Related papers: Microlocal analysis and evolution equations
The solution of pseudo initial value differential equations, either ordinary or partial (including those of fractional nature), requires the development of adequate analytical methods, complementing those well established in the ordinary…
These three lectures present some fundamental and classical aspects of microlocal analysis. Starting with the Sato's microlocalization functor and the microsupport of sheaves, we then construct a microlocal analogue of the Hochschild…
We revisit previously developed analytic models for defect evolution and adapt them appropriately for the study of semilocal string networks. We thus confirm the expectation (based on numerical simulations) that linear scaling evolution is…
This is a introductory course focusing some basic notions in pseudodifferential operators ($\Psi$DOs) and microlocal analysis. We start this lecture notes with some notations and necessary preliminaries. Then the notion of symbols and…
Using the theory of evolutionary equations, we consider abstract differential equations including non-local integral operators. After providing a condition for the well-posedness of the addressed equation we consider a numerical method of…
In this work we begin a theoretical and numerical investigation on the spectra of evolution operators of neutral renewal equations, with the stability of equilibria and periodic orbits in mind. We start from the simplest form of linear…
The topic of these notes could be easily expanded into a full one-semester course. Nevertheless, we shall try to give some flavour along with theoretical bases of spectral and pseudo-spectral methods. The main focus is made on Fourier-type…
Using energy methods, we prove some power-law and exponential decay estimates for classical and nonlocal evolutionary equations. The results obtained are framed into a general setting, which comprise, among the others, equations involving…
We introduce an encoder-only approach to learn the evolution operators of large-scale non-linear dynamical systems, such as those describing complex natural phenomena. Evolution operators are particularly well-suited for analyzing systems…
We suggest the method for group classification of evolution equations admitting nonlocal symmetries which are associated with a given evolution equation possessing nontrivial Lie symmetry. We apply this method to second-order evolution…
We introduce a general context involving a presheaf A and a subpresheaf B of A. We show that all previously considered cases of local analysis of generalized functions (defined from duality or algebraic techniques) can be interpretated as…
These are lecture notes for a minicourse on applications of microlocal analysis in inverse problems, given in Helsinki and Shanghai in June 2019.
We develop efficient group-theoretical approach to the problem of classification of evolution equations that admit non-local transformation groups (quasi-local symmetries), i.e., groups involving integrals of the dependent variable. We…
We introduce a special class of real semiflows, which is used to define a general type of evolution semigroups, associated to not necessarily exponentially bounded evolution families. Giving spectral characterizations of the corresponding…
This course introduces the use of semigroup methods in the solution of linear and nonlinear (quasi-linear) hyperbolic partial differential equations, with particular application to wave equations and Hermitian hyperbolic systems. Throughout…
Recently, computational modelling became a very important research tool that enables us to study problems that for decades evaded scientific analysis. Evolutionary systems are certainly examples of such problems: they are composed of many…
We expand our group classification of quasilinear evolution equations (Acta Appl.Math., v.69, 2001) to the case of general evolution equation in one spatial variable. This enables obtaining several new classes of evolution equations with…
We shall explain how the idea of microlocal analysis of the seventies has been reformulated in the framework of sheaf theory in the eighties and then applied to various branches of mathematics, such as linear partial differential equations…
This paper surveys a few aspects of the global theory of wave equations. This material is structured around the contents of a minicourse given by the second author during the CMI/ETH Summer School on evolution equations during the Summer of…
We survey some recent progress on modulation spaces and the well-posedness results for a class of nonlinear evolution equations by using the frequency-uniform localization techniques.