On the Existence, Blowup and Large Time Behavior of the Zakharov System
We present in this book a wide-range survey of important topics on mathematical theories of Zakharov system Zakharov (Sov Phys JETP 35:908-914, 1972, [196]) established in 1972, with particular emphasis on various modern developments. It was Sulem and Sul
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On the Existence, Blowup and Large Time Behavior of the Zakharov System
We present in this book a wide-range survey of important topics on mathematical theories of Zakharov system [196] established in 1972, with particular emphasis on various modern developments. It was Sulem and Sulem [180] who originally issued in the existence and uniqueness of global solutions to one dimensional Zakharov system. Global existence of the initial-boundary value problem for 1D Zakharov system was first obtained by Guo and Shen [85]. From then on, many authors have devoted to the mathematical theories of such system, especially the local or global well-posedness theory of the Cauchy problem. Using logarithm type Sobolev inequality, Added and Added [2] verified the existence of global smooth solutions under small conditions of the initial data in 2D. Finite difference schemes of Zakharov system were studied numerically by Guo [75, 76], Guo and Chang [32, 83], Chang, Guo and Jiang [33]. Existence of global attractors and estimates of finite Hausdorff dimensions were explored by Flahaut [54], Goubet and Moise [73], Chueshov and Shcherbina [36] as well as Dai and Guo [45], Guo and Li [127, 128]. Ozawa and Tsutsumi [159] in 1992 established the local existence of solution in H 2 (Rd ) × H 1 (Rd ) and investigated the smoothness effect of the solution. Bourgain and Colliander [25] obtained local well-posedness result in the energy space H 1 × L 2 , and also global existence in energy spaces for small initial data. Glangetas and Merle [70, 71] and Merle [146–149] investigated the blowup problem of 2D and 3D Zakharov system. Concerning the scattering issues of Zakharov system, the final value problems were studied in [69, 160, 173], and the scattering theories of the Cauchy problem in 3D were obtained by Guo and Nakanishi [98] with radial initial data, Guo, Lee, Nakanishi and Wang [97] without radial condition but with the degree condition of angular regularity (see also [101] for a dichotomy between scattering and growup), Hani, Pusateri and Shatah [105] with sufficiently smooth initial data. See also the work of Bejenaru, Guo, Herr and Nakanishi [16] and Kato, Tsugawa [117] for dimension four or higher dimensions. In this chapter, we introduce some classical results for Zakharov system in energy spaces, which include the existence and uniqueness theory of local (or global) smooth © Springer Science+Business Media Singapore and Science Press 2016 B. Guo et al., The Zakharov System and its Soliton Solutions, DOI 10.1007/978-981-10-2582-2_2
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2 On the Existence, Blowup and Large Time Behavior of the Zakharov System
solutions, the blowup theory in 2D, the scattering theory in 3D, and the existence theory of global attractors in 1D. These results are not only interested in mathematics, but also in many important applications.
2.1 Existence and Uniqueness Theory of the Zakharov System In this section, we are concerned with the existence and uniqueness theory of the Cauchy problem for the standard Zakharov system, which reads iEt + E
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