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April  2011, 30(1): 77-113. doi: 10.3934/dcds.2011.30.77

## The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms

 1 University of Wisconsin-Madison, Mathematics Department, 480 Lincoln Dr. Madison, WI 53706-1388, United States

Received  December 2009 Revised  May 2010 Published  February 2011

In this paper, we study the generic behavior of the solutions to a large class of evolution equations. The Schrödinger evolution is considered as an application.
Citation: Sergey A. Denisov. The generic behavior of solutions to some evolution equations: Asymptotics and Sobolev norms. Discrete & Continuous Dynamical Systems - A, 2011, 30 (1) : 77-113. doi: 10.3934/dcds.2011.30.77
##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", National Bureau of Standards Applied Mathematics Series, 55 ().   Google Scholar [2] A. Böttcher and B. Silbermann, "Introduction to Large Truncated Toeplitz Matrices,", Springer. 1998., (1998).   Google Scholar [3] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential,, J. Anal. Math., 77 (1999), 315.  doi: 10.1007/BF02791265.  Google Scholar [4] M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials,, Geom. Funct. Anal., 12 (2002), 1174.  doi: 10.1007/s00039-002-1174-9.  Google Scholar [5] S. Denisov, Continuous analogs of polynomials orthogonal on the unit circle. Krein systems,, Int. Math. Res. Surveys, 2006 (2006).   Google Scholar [6] S. Denisov, An evolution equation as the WKB correction in long-time asymptotics of Schrödinger dynamics,, Comm. Partial Differential Equations, 33 (2008), 307.  doi: 10.1080/03605300701249655.  Google Scholar [7] O. Jørsboe and L. Mejlbro, "The Carleson-Hunt Theorem on Fourier Series,", Lecture Notes in Mathematics \textbf{911}, 911 (1982).   Google Scholar [8] A. Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators,, Duke Math. J., 94 (1998), 619.  doi: 10.1215/S0012-7094-98-09425-X.  Google Scholar [9] G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps,, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411.   Google Scholar [10] B. Perthame, Mathematical tools for kinetic equations,, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205.   Google Scholar [11] B. Simon, "Orthogonal Polynomials on the Unit Circle,", Parts 1 and 2. American Mathematical Society Colloquium Publications, 54 (2005).   Google Scholar [12] W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations,, J. Funct. Anal., 254 (2008), 2926.  doi: 10.1016/j.jfa.2007.11.012.  Google Scholar

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##### References:
 [1] M. Abramowitz and I. Stegun, "Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables,", National Bureau of Standards Applied Mathematics Series, 55 ().   Google Scholar [2] A. Böttcher and B. Silbermann, "Introduction to Large Truncated Toeplitz Matrices,", Springer. 1998., (1998).   Google Scholar [3] J. Bourgain, On growth of Sobolev norms in linear Schrödinger equations with smooth time dependent potential,, J. Anal. Math., 77 (1999), 315.  doi: 10.1007/BF02791265.  Google Scholar [4] M. Christ and A. Kiselev, Scattering and wave operators for one-dimensional Schrödinger operators with slowly decaying nonsmooth potentials,, Geom. Funct. Anal., 12 (2002), 1174.  doi: 10.1007/s00039-002-1174-9.  Google Scholar [5] S. Denisov, Continuous analogs of polynomials orthogonal on the unit circle. Krein systems,, Int. Math. Res. Surveys, 2006 (2006).   Google Scholar [6] S. Denisov, An evolution equation as the WKB correction in long-time asymptotics of Schrödinger dynamics,, Comm. Partial Differential Equations, 33 (2008), 307.  doi: 10.1080/03605300701249655.  Google Scholar [7] O. Jørsboe and L. Mejlbro, "The Carleson-Hunt Theorem on Fourier Series,", Lecture Notes in Mathematics \textbf{911}, 911 (1982).   Google Scholar [8] A. Kiselev, Stability of the absolutely continuous spectrum of the Schrödinger equation under slowly decaying perturbations and a.e. convergence of integral operators,, Duke Math. J., 94 (1998), 619.  doi: 10.1215/S0012-7094-98-09425-X.  Google Scholar [9] G. Nenciu, Adiabatic theory: Stability of systems with increasing gaps,, Ann. Inst. H. Poincare Phys. Theor., 67 (1997), 411.   Google Scholar [10] B. Perthame, Mathematical tools for kinetic equations,, Bull. Amer. Math. Soc. (N.S.), 41 (2004), 205.   Google Scholar [11] B. Simon, "Orthogonal Polynomials on the Unit Circle,", Parts 1 and 2. American Mathematical Society Colloquium Publications, 54 (2005).   Google Scholar [12] W.-M. Wang, Bounded Sobolev norms for linear Schrödinger equations under resonant perturbations,, J. Funct. Anal., 254 (2008), 2926.  doi: 10.1016/j.jfa.2007.11.012.  Google Scholar
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