This is an expanded version, enriched by references, of my inaugural speech held on November 7, 2001 at the Real Academia de Ciencas Exactas, Físicas y Naturales in Madrid. It explains in a nontechnical way, accessible to a general scientific community, some of the motivation and basic ideas of my research of the last twenty years on a functional-analytical approach to nonlinear parabolic problems.
An abstract parabolic equation with sectorial operator and continuous nonlinearity is studied in this paper. In particular, the asymptotic behavior of solutions is described within the framework of the theory of global attractors. Examples included in the final part of the paper illustrate the presented ideas.
We consider abstract parabolic problems in ordered Banach spaces and give conditions under which they have global attractors. Our approach is via comparison of solutions. Within this approach abstract comparison principles are obtained and bounds on the attractors are given by order intervals in Banach spaces. These results are applied to ordinary differential equations and to parabolic equations for which the main part is given by a sum of fractional powers of sectorial operators having increasing...
The Cauchy problem for a semilinear abstract parabolic equation is considered in a fractional power scale associated with a sectorial operator appearing in the linear main part of the equation. Existence of local solutions is proved for non-Lipschitz nonlinearities satisfying a certain critical growth condition.
A class of quasi-variational inequalities (QVI) of elliptic type is studied in reflexive Banach spaces. The concept of QVI was earlier introduced by A. Bensoussan and J.-L. Lions [2] and its general theory has been developed by many mathematicians, for instance, see [6, 7, 9, 13] and a monograph [1]. In this paper we give a generalization of the existence theorem established in [14]. In our treatment we employ the compactness method along with a concept of convergence of nonlinear multivalued operators...
Recently, we established some generalizations of the theory of Lagrange multipliers arising from nonlinear programming in Banach spaces, which enable us to treat not only elliptic problems but also parabolic problems in the same generalized framework. The main objective of the present paper is to discuss a typical time-dependent double obstacle problem as a new application of the above mentioned generalization. Actually, we describe it as a usual parabolic variational inequality and then characterize...
We consider an abstract version of the Cauchy-Kowalewski Problem with the right hand side being free from the Lipschitz type conditions and prove the existence theorem.
Žemlička, Jan: Structure of steady rings. Zemek, Martin: On some aspects of subdifferentiality of functions on Banach spaces. Hlubinka, Daniel: Construction of Markov kernels with application for moment problem solution. Somberg, Petr: Properties of the BGG resolution on the spheres. Krump, Lukáš: Construction of Bernstein-Gelfand-Gelfand for almost hermitian symmetric structures. Kolář, Jan: Simultaneous extension operators. Porosity.
The reduced basis method is a model reduction technique yielding substantial savings of computational time when a solution to a parametrized equation has to be computed for many values of the parameter. Certification of the approximation is possible by means of an a posteriori error bound. Under appropriate assumptions, this error bound is computed with an algorithm of complexity independent of the size of the full problem. In practice, the evaluation of the error bound can become very sensitive...
Motivated by the theory of superconductivity and more precisely by the problem of the onset of superconductivity in dimension two, many papers devoted to the analysis in a semi-classical regime of the lowest eigenvalue of the Schrödinger operator with magnetic field have appeared recently. Here we would like to mention the works by Bernoff-Sternberg, Lu-Pan, Del Pino-Felmer-Sternberg and Helffer-Morame and also Bauman-Phillips-Tang for the case of a disc. In the present paper we settle one important...
We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...
We present an alternative framework for designing efficient numerical schemes for non-conservative hyperbolic systems. This approach is based on the design of entropy conservative discretizations and suitable numerical diffusion operators that mimic the effect of underlying viscous mechanisms. This approach is illustrated by considering two model non-conservative systems: Lagrangian gas dynamics in non-conservative form and a form of isothermal Euler equations. Numerical experiments demonstrating...
Asymptotics with sharp remainder estimates are recovered for number of eigenvalues of operator crossing level as runs from to , . Here is periodic matrix operator, matrix is positive, periodic with respect to first copy of and decaying as second copy of goes to infinity, either belongs to a spectral gap of or is one its ends. These problems are first treated in papers of M. Sh. Birman, M. Sh. Birman-A. Laptev and M. Sh. Birman-T. Suslina.