Generator Of Semigroup Research Articles

The holomorphic functional calculus approach to operator semigroups

In this article we construct a holomorphic functional calculus for operators of half-plane type and show how key facts of semigroup theory (Hille-Yosida and Gomilko-Shi-Feng generation theorems, Trotter-Kato apprximation theorem, Euler approximation formula, Gearhart-Prüss theorem) can be elegantly obtained in this framework. Then we discuss the notions of bounded H∞-calculus and m-bounded calculus on half-planes and their relation to weak bounded variation conditions over vertical lines for powers of the resolvent. Finally we discuss the Hilbert space case, where semigroup generation is characterised by the operator having a strong m-bounded calculus on a half-plane.

Global attractors for semigroup actions

In this paper the notions of global attractor and global uniform attractor are extended to the setting of general semigroup actions. We study the necessary and sufficient conditions for the existence of the global attractor for semigroups acting on noncompact metric spaces. We present conditions for the equivalence between the global attractor and global uniform attractor.

On semigroups of nonnegative functions and positive operators

We give extensions of results on nonnegative matrix semigroups which deduce finiteness or boundedness of such semigroups from the corresponding local properties, e.g., from finiteness or boundedness of values of certain linear functionals applied to them. We also consider more general semigroups of functions.

A direct approach to generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators in weighted [formula omitted] spaces

Generation of analytic semigroups by generalized Ornstein–Uhlenbeck operators A=−Δ+∇Φ⋅∇−G⋅∇+V in the weighted space Lp(RN,e−Φ(x)dx) is shown. The results improve the recent results established by Metafune–Prüss–Rhandi–Schnaubelt (2005) [8] (V≡0) and Kojima–Yokota (2010) [6] (V≢0). The proofs of those are based on a generation result acting on the unweighted space Lp(RN)[8, Theorem 3.4], while ours is carried out by an application of Lp-theory for second-order elliptic operators [13, Theorem 1.3] obtained by Sobajima (2012). Consequently, the additional condition |divG|≤ε(|∇Φ|2+V)+Cε assumed in both [8,6] is completely removed.

Singular degenerate problems occurring in biosorption process

The boundary value problems for singular degenerate arbitrary order differential-operator equations with variable coefficients are considered. The uniform coercivity properties of ordinary and partial differential equations with small parameters are derived in abstract spaces. It is shown that corresponding differential operators are positive and also are generators of analytic semigroups. In application, well-posedeness of the Cauchy problem for an abstract parabolic equation and systems of parabolic equations are studied in mixed spaces. These problems occur in fluid mechanics and environmental engineering. MSC:34G10, 35J25, 35J70.

Complete intersection vanishing ideals on degenerate tori over finite fields

We study the complete intersection property and the algebraic invariants (index of regularity, degree) of vanishing ideals on degenerate tori over finite fields. We establish a correspondence between vanishing ideals and toric ideals associated to numerical semigroups. This correspondence is shown to preserve the complete intersection property, and allows us to use some available algorithms to determine whether a given vanishing ideal is a complete intersection. We give formulae for the degree, and for the index of regularity of a complete intersection in terms of the Frobenius number and the generators of a numerical semigroup.

Quantum Speed Limits in Open System Dynamics

Bounds to the speed of evolution of a quantum system are of fundamental interest in quantum metrology, quantum chemical dynamics, and quantum computation. We derive a time-energy uncertainty relation for open quantum systems undergoing a general, completely positive, and trace preserving evolution which provides a bound to the quantum speed limit. When the evolution is of the Lindblad form, the bound is analogous to the Mandelstam-Tamm relation which applies in the unitary case, with the role of the Hamiltonian being played by the adjoint of the generator of the dynamical semigroup. The utility of the new bound is exemplified in different scenarios, ranging from the estimation of the passage time to the determination of precision limits for quantum metrology in the presence of dephasing noise.

Regular Poles and β-Numbers in the Theory of Holomorphic Semigroups

We introduce the notion of regular (boundary) poles for infinitesimal generators of semigroups of holomorphic self-maps of the unit disc. We characterize such regular poles in terms of β-points (i.e., pre-images of values with positive Carleson–Makarov β-numbers) of the associated semigroup and of the associated Konigs intertwining function. We also define a natural duality operation in the cone of infinitesimal generators and show that the regular poles of an infinitesimal generator correspond to the regular null poles of the dual generator. Finally we apply such a construction to study radial multi-slits and give an example of a nonisolated radial slit whose tip does not have not a positive Carleson–Makarov β-number.

Strong fragmentation and coagulation with power-law rates

Existence of global classical solutions to fragmentation and coagulation equations with unbounded coagulation rates has been recently proved for initial conditions with finite higher order moments. These results cannot be directly generalized to the most natural space of solutions with finite mass and number of particles due to the lack of precise characterization of the domain of the generator of the fragmentation semigroup. In this paper we show that such a generalization is possible in the case when both fragmentation and coagulation are described by power-law rates which are commonly used in the engineering practice. This is achieved through direct estimates of the resolvent of the fragmentation operator, which in this case is explicitly known, proving that it is sectorial and carefully intertwining the corresponding intermediate spaces with appropriate weighted L1 spaces.

A non-local evolution equation model of cell–cell adhesion in higher dimensional space

A model for cell–cell adhesion, based on an equation originally proposed by Armstrong et al. [A continuum approach to modelling cell–cell adhesion, J. Theor. Biol. 243 (2006), pp. 98–113], is considered. The model consists of a nonlinear partial differential equation for the cell density in an N-dimensional infinite domain. It has a non-local flux term which models the component of cell motion attributable to cells having formed bonds with other nearby cells. Using the theory of fractional powers of analytic semigroup generators and working in spaces with bounded uniformly continuous derivatives, the local existence of classical solutions is proved. Positivity and boundedness of solutions is then established, leading to global existence of solutions. Finally, the asymptotic behaviour of solutions about the spatially uniform state is considered. The model is illustrated by simulations that can be applied to in vitro wound closure experiments.

Transformation semigroups associated to Γ-semigroups

The concept of Γ-semigroups is a generalization of semigroups. In this paper, we associate two transformation semigroups to a Γ-semigroup and we call them the left and right transformation semigroups. We prove some relationships between the ideals of a Γ-semigroup and the ideals of its left and right transformation semigroups. Finally, we study some relationships between Green’s equivalence relations of a Γ-semigroup and its left (right) transformation semigroup.

The weighted estimate for the commutator of the generalized fractional integral

(Communicated by S. Koumandos) Abstract. Let L be the infinitesimal generator of an analytic semigroup on L 2 (R n ) with Gaus- sian kernel bound, and let L −α/2 be the fractional integral of L for 0 (α + β)/n.

B-separable anisotropic convolution operators and applications

In this paper, the B-separability properties of anisotropic convolution-operators are investigated. In particular, the sharp estimates for the resolvent are established. Thus these operators are positive and also are generators of analytic semigroups. Moreover, we find sufficient conditions that guarantee the maximal B-regularity of the Cauchy problem for a parabolic convolution equation. Finally, the convolution equation is applied to establish maximal regularity properties for an anisotropic integro-differential equations and their infinite systems.

Approximation of a semigroup model of anomalous diffusion in a bounded set

The convergence is established for a sequence of operator semigroups, where the limiting object is the transition semigroup for a reflected stable processes. For semilinear equations involving the generators of these transition semigroups, an approximation method is developed as well. This makes it possible to derive an a priori bound for solutions to these equations, and therefore prove global existence of solutions. An application to epidemiology is also given.

Semigroups of Relations and Subdifferentials of the Minimal Time Function and of the Geodesic Distance

Given a closed subset $S$ of a Banach space $X$, we study the minimal time function to reach a point $x$ of $X$ starting from $S$. We relate this problem to the study of the geodesic distance from $x$ to $S$ associated with a Riemannian metric or a Finsler metric. It appears that both cases can be treated simultaneously and yield solutions to a Hamilton--Jacobi equation. We detect simple links between the general framework we propose and multivalued semigroups. New concepts of infinitesimal generators of such semigroups are considered.

Some Estimates for Commutators of Fractional Integrals Associated to Operators with Gaussian Kernel Bounds on Weighted Morrey Spaces

Let $L$ be the infinitesimal generator of an analytic semigroup on $L^2(\mathbb R^n)$ with Gaussian kernel bound, and let $L^{-\alpha/2}$ be the fractional integrals of $L$ for $0<\alpha<n$. In this paper, we will obtain some boundedness properties of commutators $\big[b,L^{-\alpha/2}\big]$ on the weighted Morrey spaces $L^{p,\kappa}(w)$ when the symbol $b$ belongs to $BMO(\mathbb R^n)$ or homogeneous Lipschitz space.

Spectral Analysis and Variable Structural Control of an Elastic Beam

An elastic beam system formulated by partial differential equations with initial and boundary conditions is investigated in this paper. An evolution equation corresponding with the beam system is established in an appropriate Hilbert space. The spectral analysis and semigroup generation of the system operator of the beam system are discussed. Finally, a variable structural control is proposed and a significant result that the solution of the system is exponentially stable under a variable structural control with some appropriate conditions is obtained.

Rational decay rates for a PDE heat--structure interaction: A frequency domain approach

In this paper, we consider a simplified version of a fluid--structure PDEmodel---in fact, a heat--structure interaction PDE-model. It is intended to be a first step toward a more realistic fluid--structure PDE model which has been of longstanding interest within the mathematical and biological sciences [33, p. 121], [17], [19]. This physically more sound and mathematically more challenging model will be treated in [13]. The simplified model replaces the linear dynamic Stokes equation with a linear $n$-dimensional heat equation (heat--structure interaction). The entire dynamics manifests both hyperbolic and parabolic features. Our main result is as follows: Given smooth initial data---i.e., data in the domain of the associated semigroup generator---the corresponding solutions decay at the rate $o( t^{-\frac{1}{2}}) $(see Theorem 1.3 below). The basis of our proof is the recently derived resolvent criterion in [15]. In order to apply it, however, suitable PDE-estimates need to be established for each component by also making critical use of the interface conditions. A companion paper [6] will sharpen Lemma 5.8 of the present work by use of a lengthy and technical microlocal argument as in [26,29,30,31], to obtain the optimal value $\alpha =1$; hence, the optimal decay rate $o(t^{-1})$. See Remarks 1.2,1.3.

Distributional chaos for strongly continuous semigroups of operators

Distributional chaos for strongly continuous semigroups is studied and characterized. It is shown to be equivalent to the existence of a distributionally irregular vector. Finally, a sufficient condition for distributional chaos on the point spectrum of the generator of the semigroup is presented. An application to the semigroup generated in $L^2(R)$ by a translation of the Ornstein-Uhlenbeck operator is also given.

Resolvent conditions for the control of parabolic equations

Since the seminal work of Russell and Weiss in 1994, resolvent conditions for various notions of admissibility, observability and controllability, and for various notions of linear evolution equations have been investigated intensively, sometimes under the name of infinite-dimensional Hautus test. This paper sets out resolvent conditions for null-controllability in arbitrary time: necessary for general semigroups, sufficient for analytic normal semigroups. For a positive self-adjoint operator A, it gives a sufficient condition for the null-controllability of the semigroup generated by −A which is only logarithmically stronger than the usual condition for the unitary group generated by iA. This condition is sharp when the observation operator is bounded. The proof combines the so-called “control transmutation method” and a new version of the “direct Lebeau–Robbiano strategy”. The improvement of this strategy also yields interior null-controllability of new logarithmic anomalous diffusions.