Advanced search
1 file | 734.59 KB

Smoothing properties of evolution equations via canonical transforms and comparison principle

Author
Organization
Abstract
This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations.
Keywords
FOURIER INTEGRAL-OPERATORS, NONLINEAR SCHRODINGER-EQUATIONS, DISPERSIVE EQUATIONS, REGULARITY, DECAY, CONVERGENCE, CONSTANTS, SYSTEMS

Downloads

  • 0612274.pdf
    • full text
    • |
    • open access
    • |
    • PDF
    • |
    • 734.59 KB

Citation

Please use this url to cite or link to this publication:

Chicago
Ruzhansky, Michael, and Mitsuru Sugimoto. 2012. “Smoothing Properties of Evolution Equations via Canonical Transforms and Comparison Principle.” Proceedings of the London Mathematical Society 105 (2): 393–423.
APA
Ruzhansky, M., & Sugimoto, M. (2012). Smoothing properties of evolution equations via canonical transforms and comparison principle. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY, 105(2), 393–423.
Vancouver
1.
Ruzhansky M, Sugimoto M. Smoothing properties of evolution equations via canonical transforms and comparison principle. PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY. 2012;105(2):393–423.
MLA
Ruzhansky, Michael, and Mitsuru Sugimoto. “Smoothing Properties of Evolution Equations via Canonical Transforms and Comparison Principle.” PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY 105.2 (2012): 393–423. Print.
@article{8585505,
  abstract     = {This paper describes a new approach to global smoothing problems for dispersive and non-dispersive evolution equations based on the global canonical transforms and the underlying global microlocal analysis. For this purpose, the Egorov-type theorem is established with canonical transformations in the form of a class of Fourier integral operators, and their weighted L-2-boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, a new comparison principle for evolution equations is introduced. In particular, it allows us to relate different smoothing estimates by comparing certain expressions involving their symbols. As a result, it is shown that the majority of smoothing estimates for different equations are equivalent to each other. Moreover, new estimates as well as several refinements of known results are obtained. The proofs are considerably simplified. A comprehensive analysis is presented for smoothing estimates for dispersive equations. Applications are given to the detailed description of smoothing properties of the Schrodinger, relativistic Schrodinger, wave, Klein-Gordon and other equations.},
  author       = {Ruzhansky, Michael and Sugimoto, Mitsuru},
  issn         = {0024-6115},
  journal      = {PROCEEDINGS OF THE LONDON MATHEMATICAL SOCIETY},
  keywords     = {FOURIER INTEGRAL-OPERATORS,NONLINEAR SCHRODINGER-EQUATIONS,DISPERSIVE EQUATIONS,REGULARITY,DECAY,CONVERGENCE,CONSTANTS,SYSTEMS},
  language     = {eng},
  number       = {2},
  pages        = {393--423},
  title        = {Smoothing properties of evolution equations via canonical transforms and comparison principle},
  url          = {http://dx.doi.org/10.1112/plms/pds006},
  volume       = {105},
  year         = {2012},
}

Altmetric
View in Altmetric
Web of Science
Times cited: