Fourier Analysis and Partial Differential Equations

Part I. Fourier Series and Periodic Distributions: 1. Preliminaries 2. Fourier series: basic theory 3. Periodic distributions and Sobolev spaces Part II. Applications to Partial Differential Equations: 4. Linear equations 5. Nonlinear evolution equations 6. The Korteweg-de Vries Part III. Some Nonperiodic Problems: 7. Distributions, Fourier transforms and linear equations 8. KdV, BO and friends Appendix A. Tools from the theory of ODEs Appendix B. Commutator estimates Bibliography Index.

[1]  J. Bona,et al.  Nonlocal models for nonlinear, dispersive waves , 1990 .

[2]  Daniel B. Dix.  Nonuniqueness and uniqueness in the initial-value problem for Burgers' equation , 1996 .

[3]  J. Bourgain,et al.  Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations , 1993 .

[4]  J. Antoine,et al.  Exactly solvable models of sphere interactions in quantum mechanics , 1987 .

[5]  Fundamental solutions and eigenfunction expansions for Schrödinger operators. II. Eigenfunction expansions , 1988 .

[6]  G. Folland,et al.  The uncertainty principle: A mathematical survey , 1997 .

[7]  Gustavo Ponce,et al.  On the global well-posedness of the Benjamin-Ono equation , 1991, Differential and Integral Equations.

[8]  Luis Vega,et al.  A bilinear estimate with applications to the KdV equation , 1996 .

[9]  Tosio Kato,et al.  The abstract theory of scattering , 1971 .

[10]  C. Kenig,et al.  Well‐posedness and scattering results for the generalized korteweg‐de vries equation via the contraction principle , 1993 .

[11]  R. Miura.  The Korteweg–deVries Equation: A Survey of Results , 1976 .

[12]  J. Albert.  Dispersion of low-energy waves for the generalized Benjamin-Bona-Mahony equation , 1986 .

[13]  R. Pinsky.  The Behavior of the Life Span for Solutions tout=Δu+a(x)upinRd , 1998 .

[14]  M. Tsutsumi.  Weighted Sobolev spaces and rapidly decreasing solutions of some nonlinear dispersive wave equations , 1981 .

[15]  Luis Vega,et al.  The Cauchy problem for the Korteweg-de Vries equation in Sobolev spaces of negative indices , 1993 .

[16]  H. Fujita.  On the blowing up of solutions fo the Cauchy problem for u_t=Δu+u^ , 1966 .

[17]  S. Nicaise,et al.  Nemytskij's operators and global existence of small solutions of semilinear evolution equations on nonsmooth Domains , 1997 .

[18]  A. Faminskii,et al.  GENERALIZED SOLUTIONS OF THE CAUCHY PROBLEM FOR THE KORTEWEG-DE VRIES EQUATION , 1984 .

[19]  Tosio Kato,et al.  Wave operators and similarity for some non-selfadjoint operators , 1966 .

[20]  Jerry L. Bona,et al.  Solutions of the Korteweg-de Vries equation in frac-tional order Sobolev space , 1976 .

[21]  Daniel B. Dix.  Temporal Asymptotic Behavior of Solutions of the Benjamin-Ono-Burgers Equation , 1991 .

[22]  R. Prosser.  A double scale of weighted $L^2$ spaces , 1975 .

[23]  Tosio Kato,et al.  Nonlinear evolution equations and the Euler flow , 1984 .

[24]  Tosio Kato,et al.  On the Korteweg-de Vries equation , 1979 .

[25]  Paul R. Chernoff,et al.  Pointwise Convergence of Fourier Series , 1980 .

[26]  Stuart S. Antman,et al.  The Equations for Large Vibrations of Strings , 1980 .

[27]  Hiroaki Ono.  Algebraic Solitary Waves in Stratified Fluids , 1975 .

[28]  T. Benjamin.  Internal waves of permanent form in fluids of great depth , 1967, Journal of Fluid Mechanics.

[29]  J. Albert.  On the decay of solutions of the generalized Benjamin-Bona-Mahony equation , 1989 .

[30]  Charles Fefferman,et al.  the uncertainty principle , 1983 .