Smoothing Properties of Evolution Equations via Canonical Transforms and Comparison
نویسندگان
چکیده
The 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–boundedness properties are derived. This allows us to globally reduce general dispersive equations to normal forms in one or two dimensions. Then, several new comparison techniques for evolution equations are introduced. In particular, they allow 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 of smoothing estimates for homogeneous and inhomogeneous, dispersive and also non-dispersive equations with constant coefficients. Results are presented also for equations with time dependent coefficients. Applications are given to the detailed description of smoothing properties of the Schrödinger, relativistic Schrödinger, wave, Klein-Gordon, and other equations. Critical cases of some estimates and their relation to the trace estimates are discussed.
منابع مشابه
Smoothing Estimates for Evolution Equations via Canonical Transforms and Comparison
The 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–boundedness properties are d...
متن کاملOn Properties of Third-Order Differential Equations via Comparison Principles
The objective of this paper is to offer sufficient conditions for certain asymptotic properties of the third-order functional differential equation r t x′ t γ ′′ p t x τ t 0, where studied equation is in a canonical form, that is, ∫∞ r −1/γ s ds ∞. Employing Trench theory of canonical operators, we deduce properties of the studied equations via new comparison theorems. The results obtained esse...
متن کاملComparison of acceleration techniques of analytical methods for solving differential equations of integer and fractional order
The work addressed in this paper is a comparative study between convergence of the acceleration techniques, diagonal pad'{e} approximants and shanks transforms, on Homotopy analysis method and Adomian decomposition method for solving differential equations of integer and fractional orders.
متن کاملNumerical Solution of Volterra-Fredholm Integral Equations with The Help of Inverse and Direct Discrete Fuzzy Transforms and Collocation Technique
متن کامل
Solving a Class of Partial Differential Equations by Differential Transforms Method
In this work, we find the differential transforms of the functions $tan$ and $sec$, and then we applied this transform on a class of partial differential equations involving $tan$ and $sec$.
متن کامل