Spaces of Smooth and Generalized Vectors of the Generator of an Analytic Semigroup and Their Applications

2017 ◽  
Vol 69 (4) ◽  
pp. 561-597 ◽  
Author(s):  
V. M. Horbachuk ◽  
M. L. Horbachuk
Keyword(s):  
Analytic Semigroup

scholarly journals Analytic semigroups generated by Dirichlet-to-Neumann operators on manifolds

2021 ◽  
Author(s):  
Tim Binz
Keyword(s):  
Boundary Conditions ◽  
Elliptic Operator ◽  
Compact Manifold ◽  
Analytic Semigroup ◽  
Continuous Functions ◽  
Analytic Semigroups ◽  
Abstract Theory ◽  
Neumann Operator ◽  
Wentzell Boundary Conditions ◽  
Dirichlet To Neumann Operator

AbstractWe consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $$\mathrm {C}(\partial M)$$ C ( ∂ M ) of continuous functions on the boundary $$\partial M$$ ∂ M of a compact manifold $$\overline{M}$$ M ¯ with boundary. We prove that it generates an analytic semigroup of angle $$\frac{\pi }{2}$$ π 2 , generalizing and improving a result of Escher with a new proof. Combined with the abstract theory of operators with Wentzell boundary conditions developed by Engel and the author, this yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $$\frac{\pi }{2}$$ π 2 on the space $$\mathrm {C}(\overline{M})$$ C ( M ¯ ) .

scholarly journals The Super-Diffusive Singular Perturbation Problem

Mathematics ◽  
2020 ◽  
Vol 8 (3) ◽  
pp. 403
Author(s):  
Edgardo Alvarez ◽  
Carlos Lizama
Keyword(s):  
Singular Perturbation ◽  
Unique Solution ◽  
Analytic Semigroup ◽  
Singular Perturbation Problem ◽  
Singularly Perturbed ◽  
Regularity Conditions ◽  
Cauchy Problems ◽  
Perturbation Problem ◽  
Bounded Analytic Semigroup ◽  
Abstract Cauchy Problems

In this paper we study a class of singularly perturbed defined abstract Cauchy problems. We investigate the singular perturbation problem ( P ϵ ) ϵ α D t α u ϵ ( t ) + u ϵ ′ ( t ) = A u ϵ ( t ) , t ∈ [ 0 , T ] , 1 < α < 2 , ϵ > 0 , for the parabolic equation ( P ) u 0 ′ ( t ) = A u 0 ( t ) , t ∈ [ 0 , T ] , in a Banach space, as the singular parameter goes to zero. Under the assumption that A is the generator of a bounded analytic semigroup and under some regularity conditions we show that problem ( P ϵ ) has a unique solution u ϵ ( t ) for each small ϵ > 0 . Moreover u ϵ ( t ) converges to u 0 ( t ) as ϵ → 0 + , the unique solution of equation ( P ) .

An analytic semigroup version of the Beurling-Helson theorem

1997 ◽  
Vol 225 (1) ◽  
pp. 151-165
Author(s):  
José E. Galé ◽  
Michael C. White
Keyword(s):  
Analytic Semigroup ◽  
Semigroup Version

scholarly journals Exponential-Square Integrability, Weighted Inequalities for the Square Functions Associated to Operators, and Applications

2020 ◽  
Author(s):  
Peng Chen ◽  
Xuan Thinh Duong ◽  
Liangchuan Wu ◽  
Lixin Yan
Keyword(s):  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Square Function ◽  
Weighted Norm ◽  
Square Functions ◽  
Spectral Multipliers ◽  
Eigenvalue Estimates ◽  
Preservation Condition ◽  
Square Integrability ◽  
The Laplace Operator

Abstract Let $X$ be a metric space with a doubling measure. Let $L$ be a nonnegative self-adjoint operator acting on $L^2(X)$, hence $L$ generates an analytic semigroup $e^{-tL}$. Assume that the kernels $p_t(x,y)$ of $e^{-tL}$ satisfy Gaussian upper bounds and Hölder continuity in $x$, but we do not require the semigroup to satisfy the preservation condition $e^{-tL}1 = 1$. In this article we aim to establish the exponential-square integrability of a function whose square function associated to an operator $L$ is bounded, and the proof is new even for the Laplace operator on the Euclidean spaces ${\mathbb R^n}$. We then apply this result to obtain: (1) estimates of the norm on $L^p$ as $p$ becomes large for operators such as the square functions or spectral multipliers; (2) weighted norm inequalities for the square functions; and (3) eigenvalue estimates for Schrödinger operators on ${\mathbb R}^n$ or Lipschitz domains of ${\mathbb R}^n$.

Asymptotic estimates for a semigroup related to compressible viscous fluid flow

Analysis ◽  
2007 ◽  
Vol 27 (1) ◽  
Author(s):  
Gerhard Ströhmer
Keyword(s):  
Fluid Flow ◽  
Viscous Fluid ◽  
Linear Stability ◽  
Analytic Semigroup ◽  
Spherically Symmetric ◽  
Asymptotic Estimates ◽  
Governing Equations ◽  
Symmetric Equilibrium ◽  
Compressible Viscous Fluid ◽  
Viscous Fluid Flow

The paper is related to the question of stability for the motionless spherically symmetric equilibrium states of viscous, barotropic, self-gravitating fluids. It considers a perturbation of the linearization of the governing equations of this problem, taking a step in the derivation of estimates which will allow us to prove non-linear stability of the equilibria. The perturbed operator, like the linearization considered earlier, generates an analytic semigroup, which allows us to derive asymptotic estimates as

The fragmentation equation with size diffusion: Well posedness and long-term behaviour

2021 ◽  
pp. 1-34 ◽  
Author(s):  
PH. LAURENÇOT ◽  
CH. WALKER
Keyword(s):  
Linear Operator ◽  
Stationary Solution ◽  
Analytic Semigroup ◽  
Well Posedness ◽  
Exponential Rate ◽  
Initial Value ◽  
One Dimensional ◽  
Fragmentation Rate ◽  
Fragmentation Equation

The dynamics of the fragmentation equation with size diffusion is investigated when the size ranges in $(0,\infty)$ . The associated linear operator involves three terms and can be seen as a nonlocal perturbation of a Schrödinger operator. A Miyadera perturbation argument is used to prove that it is the generator of a positive, analytic semigroup on a weighted $L_1$ -space. Moreover, if the overall fragmentation rate does not vanish at infinity, then there is a unique stationary solution with given mass. Assuming further that the overall fragmentation rate diverges to infinity for large sizes implies the immediate compactness of the semigroup and that it eventually stabilizes at an exponential rate to a one-dimensional projection carrying the information of the mass of the initial value.

scholarly journals Notes on boundedness of spectral multipliers on Hardy spaces associated to operators

2011 ◽  
Vol 203 ◽  
pp. 109-122
Author(s):  
Bui The Anh
Keyword(s):  
Hardy Space ◽  
Hardy Spaces ◽  
Analytic Semigroup ◽  
Adjoint Operator ◽  
Upper Bounds ◽  
Space Of Homogeneous Type ◽  
Homogeneous Type ◽  
Spectral Multipliers ◽  
Spectral Multiplier ◽  
Suitable Function

AbstractLetLbe a nonnegative self-adjoint operator onL2(X), whereXis a space of homogeneous type. Assume thatLgenerates an analytic semigroupe–tlwhose kernel satisfies the standard Gaussian upper bounds. We prove that the spectral multiplierF(L) is bounded onfor 0&lt; p&lt; 1, the Hardy space associated to operatorL, whenFis a suitable function.

scholarly journals Existence of Mild Solutions for a Class of Fractional Evolution Equations with Compact Analytic Semigroup

2012 ◽  
Vol 2012 ◽  
pp. 1-15 ◽  
Author(s):  
He Yang
Keyword(s):  
Analytic Semigroup ◽  
Evolution Equations ◽  
Mild Solutions ◽  
Fractional Power ◽  
Growth Conditions ◽  
Local Growth ◽  
Fractional Evolution Equations ◽  
Nonlinear Part ◽  
Compact Analytic Semigroup

This paper deals with the existence of mild solutions for a class of fractional evolution equations with compact analytic semigroup. We prove the existence of mild solutions, assuming that the nonlinear part satisfies some local growth conditions in fractional power spaces. An example is also given to illustrate the applicability of abstract results.

ANALYTICITY, HYPERBOLICITY AND UNIFORM STABILITY OF SEMIGROUPS ARISING IN MODELS OF COMPOSITE BEAMS

2000 ◽  
Vol 10 (04) ◽  
pp. 555-580 ◽  
Author(s):  
SCOTT W. HANSEN ◽  
IRENA LASIECKA
Keyword(s):  
Boundary Conditions ◽  
Analytic Semigroup ◽  
Inner Core ◽  
Sandwich Beam ◽  
Composite Beams ◽  
Rotational Inertia ◽  
Viscous Resistance ◽  
Stability Properties ◽  
Exponentially Stable ◽  
The Stability

We examine the stability properties of a sandwich beam consisting of two outer layers and a thin core. The outer layers are modeled as Euler Bernoulli beams and the inner core provides both elastic and viscous resistance to shearing. We show for both clamped and hinged boundary conditions that (i) if rotational inertia terms are neglected, the model is described by an analytic semigroup, and (ii) if rotational inertia is retained in the outer layers, the model is uniformly exponentially stable.

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