Generator Of Semigroup Research Articles

A class of relatively bounded perturbations for generators of bounded analytic semigroups in Banach spaces

Generation of analytic semigroups on Banach space X by −(A+kB) is shown, where A is a negative generator of a bounded analytic semigroup on X, B is a closed operator in X belonging to a class related to A and k∈C is a parameter satisfying Rek>c for some c∈R. The proof may be regarded as a modification of the perturbation argument established by Okazawa [8]. As applications, the generation of non-contractive analytic semigroups by Schrödinger operators with inverse square potentials in Lp(RN) is discussed.

Abstract elliptic operators appearing in atmospheric dispersion

In this paper, the boundary value problem for the differential-operator equation with principal variable coefficients is studied. Several conditions for the separability and regularity in abstract -spaces are given. Moreover, sharp uniform estimates for the resolvent of the corresponding elliptic differential operator are shown. It is implies that this operator is positive and also is a generator of an analytic semigroup. Then the existence and uniqueness of maximal regular solution to nonlinear abstract parabolic problem is derived. In an application, maximal regularity properties of the abstract parabolic equation with variable coefficients and systems of parabolic equations are derived in mixed -spaces. MSC:34G10, 34B10, 35J25.

ON SOLUTIONS IN THE HYDRODYNAMIC APPROXIMATION OF SOLAR AND STELLAR WINDS WITH VISCOSITY

In this article we present some results in existence and uniqueness of strong and classical solutions of the hydrodynamic equations modeling solar and stellar winds. The system of Navier-Stokes equations for solar and stellar winds is considered in its corresponding differential evolution equation form (d/dt+A)υ(t) = F(υ(t), t), where F is a given non-linear function and -A is the infinitesimal generator of the analytic semigroup arises by the hydrodynamic Stokes operator.

Infinitely delayed stochastic evolution equations on UMD Banach spaces

We prove existence and uniqueness of solutions for a class of infinitely delayed stochastic evolution equations with multiplicative noise term{dU(t)=(AU(t)+F(t,Ut))dt+G(t,Ut)dW(t),t∈[0,T0],U(0)=X:Ω→E,U0=Φ:(−∞,0]×Ω→E, where A is the generator of an analytic semigroup on a UMD Banach space E and F and G are functions from the history of the system satisfying Lipschitz conditions. This paper is based on recent work of van Neerven et al., developing the theory of abstract stochastic evolution equations in UMD Banach spaces.

Subordination in the sense of Bochner of L p -semigroups and associated Markov processes

We show that the subordination induced by a convolution semigroup (subordination in the sense of Bochner) of a C0-semigroup of sub-Markovian operators on an Lp space is actually associated to the subordination of a right (Markov) process. As a consequence, we solve the martingale problem associate with the Lp-infinitesimal generator of the subordinate semigroup. We also prove quasi continuity properties for the elements of the domain of the Lp-generator of the subordinate semigroup. It turns out that an enlargement of the base space is necessary. A main step in the proof is the preservation under such a subordination of the property of a Markov process to be a Borel right process. We use several analytic and probabilistic potential theoretical tools.

Periodicity in the [formula omitted]-norm for partial functional differential equations in fading memory spaces

The aim of this work is to study the existence of periodic solutions in the α-norm for some partial functional differential equations of neutral type in fading memory spaces. We assume that a linear part is densely defined and is the generator of an analytic semigroup. The delayed part is assumed to be periodic with respect to the first argument. In the nonhomogeneous linear case, we show that the existence of a bounded solution in R+ implies the existence of the periodic solution. In the nonlinear case, we use two approaches, the first one is based on the ultimate boundedness of solutions and the second one is based on the multivalued fixed point theorem.

Separable convolution-elliptic operators with parameters

Abstract The maximal regularity properties of parameter dependent abstract convolution-elliptic equations are investigated. Here find sufficient conditions that guarantee the uniform separability of these problems in Lp spaces. It is established that the corresponding convolution-elliptic operator is sectorial and is also a generator of an analytic semigroup. Finally, these results applied to obtain the uniform maximal regularity for the Cauchy problem for abstract parabolic equation in mixed L 𝐩 ${L_{\mathbf {p}}}$ norms, boundary value problems for anisotropic integro-differential equations and infinite systems of elliptic integro-differential equations with parameters.

Affine infinitesimal generators of semigroups on the polydisk

This paper concerns affine infinitesimal generators of continuous semigroups of holomorphic functions on the polydisk. An easy-to-check necessary and sufficiently condition for an affine vector field to be an infinitesimal generator is given.

Honesty in discrete, nonlocal and randomly position structured fragmentation model with unbounded rates

In the process of discrete and nonlocal aggregation, the major problem arises when each fragmentation rate becomes infinite at infinity. In this paper, a discrete Cauchy problem describing multiple fragmentation processes is investigated by means of parameter-dependent operators together with the theory of substochastic semigroups with a parameter. We focus on the case where fragmentation rates are size and position dependent and where new particles are spatially randomly distributed according to a certain probabilistic law. Unlike [8], where the discrete model with bounded fragmentation rates is treated, we use, in this paper, Katoʼs theorem in L1[2] and the dominated convergence theorem [4] to show the existence of the infinitesimal generator of a positive semigroup of contractions and give sufficient conditions for honesty in the case of unbounded fragmentation rates. Essentially, we demonstrate that, even in the discrete and nonlocal case, the process is conservative if at infinity daughter particles tend to go back into the system with a high probability.

Linear and nonlinear convolution elliptic equations

In this paper, the separability properties of elliptic convolution operator equations are investigated. It is obtained that the corresponding convolution-elliptic operator is positive and also is a generator of an analytic semigroup. By using these results, the existence and uniqueness of maximal regular solution of the nonlinear convolution equation is obtained in spaces. In application, maximal regularity properties of anisotropic elliptic convolution equations are studied. MSC:34G10, 45J05, 45K05.

Functional calculus for semigroup generators via transference

In this article we apply a recently established transference principle in order to obtain the boundedness of certain functional calculi for semigroup generators. In particular, it is proved that if −A generates a C0-semigroup on a Hilbert space, then for each τ>0 the operator A has a bounded calculus for the closed ideal of bounded holomorphic functions on a (sufficiently large) right half-plane that satisfy f(z)=O(e−τRe(z)) as |z|→∞. The bound of this calculus grows at most logarithmically as τ↘0. As a consequence, f(A) is a bounded operator for each holomorphic function f (on a right half-plane) with polynomial decay at ∞. Then we show that each semigroup generator has a so-called (strong) m-bounded calculus for all m∈N, and that this property characterizes semigroup generators. Similar results are obtained if the underlying Banach space is a UMD space. Upon restriction to so-called γ-bounded semigroups, the Hilbert space results actually hold in general Banach spaces.

Formal adjoint operators and asymptotic formula for solutions of autonomous linear integral equations

For autonomous linear integral equations, we establish an explicit asymptotic representation formula of solutions, developing the spectral analysis for the generator of the solution semigroup as well as the one for the formal adjoint operator associated with the generator.

The minimal number of generators of a finite semigroup

The rank of a finite semigroup is the smallest number of elements required to generate the semigroup. A formula is given for the rank of an arbitrary (non necessarily regular) Rees matrix semigroup over a group. The formula is expressed in terms of the dimensions of the structure matrix, and the relative rank of a certain subset of the structure group obtained from subgroups generated by entries in the structure matrix, which is assumed to be in Graham normal form. This formula is then applied to answer questions about minimal generating sets of certain natural families of transformation semigroups. In particular, the problem of determining the maximum rank of a subsemigroup of the full transformation monoid (and of the symmetric inverse semigroup) is considered.

Goldstein-wentzell boundary conditions: Recent results with jerry and gisèle goldstein

We present a survey of recent results concerning heat and telegraph equations, equipped with Goldstein-Wentzell boundary conditions (already known as general Wentzell boundary conditions). We focus on the generation of analytic semigroups and continuous dependence of the solutions of the associated Cauchy problems from the boundary conditions.

Growth bound for the shift semigroup on weighted Sobolev spaces

We present sufficient and necessary conditions for the strong continuity of the shift semigroup on weighted Sobolev spaces explicitly in terms of the weight function. Further, we discuss properties of the spectrum and the point spectrum of the generator of the semigroup. We show how the spectrum and point spectrum of the generator and the growth bound of the semigroup depend on the weight function.

Skew product semiflows and Morse decomposition

This paper is devoted to the investigation of the dynamics of non-autonomous differential equations. The description of the asymptotic dynamics of non-autonomous equations lies on dynamical structures of some associated limiting non-autonomous – and autonomous – differential equations (one for each global solution in the attractor of the driving semigroup of the associated skew product semiflow). In some cases, we have infinitely many limiting problems (in contrast with the autonomous – or asymptotically autonomous – case for which we have only one limiting problem; that is, the semigroup itself). We concentrate our attention in the study of the Morse decomposition of attractors for these non-autonomous limiting problems as a mean to understand some of the asymptotics of our non-autonomous differential equations. In particular, we derive a Morse decomposition for the global attractors of skew product semiflows (and thus for pullback attractors of non-autonomous differential equations) from a Morse decomposition of the attractor for the associated driving semigroup. Our theory is well suited to describe the asymptotic dynamics of non-autonomous differential equations defined on the whole line or just for positive times, or for differential equations driven by a general semigroup.

Almost periodic solutions for nonlinear delay evolutions with nonlocal initial conditions

We consider the nonlinear delay differential evolution equation $$\left\{\begin{array}{ll} u'(t) \in Au(t) + f(t, u_t), \quad \quad t \in \mathbb{R}_+,\\ u(t) = g(u)(t),\qquad \qquad \quad t \in [-\tau, 0], \end{array} \right.$$ where τ ≥ 0, X is a real Banach space, A is the infinitesimal generator of a nonlinear semigroup of contractions whose Lipschitz seminorm decays exponentially as $${t \mapsto {\rm{e}}^{-\omega t}}$$ when $${t \to + \infty}$$ and $${f : {\mathbb{R}}_+ \times C([-\tau, 0]; \overline{D(A)}) \to X}$$ is jointly continuous. We prove that if f Lipschitz with respect to its second argument and its Lipschitz constant l satisfies the condition $${\ell{\rm{e}}^{\omega\tau} < \omega, g : C_b([-\tau, +\infty); \overline{D(A)}) \to C([-\tau, 0]; \overline{D(A)})}$$ is nonexpansive and (I – A)−1 is compact, then the unique C 0-solution of the problem above is almost periodic.

Ergodic and mixing quantum channels in finite dimensions

The paper provides a systematic characterization of quantum ergodic and mixing channels in finite dimensions and a discussion of their structural properties. In particular, we discuss ergodicity in the general case where the fixed point of the channel is not a full-rank (faithful) density matrix. Notably, we show that ergodicity is stable under randomizations, namely that every random mixture of an ergodic channel with a generic channel is still ergodic. In addition, we prove several conditions under which ergodicity can be promoted to the stronger property of mixing. Finally, exploiting a suitable correspondence between quantum channels and generators of quantum dynamical semigroups, we extend our results to the realm of continuous-time quantum evolutions, providing a characterization of ergodic Lindblad generators and showing that they are dense in the set of all possible generators.

Operator calculus on the class of Sato’s hyperfunctions

We construct a functional calculus for generators of analytic semigroups of operators on a Banach space. The symbol class of the calculus consists of hyperfunctions with a compact support in $[0,\infty)$. Domain of constructed calculus is dense in the Banach space.

A semigroup approach to functional differential evolution equations

We study functional differential evolution equations of the form , where − A is infinitesimal generator of an analytic semigroup in a Banach space E (with or without order) and the given right‐hand side modelling delay. In many cases, E is a Banach space of sections, such as vector field or differential forms of a (real or complex) vector bundle Ω (of possibly infinite dimension) over a locally convex manifold, for example, a Carathéodory–Finsler manifold. The operator A may in particular be generated by a Dirichlet form acting on an ordered Hilbert space. As an application, we consider a problem from thermo‐magnetohydrodynamics. Copyright © 2013 John Wiley &amp; Sons, Ltd.