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ОбложкаErdogan M.B. Dispersive partial differential equations: wellposedness and applications / M.B.Erdogan, N.Tzirakis. - Cambridge: Cambridge university press, 2016. - xvi, 186 p. - (Student texts / London mathematical society; 86). - Bibliogr.: p.175-183. - Ind.: p.185-186. - ISBN 978-1-107-14904-5
Шифр: (И/В16-Е66) 02
 

Место хранения: 02 | Отделение ГПНТБ СО РАН | Новосибирск

Оглавление / Contents
 
   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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