Dispersive Equations and Nonlinear Waves

Transcription

1 Herbert Koch Daniel Tataru Monica Vi an Dispersive Equations and Nonlinear Waves Generalized Korteweg-de Vries, Nonlinear Schrodinger, Wave and Schrodinger Maps ^ Birkhauser

2 Contents Preface xi Nonlinear Dispersive Equations 1 Herbert Koch 1 Introduction 3 2 Stationary phase and dispersive estimates Examples and dispersive estimates The Schrodinger equation The Airy function and the Airy equation Laplacian and related operators Gaussians, heat and Schrodinger equation The half-wave equation The Klein-Gordon half wave The Kadomtsev-Petviashvili equation 20 3 Strichartz estimates and small data for the nonlinear Schrodinger equation Strichartz estimates for the Schrodinger equation Strichartz estimates for the Airy equation The Kadomtsev-Petviashvili equation The (half-) wave equation and the Klein-Gordon equation The endpoint Strichartz estimate Small data solutions to the nonlinear Schrodinger equation Initial data in I? Initial data in H1 for d > Initial data in ff1(md) 39 V

3 vi Contents 4 Functions of bounded p-variation Functions of bounded p-variation and the spaces Up and Vp Duality and the Riemann-Stieltjes integral Step functions are dense Convolution and regularization More duality Consequences of Minkowski's inequality The bilinear form as integral Differential equations with rough paths The Brownian motion Adapted function spaces Strichartz estimates Estimates by duality High modulation estimates 71 5 Convolution of measures on hypersurfaces, bilinear estimates, and local smoothing 73 6 Well-posedness for nonlinear dispersive equations Adapted function spaces approach for a model problem The (generalized) KdV equation The derivative nonlinear Schrodinger equation The Kadomtsev-Petviashvili II equation Appendix A: Young's inequality and interpolation Complex interpolation: The Riesz-Thorin theorem Appendix B: Bessel functions Appendic C: The Fourier transform The Fourier transform in L The Fourier transform of Schwartz functions Tempered distributions 130 Bibliography 135

4 Contents vii Geometric Dispersive Evolutions 139 Daniel Tataru 1 Introduction Maps into manifolds The tangent bundle and covariant differentiation Special targets Sobolev spaces S2 and targets: homotopy classes and equivariance Frames and gauge freedom Geometric pde's Harmonic maps The harmonic heat flow Wave maps Schrodinger maps Wave maps Small data heuristics A perturbative set-up The Strichartz norms The null structure The null frame spaces The paradifferential equation and renormalization Function spaces Frequency envelopes Linear analysis in the S and N spaces Multilinear estimates Renormalization The small data result The a priori estimate Global existence and regularity Weak Lipschitz dependence on the initial data Rough solutions and continuous dependence on the initial data Energy dispersion Energy dispersion and multilinear estimates Compare the initial data of (j> and $ Compare the low frequencies of <j> and <j> Compare the high frequencies Energy and Morawetz estimates Notations 185

5 viii Contents The energy-momentum tensor Energy estimates The energy of self-similar maps Morawetz estimates The threshold theorem Further developments Schrodinger maps Frames and gauges Function spaces The small data result Bounds for the harmonic heat flow Bounds for the Schrodinger map flow Rough solutions and continuous dependence Further developments Other targets Large data Near soliton behavior 216 Bibliography 219 Dispersive Equations 223 Monica Vi an 1 Notation Dispersive and Strichartz estimates The linear Schrodinger equation The Airy equation The linear wave equation From dispersive to Strichartz estimates Bilinear Strichartz and local smoothing estimates An inverse Strichartz inequality A linear profile decomposition Stability theory for the energy-critical NLS A large data critical problem A Palais-Smale type condition Existence of minimal blowup solutions and their properties 271

6 Contents ix 9 Long-time Strichartz estimates and applications A long-time Strichartz inequality The rapid frequency cascade scenario Frequency-localized interaction Morawetz inequalities and applications A frequency-localized interaction Morawetz inequality The quasi-soliton scenario Appendix A: Background material Compactness in IP Littlewood-Paley theory Fractional calculus A paraproduct estimate 306 Bibliography 309