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Front Matter |
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Abstract
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Abstract
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Topologies in the Set of Rapidly Decreasing Distributions |
Jan Kisyński |
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Abstract
Two topologies are introduced in the set of rapidly decreasing distributions on Euclidean space. One of these turns the standard convergence structure carried by this set into a topological convergence structure. The other allows the set in question to be topologically identified with the multiplier space for the Schwartz space of rapidly decreasing functions.
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Abstract
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The Method of Chernoff Approximation |
Yana A. Butko |
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This survey describes the method of approximation of operator semigroups, based on the Chernoff theorem. We outline recent results in this domain as well as clarify relations between constructed approximations, stochastic processes, numerical schemes for PDEs and SDEs, path integrals. We discuss Chernoff approximations for operator semigroups and Schrödinger groups. In particular, we consider Feller semigroups in ., (semi)groups obtained from some original (semi)groups by different procedures: additive perturbations of generators, multiplicative perturbations of generators (which sometimes corresponds to a random time-change of related stochastic processes), subordination of semigroups/processes, imposing boundary/external conditions (e.g., Dirichlet or Robin conditions), averaging of generators, “rotation” of semigroups. The developed techniques can be combined to approximate (semi)groups obtained via several iterative procedures listed above. Moreover, this method can be implemented to obtain approximations for solutions of some time-fractional evolution equations, although these solutions do not possess the semigroup property.
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Laplacians with Point Interactions—Expected and Unexpected Spectral Properties |
Amru Hussein,Delio Mugnolo |
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Abstract
We study the one-dimensional Laplace operator with point interactions on the real line identified with two copies of the half-line .. All possible boundary conditions that define generators of .-semigroups on . are characterized. Here, the Cayley transform of the matrices that describe the boundary conditions plays an important role and using an explicit representation of the Green’s functions, it allows us to study invariance properties of semigroups.
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Remarks on a Characterization of Generators of Bounded ,-Semigroups |
Sylwia Kosowicz |
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Abstract
We provide and discuss an integral resolvent criterion for generation of bounded .-semigroups on Banach spaces.
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Semigroups Associated with Differential-Algebraic Equations |
Sascha Trostorff |
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We consider differential-algebraic equations in infinite dimensional state spaces, and study under which conditions we can associate a .-semigroup with such equations. We determine the right space of initial values and characterise the existence of a .-semigroup in the case of operator pencils with polynomially bounded resolvents.
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Positive Degenerate Holomorphic Groups of the Operators and Their Applications |
Sophiya A. Zagrebina,Natalya N. Solovyova |
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In this paper, we study degenerate holomorphic groups of the operators generated by the linear and continuous operators . and ., where the operator . is (., .)-bounded. Necessary and sufficient conditions for the positivity of such groups are found. Using these groups, we obtain positive solutions to linear homogeneous and inhomogeneous Sobolev type equations. As an example, positive degenerate holomorphic groups are considered in Sobolev spaces of sequences.
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Abstract
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Microscopic Selection of Solutions to Scalar Conservation Laws with Discontinuous Flux in the Contex |
Boris Andreianov,Massimiliano D. Rosini |
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Abstract
In the context of road traffic modeling we consider a scalar hyperbolic conservation law with the flux (fundamental diagram) which is discontinuous at ., featuring variable velocity limitation. The flow maximization criterion for selection of a unique admissible weak solution is generally admitted in the literature, however justification for its use can be traced back to the irrelevant vanishing viscosity approximation. We seek to assess the use of this criterion on the basis of modeling proper to the traffic context. We start from a first order microscopic follow-the-leader (FTL) model deduced from basic interaction rules between cars. We run numerical simulations of FTL model with large number of agents on truncated Riemann data, and observe convergence to the flow-maximizing Riemann solver. As an obstacle towards rigorous convergence analysis, we point out the lack of order-preservation of the FTL semigroup.
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Newton’s Method for the McKendrick-von Foerster Equation |
Agnieszka Bartłomiejczyk,Monika Wrzosek |
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Abstract
In the paper we study an age-structured model which describes the dynamics of one population with growth, reproduction and mortality rates. We apply Newton’s method to the McKendrick-von Foerster equation in the semigroup setting. We prove its first- and second-order convergence.
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Singular Thermal Relaxation Limit for the Moore-Gibson-Thompson Equation Arising in Propagation of A |
Marcelo Bongarti,Sutthirut Charoenphon,Irena Lasiecka |
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Abstract
Moore-Gibson-Thompson (MGT) equations, which describe acoustic waves in a heterogeneous medium, are considered. These are the third order in time evolutions of a predominantly hyperbolic type. MGT models account for a finite speed propagation due to the appearance of thermal relaxation coefficient . in front of the third order time derivative. Since the values of . are relatively small and often negligible, it is important to understand the asymptotic behavior and characteristics of the model when .. This is a particularly delicate issue since the . dynamics is governed by a generator which is singular as . It turns out that the limit dynamics corresponds to the linearized Westervelt equation which is of a parabolic type. In this paper, we provide a rigorous analysis of the asymptotics which includes strong convergence of the corresponding evolutions over infinite horizon. This is obtained by studying convergence rates along with the uniform exponential stability of the third order evolutions. Spectral analysis for the MGT-equation along with a discussion of spectral uppersemicontinuity for both equations (MGT and linearized Westervelt) will also be provided.
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Applications of the Kantorovich–Rubinstein Maximum Principle in the Theory of Boltzmann Equations |
Henryk Gacki,Roksana Brodnicka |
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Abstract
A generalized version of the Tjon–Wu equation is considered. It describes the evolution of the energy distribution in a model of gas in which simultaneous collisions of many particles are permitted. Using the technique of the Kantorovich–Rubinstein maximum principle concerning the properties of probability metrics we show that the stationary solution is asymptotically stable with respect to the Kantorovich–Wasserstein distance. Our research was stimulated by the problem of stability of solutions of the same equation which was derived by Lasota (see [.]). This stability result was based on the technique of Zolotarev norms. Lasota shows that the stationary solution is exponentially stable in the Zolotarev norm of order 2.
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Propagators of the Sobolev Equations |
Evgeniy V. Bychkov |
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Abstract
In this paper a initial-boundary value problem for the Sobolev equation is investigated. This problem is a part of more general mathematical model of wave propagation in uniform incompressible rotating with constant angular velocity . fluid. The studied problem may be obtained from it if we direct . axis collinear to .. In addition an initial-boundary value problem for the Sobolev equation represents a model describing oscillations in stratified fluid. The solution of this problem is called an inertial (gyroscopic) wave because it arises by virtue of the Archimedes law and under influence of inertial forces. The paper shows that the relative spectrum of the pencil of operators entering the Sobolev equation is bounded. Then, based on the theory of relatively polynomially bounded pencils of operators and the theory of Sobolev type equations of higher order, the propagators of the Sobolev equation, given in a cylinder and in a parallelepiped are constructed.
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The Fourth Order Wentzell Heat Equation |
Gisèle Ruiz Goldstein,Jerome A. Goldstein,Davide Guidetti,Silvia Romanelli |
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Abstract
We consider the Wentzell Laplacian and its square in general domains. Applications to the Cauchy problems associated with Wentzell heat, wave and plate equations are presented.
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Nonlinear Semigroups and Their Perturbations in Hydrodynamics. Three Examples |
Piotr Kalita,Grzegorz Łukaszewicz,Jakub Siemianowski |
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Abstract
In this paper we present some mutual relations between semigroup theory in the context of the theory of infinite dimensional dynamical systems and the mathematical theory of hydrodynamics. These mutual relations prove to be very fruitful, enrich both fields and help to understand behaviour of solutions of both infinite dimensional dynamical systems and hydrodynamical equations. We confine ourselves to present these connections on some recent developments in the important problem of heat transport in incompressible fluids which features all main aspects of chaotic dynamics. To be specific, we consider the Rayleigh–Bénard problem for the two and three-dimensional Boussinesq systems for the Navier–Stokes and micropolar fluids and two-dimensional thermomicropolar fluid. Each of the three examples is remarkably distinct from the other two in the context of the semigroup theory.
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Method of Lines for a Kinetic Equation of Swarm Formation |
Adrian Karpowicz,Henryk Leszczyński |
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Abstract
Kinetic equations with drift and without drift are approximated by the method of lines. Its stability is proved in .. Because integrals are calculated over unbounded domains, we apply Gauss-Hermite quadratures.
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Degenerate Matrix Groups and Degenerate Matrix Flows in Solving the Optimal Control Problem for Dyna |
Alevtina V. Keller,Minzilia A. Sagadeeva |
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Abstract
Dynamic interindustry balance models are described by differential equations of the first order, and the matrix at the derivative, which is a matrix of specific capital expenditures, can be degenerated. The stationary case, including optimal control problems, is well studied for such models. We consider the non-stationary case, where one of the matrices is multiplied by the scalar function that depends on time. In the stationary case, the results of the study of optimal control problems for degenerate balance interindustry models are presented by methods of the theory of degenerate matrix groups. Highlight that there is the importance of considering this problem for Leontief type models. Only for this particular case there is a convergence of numerical solutions to the optimal control problem to the exact one. We introduce the concept of flow of degenerate matrices and use them to construct the solutions of the non-stationary Leontief type system. We use these solutions in order to investigate an optimal control problem for non-stationary Leontief type systems of the specified type and we have proved the existence of a unique solution to the problem. The obtained results are illust
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Degenerate Holomorphic Semigroups of Operators in Spaces of ,-“Noises” on Riemannian manifolds |
Olga G. Kitaeva,Dmitriy E. Shafranov,Georgy A. Sviridyuk |
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Abstract
We investigate the degenerate holomorphic resolving semigroups for the linear stochastic Sobolev type models with relatively .-sectorial operator in the spaces of smooth differential forms defined on a smooth compact oriented Riemannian manifold without boundary. To this end, in the space of differential forms, we use the pseudo-differential Laplace–Beltrami operator instead of the usual Laplace operator. The Cauchy condition and the Showalter–Sidorov condition are used as the initial conditions for an abstract Sobolev type model. Since “white noise” of the model is non-differentiable in the usual sense, we use the derivative of stochastic process in the Nelson–Gliklikh sense. In order to investigate the stability of solutions, we establish that there exist the exponential dichotomies splitting the space of solutions into stable and unstable invariant subspaces. As an example, we use a stochastic version of the Dzektser equation in the space of differential forms defined on the 2-dimensional torus, which is a smooth compact oriented Riemannian surface without boundary.
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发表于 2025-3-21 19:16:45
