Preface page ................................................ ix
Notation ................................................... xiv
1 Preliminaries and tools ...................................... 1
Exercises .................................................... 8
2 Linear dispersive equations ................................. 12
2.1 Estimates on the real line ............................. 14
2.2 Estimates on the torus ................................. 22
2.3 The Talbot effect ...................................... 32
Exercises ................................................... 46
3 Methods for establishing wellposedness ...................... 49
3.1 The energy method ...................................... 50
3.1.1 A priori bounds ................................. 51
3.1.2 Existence and uniqueness ........................ 52
3.1.3 Growth bounds for KdV with potential ............ 59
3.2 Oscillatory integral method ............................ 60
3.3 Restricted norm method ................................. 64
3.3.1 L2 solutions of KdV on the real line ............ 64
3.3.2 Low regularity solutions of KdV on the torus .... 70
3.3.3 Forced and damped KdV with a potential .......... 80
3.4 Differentiation by parts on the torus: unconditional
wellposedness .......................................... 82
3.5 Local theory for NLS on the torus ...................... 90
3.5.1 L2 wellposedness of cubic NLS on the torus ...... 91
3.5.2 Hs local wellposedness of the quintic NLS on
the torus ....................................... 93
3.6 Illposedness results ................................... 97
Exercises .................................................. 107
4 Global dynamics of nonlinear dispersive PDEs ............... 111
4.1 Smoothing for nonlinear dispersive PDEs on the torus .. 112
4.1.1 Cubic NLS on the torus ......................... 112
4.1.2 The KdV equation on the torus .................. 117
4.1.3 Proof of Proposition 4.7 ....................... 127
4.2 High-low decomposition method ......................... 129
4.3 The I-method for the quintic NLS equation on the
torus ................................................. 135
Exercises .................................................. 151
5 Applications of smoothing estimates ........................ 154
5.1 Bounds for higher order Sobolev norms ................. 154
5.2 Almost everywhere convergence to initial data ......... 160
5.3 Nonlinear Talbot effect ............................... 162
5.4 Global attractors for dissipative and dispersive
PDEs .................................................. 164
5.4.1 The global attractor is trivial for large
damping ........................................ 169
5.4.2 Bounds on the forced KdV equation .............. 171
Exercises .................................................. 172
References ................................................. 175
Index ...................................................... 185
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