本書著重研究幾種重要類型的Zakharov方程在能量空間中的一些經(jīng)典結(jié)果，其中包括一維及高維問題的適定性結(jié)果、爆破問題和長(zhǎng)時(shí)間行為、高維非均勻介質(zhì)中的Zakharov方程、klein-gordon-Zakharov方程、離子聲Zakharov方程及磁場(chǎng)Zakharov方程的相關(guān)數(shù)學(xué)理論研究成果。
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V.E. Zakharov is a famous Russian （Soviet） mathematical physicist. He made excellent contributions to plasma physics， turbulence and soliton theory. In 1962， he considered the collapse of Langmiur waves， and in 1972， he proposed a system coupled by the electric field and the perturbation of particles when studying the interaction between plasma and laser. From the viewpoint of mathematics， such system （which was known as Zakharov system） is the nonlinear Schrodinger equation coupled with the wave equation with strong nonlinearity. Zakharov computed the soliton solution for this system， and explained clearly the unresolved phenomenon for a long time in laser target shooting， namely， the density hollow phenomenon near the critical surface. Hence， this system caused great concern among the intemational physics community. Since then， many research works have been done on Zakharov system both in mathematics and physics， and many innovative， significant achievements which are of great impact on other research fields have been obtained. In mathematics， a lot of researchers， for example， I. Bejenaru， J. Bourgain， J. Colliander， L. Glangetas， B. Guo， Z. Guo， Z. Hani， S. Herr， J. Holmer， C. Kenig， N. Masmoudi， F. Merle， K. Nakanishi， T. Ozawa， G. Ponce， H. Pecher， F. Pusateri， C. Sulem， P.L. Sulem， D. Tataru， Y. Tsutsumi， N. Tzirakis， J. Shatah， and L. Vega made a series of important works on problems about the existence of global solution， uniqueness， blow-up， low regularity and singular limit.
This book introduces the mathematical theories， research methods and results for Zakharov type system， including the existence of global solution in energy space， uniqueness， blow-up， low regularity， large time behavior and the singular limit. As the developments in this literature are quite qruck， we are not aiming to cover all the important results for such system. However， through the contents introduced in the book， we hope the readers can quickly trace the main subject of Zakharov system and conduct related research on this system.
Owing to the limited time and knowledge of the authors， there must be some improper errors and omissions in the book. Any suggestions and comments are welcome.
Contents
1Physical Background of Zakharov Equations and Its Soliton1
1.1 Transport Process in a Plasma 1
1.2System of Equations for Two-F1uid Dynamics 6
1.3Solitons in Plasmas11
1.3.1Soliton in Ion Acoustic Wave 12
1.3.2 Langmuir Soliton14
1.3.3Ls Soliton 16
1.3.4 The Light Soliton17
1.3.5 Solitons of Simplified Two-Fluid System18
2 0n the Existence, Blowup and Large Time Behavior of the Zakharov System21
2.1Existence and Uniqueness Theory of the Zakharov System22
2.1.1Weak Solution Theory of Zakharov System 23
2.1.2Local Smooth Solution to Zakharov System26
2.1.3Global Smooth Solution to Zakharov System35
2.2Blowup Phenomenon of the Zakharov System 40
2.2.1 Existence of Self-Similar Blowup Solutions to Zakharov System 40
2.2.2 Auxiliary Propositions and Lemmas44
2.2.3 Existence and Uniqueness of Radially Symmetric Solutions 47
2.2.4 Concentration Phenomenon of the Blowup Solutions62
2.2.5Nonexistence of Blowup Solutions with Minimum Mass73
2.3Scattering for the Zakharov System in 3D 76
2.3.1Reduction of the System and Linear Decay Estimates77
2.3.2 Energy Estimate81
2.3.3 Decay Estimate for the Wave Equation85
2.3.4 Weighted Estimates for the Wave Component91
2.3.5 Weighted Estimates for the Scl 95
2.4Global Attractors of Dissipative Zakharov System106
2.4.1Uniform a Priori Estimates 109
2.4.2 Existence of Global Attractor 121
3 Studies on Generalized Zakharov System 129
3.1Zakharov System in Nonhomogeneous Medium129
3.1.1A Priori Estimates130
3.1.2 Existence and Uniqueness of Global Smooth Solutions 139
3.2 Klein-Gordon-Zakharov System 143
3.3Zakharov System in Two Dimensional Ion-Acoustic Waves 152
3.4 Zakharov Systems with Magnetic Field Effect 164
3.4.1 Reduction of Zakharov System with a Magnetic Field 164
3.4.2Conservation Laws and Existence of Weak Solutions 167
3.4.3 Regularized System for the Magnetic Zakharov System171
3.4.4 Local Existence Theory of Zakharov System in Cold Plasmas174
3.4.5Local Existence Theory for Zakharov System in Hot Plasmas180
3.4.6 Global Existence of Smooth Solutions 188
3.4.7 Convergence Behavior of Zakharov System with Magnetic Field Effect 190
3.5Global Well-Posedness for the Quantum Zakharov System 193
3.5.1 The Main Results194
3.5.2Some Energy Estimates for the Solution196
3.5.3Proof of the Global Well-Posedness Result203
3.5.4 Proof of the Classic Limit Behavior211
4 Low Regularity Theories of Zakharov System217
4.1Preliminaries218
4.1.1 Work Space 218
4.1.2 Linear Estimates 220
4.2 Global Well-Posedness for One Dimensional Zakharov System226
4.2.1Main Results and Introduction of the Strategy 226
4.2.2 Estimates for Groups and Duhamel Terms 230
4.2.3Proof of Global Well-Posedness 237
4.2.4 Multilinear Estimates240
4.3 Low Regularity for Zakharov System in Higher Dimension247
4.3.1 Reduction of the System247
4.3.2Estimates of Nonlinear Terms249
4.3.3Well-Posedness of Zakharov System in Higher Dimensions263
4.4Well-Posedness of Two Dimensional Zakharov System268
4.4.1Local Well-Posedness Result 268
4.4.2 Proof of the Main Theorem273
4.4.3 Proof of Multilinear Estimates278
5 Singular Limit of Klein-Gordon-Zakharov System with Infinite Propagation Speed 293
5.1 Introduction 293
5.2 Preliminary Knowledge296
5.2.1 Notations and the Frequency Decomposition 296
5.2.2 Local Well-Posedness Result 298
5.2.3 Reduction of the System299
5.2.4 Strichartz Norms, Fourier Restriction Norms and Related Estimates301
5.3Bilinear Estimates for Regular Interactions and Non-resonant Interactions305
5.4 Energy Estimate on the Resonant Components 314
5.5 Convergence Results319
5.5.1 Limit Behavior of the Klein-Gordon-Zakharov System 320
5.5.2 Uniform Bounds and Two Lemmas321
5.6 Proof of the Main Results 323
5.7Convergence in the Energy Space with Small Initial Data 328
References335
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