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## Details

Genre/Form: | Electronic books |
---|---|

Additional Physical Format: | Print version: Wang, Baoxiang. HARMONIC ANALYSIS METHOD FOR NONLINEAR EVOLUTION EQUATIONS, I. Singapore : World Scientific Publishing Company, ©2011 |

Material Type: | Document, Internet resource |

Document Type: | Internet Resource, Computer File |

All Authors / Contributors: |
Baoxiang Wang; Zhaohui Huo; Chengchun Hao |

ISBN: | 9789814360746 9814360740 |

OCLC Number: | 877767902 |

Notes: | 8.2 Basic surgery tools for the Boltzmann operator. |

Description: | 1 online resource (298 pages) |

Contents: | Preface; Contents; 1. Fourier multiplier, function space X; 1.1 Schwartz space, tempered distribution, Fourier transform; 1.2 Fourier multiplier on L; 1.3 Dyadic decomposition, Besov and Triebel spaces; 1.4 Embeddings on X; 1.5 Differential-difference norm on X; 1.6 Homogeneous space; 1.7 Bessel (Riesz) potential spaces H (H); 1.8 Fractional Gagliardo-Nirenberg inequalities; 1.8.1 GN inequality in; 1.8.2 GN inequality in; 2. Navier-Stokes equation; 2.1 Introduction; 2.1.1 Model, energy structure; 2.1.2 Equivalent form of NS; 2.1.3 Critical spaces. 2.2 Time-space estimates for the heat semi-group2.2.1 L L estimate for the heat semi-group; 2.2.2 Time-space estimates for the heat semi-group; 2.3 Global well-posedness in L of NS in 2D; 2.4 Well-posedness in L of NS in higher dimensions; 2.5 Regularity of solutions for NS; 2.5.1 Gevrey class and function space E; 2.5.2 Estimates of heat semi-group in E; 2.5.3 Bilinear estimates in E; 2.5.4 Gevrey regularity of NS equation; 3. Strichartz estimates for linear dispersive equations; 3.1 L ' L estimates for the dispersive semi-group; 3.2 Strichartz inequalities: dual estimate techniques. 3.3 Strichartz estimates at endpoints4. Local and global wellposedness for nonlinear dispersive equations; 4.1 Why is the Strichartz estimate useful; 4.2 Nonlinear mapping estimates in Besov spaces; 4.3 Critical and subcritical NLS in H; 4.3.1 Critical NLS in H; 4.3.2 Wellposedness in H; 4.4 Global wellposedness of NLS in L and H; 4.5 Critical and subcritical NLKG in H; 5. The low regularity theory for the nonlinear dispersive equations; 5.1 Bourgain space; 5.2 Local smoothing effect and maximal function estimates; 5.3 Bilinear estimates for KdV and local well-posedness. 5.4 Local well-posedness for KdV in H5.5 I-method; 5.6 Schrödinger equation with derivative; 5.7 Some other dispersive equations; 6. Frequency-uniform decomposition techniques; 6.1 Why does the frequency-uniform decomposition work; 6.2 Frequency-uniform decomposition, modulation spaces; 6.2.1 Basic properties on modulation spaces; 6.3 Inclusions between Besov and modulation spaces; 6.4 NLS and NLKG in modulation spaces; 6.4.1 Schrödinger and Klein-Gordon semigroup in modulation spaces; 6.4.2 Strichartz estimates in modulation spaces; 6.4.3 Wellposedness for NLS and NLKG. 6.5 Derivative nonlinear Schrödinger equations6.5.1 Global linear estimates; 6.5.2 Frequency-localized linear estimates; 6.5.3 Proof of global wellposedness for small rough data; 7. Conservations, Morawetz' estimates of nonlinear Schrödinger equations; 7.1 Nöther's theorem; 7.2 Invariance and conservation law; 7.3 Virial identity and Morawetz inequality; 7.4 Morawetz' interaction inequality; 7.5 Scattering results for NLS; 8. Boltzmann equation without angular cutoff; 8.1 Models for collisions in kinetic theory; 8.1.1 Transport model; 8.1.2 Boltzmann model; 8.1.3 Cross section. |

### Abstract:

This monograph provides a comprehensive overview on a class of nonlinear evolution equations, such as nonlinear Schrödinger equations, nonlinear Klein-Gordon equations, KdV equations as well as Navier-Stokes equations and Boltzmann equations. The global wellposedness to the Cauchy problem for those equations is systematically studied by using the harmonic analysis methods. This book is self-contained and may also be used as an advanced textbook by graduate students in analysis and PDE subjects and even ambitious undergraduate students.

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