries (KdV) equation. For more details we refer to the papers [20, 21] about the two-dimensional ZK equation appearing here in physical circumstances. Even though the ZK equation is not completely integrable, there still exist two conserved quantities for the flow of the ZK,
u2 (x, y, t)dxdy =
M (u)(t) = R2
and
E(u)(t) = R2
R2
u20 (x, y)dxdy = M (u0 )
1 1 1 |∇u|2 − ∂x u∂y u − u3 dxdy = E(u0 ). 2 2 3
Received September 10, 2019, accepted January 13, 2020 Supported by China Postdoctoral Science Foundation (Grant No. 2019M650872)
(1.2)
(1.3)
970
Shan M. J.
Faminskii [6] firstly obtained the local well-posedness for the two-dimensional ZK equation in the energy space H 1 (R2 ) by making use of local smoothing effects together with a maximal function estimate for the linearized equation. This method was inspired by Kenig, Ponce and Vega who dealt with the local well-posedness for the KdV equation in [13]. With the help of the L2 and H 1 conservation laws, he proved global well-posedness for the ZK equation additionally. Following this idea, Linares and Pastor [22] optimized the proof of Faminskii to show local well-posedness in H s (R2 ) for s > 34 . Gr¨ unrock and Herr [9] along with Molinet and Pilod [24] proved local well-posedness in a larger data space H s (R2 ) for s > 12 by taking advantage of the Fourier restriction norm method and a kind of sharp Strichartz estimates. On the basis of the method of Gr¨ unrock and Herr, in [28] we recently improved the local well-posedness 1 2 2 result to B2,1 (R ) via applying frequency decomposition as well as atomic spaces introduced by Koch and Tataru. Actually, it
Data Loading...
