| Preface | p. vii |
| Fourier multiplier, function space Xp,qs | p. 1 |
| Schwartz space, tempered distribution, Fourier transform | p. 1 |
| Fourier multiplier on Lp | p. 4 |
| Dyadic decomposition, Besov and Triebel spaces | p. 8 |
| Embeddings on Xp,qs | p. 13 |
| Differential-difference norm on Xp,qs | p. 16 |
| Homogeneous space &Xdot;p,qs | p. 19 |
| Bessel (Riesz) potential spaces Hps(&Hdot;ps) | p. 22 |
| Fractional Gagliardo-Nirenberg inequalities | p. 25 |
| GN inequality in &Bdot;p,qs | p. 25 |
| GN inequality in &Fdot;p,qs | p. 29 |
| Navier-Stokes equation | p. 33 |
| Introduction | p. 34 |
| Model, energy structure | p. 34 |
| Equivalent form of NS | p. 34 |
| Critical spaces | p. 35 |
| Time-space estimates for the heat semi-group | p. 35 |
| Lr Lp estimate for the heat semi-group | p. 35 |
| Time-space estimates for the heat semi-group | p. 36 |
| Global well-posedness in L2 of NS in 2D | p. 39 |
| Well-posedness in Ln of NS in higher dimensions | p. 42 |
| Regularity of solutions for NS | p. 44 |
| Gevrey class and function space E2,1s | p. 44 |
| Estimates of heat semi-group in E2,1s | p. 46 |
| Bilinear estimates in E2,1s | p. 47 |
| Gevrey regularity of NS equation | p. 48 |
| Strichartz estimates for linear dispersive equations | p. 51 |
| Lp' Lp estimates for the dispersive semi-group | p. 52 |
| Strichartz inequalities: dual estimate techniques | p. 63 |
| Strichartz estimates at endpoints | p. 68 |
| Local and global wellposedness for nonlinear dispersive equations | p. 75 |
| Why is the Strichartz estimate useful | p. 75 |
| Nonlinear mapping estimates in Besov spaces | p. 78 |
| Critical and subcritical NLS in Hs | p. 83 |
| Critical NLS in Hs | p. 83 |
| Wellposedness in Hs | p. 84 |
| Global wellposedness of NLS in L2 and H1 | p. 87 |
| Critical and subcritical NLKG in Hs | p. 88 |
| The low regularity theory for the nonlinear dispersive equations | p. 91 |
| Bourgain space | p. 91 |
| Local smoothing effect and maximal function estimates | p. 99 |
| Bilinear estimates for KdV and local well-posedness | p. 104 |
| Local well-posedness for KdV in H-3/4 | p. 113 |
| I-method | p. 130 |
| Schrödinger equation with derivative | p. 145 |
| Some other dispersive equations | p. 152 |
| Frequency-uniform decomposition techniques | p. 157 |
| Why does the frequency-uniform decomposition work | p. 157 |
| Frequency-uniform decomposition, modulation spaces | p. 159 |
| Basic properties on modulation spaces | p. 160 |
| Inclusions between Besov and modulation spaces | p. 164 |
| NLS and NLKG in modulation spaces | p. 176 |
| Schrödinger and Klein-Gordon semigroup in modulation spaces | p. 176 |
| Strichartz estimates in modulation spaces | p. 179 |
| Wellposedness for NLS and NLKG | p. 183 |
| Derivative nonlinear Schrödinger equations | p. 187 |
| Global linear estimates | p. 191 |
| Frequency-localized linear estimates | p. 192 |
| Proof of global wellposedness for small rough data | p. 197 |
| Conservations, Morawetz' estimates of nonlinear Schrödinger equations | p. 205 |
| Nöther's theorem | p. 205 |
| Invariance and conservation law | p. 212 |
| Virial identity and Morawetz inequality | p. 214 |
| Morawetz' interaction inequality | p. 220 |
| Scattering results for NLS | p. 222 |
| Boltzmann equation without angular cutoff | p. 227 |
| Models for collisions in kinetic theory | p. 228 |
| Transport model | p. 228 |
| Boltzmann model | p. 229 |
| Cross section | p. 233 |
| Basic surgery tools for the Boltzmann operator | p. 235 |
| Properties of Boltzmann collision operator without cutoff | p. 240 |
| Regularity of solutions for spatially homogeneous case | p. 246 |
| Notations | p. 259 |
| Definition of scattering operator | p. 261 |
| Some fundamental results | p. 263 |
| Gagliardo-Nirenberg inequality in Sobolev spaces | p. 263 |
| Convexity Hölder inequality in sequence spaces lps | p. 263 |
| Inclusion between homogeneous Triebel-Lizorkin spaces | p. 264 |
| Riesz-Thorin interpolation theorem | p. 265 |
| Hardy-Littlewood-Sobolev inequality | p. 266 |
| Van der Corput lemma | p. 266 |
| Littlewood-Paley square function theorem | p. 266 |
| Complex interpolation in modulation spaces | p. 267 |
| Christ-Kiselev lemma | p. 267 |
| Bibliography | p. 269 |
| Index | p. 281 |
| Table of Contents provided by Ingram. All Rights Reserved. |
