Generator Of Analytic Semigroup Research Articles

Generation of analytic semigroups by elliptic operators with unbounded diffusion, drift and potential terms

Generation of analytic semigroups by elliptic operators with unbounded diffusion, drift and potential terms

Half-Space Model Problem for Navier-Lamé Equations with Surface Tension

Recently, we have seen the phenomena in use of partial differential equations (PDEs) especially in fluid dynamic area. The classical approach of the analysis of PDEs were dominated in early nineteenth century. As we know that for PDEs the fundamental theoretical question is whether the model problem consists of equation and its associated side condition is well-posed. There are many ways to investigate that the model problems are well-posed. Because of that reason, in this paper we consider the <img src=image/13426244_01.gif>-boundedness of the solution operator families for Navier-Lamé equation by taking into account the surface tension in a bounded domain of <img src=image/13426244_02.gif>-dimensional Euclidean space (<img src=image/13426244_02.gif>≥ 2) as one way to study the well-posedess. We investigate the <img src=image/13426244_01.gif>-boundedness in half-space domain case. The <img src=image/13426244_01.gif>-boundedness implies not only the generation of analytic semigroup but also the maximal <img src=image/13426244_03.gif> regularity for the initial boundary value problem by using Weis's operator valued Fourier multiplier theorem for time dependent problem. It was known that the maximal <img src=image/13426244_03.gif> regularity class is the powerful tool to prove the well-posesness of the model problem. This result can be used for further research for example to analyze the boundedness of the solution operators of the model problem in bent-half space or general domain case.

Generation of analytic semigroups for some generalized diffusion operators in $$L^p$$-spaces

We consider five different abstract Cauchy problems where the operator is of fourth order. For each problem, using semigroups techniques and functional calculus, we study the invertibility and the spectral properties of the fourth order operator to show that it generates an analytic semigroup.

An analytic semigroup generated by a fractional differential operator

We study a space-fractional diffusion problem, where the non-local diffusion flux involves the Caputo derivative of the diffusing quantity. We prove the unique existence of regular solutions to this problem by means of the semigroup theory. We show that the operator defined as divergence of product of a positive function with the Caputo derivative is a generator of analytic semigroup.

B-Coercive Convolution Equations in Weighted Function Spaces and Their Applications

We study the properties of B-separability for elliptic convolution operators in weighted Besov spaces and establish sharp estimates for the resolvents of the convolution operators. As a result, it is shown that these operators are positive and, in addition, play the role of negative generators of analytic semigroups. Moreover, the maximal B-regularity properties are established for the Cauchy problem for a parabolic convolution equation. Finally, these results are applied to obtain the maximal regularity properties for anisotropic integrodifferential equations and a system of infinitely many convolution equations.

On the $\mathcal{R}$-boundedness of solution operator families for two-phase Stokes resolvent equations

The aim of this paper is to show the existence of $\mathcal{R}$-bounded solution operator families for two-phase Stokes resolvent equations in $\dot\Omega =\Omega _+\cup\Omega _-$, where $\Omega _\pm$ are uniform $W_r^{2-1/r}$ domains of $N$-dimensional Euclidean space ${\mathbf{R}^N}$ ($N\geq 2$, $N < r < \infty$). More precisely, given a uniform $W_r^{2-1/r}$ domain $\Omega $ with two boundaries $ \Gamma _\pm$ satisfying $ \Gamma _+\cap \Gamma _-=\emptyset$, we suppose that some hypersurface $ \Gamma $ divides $\Omega $ into two sub-domains, that is, there exist domains $\Omega _\pm\subset\Omega $ such that $ \Omega _+\cap\Omega _-=\emptyset$ and $\Omega \setminus \Gamma =\Omega _+\cup\Omega _-, $ where $ \Gamma \cap \Gamma _+=\emptyset$, $ \Gamma \cap \Gamma _-=\emptyset$, and the boundaries of $\Omega _\pm$ consist of two parts $ \Gamma $ and $ \Gamma _\pm$, respectively. The domains $\Omega _\pm$ are filled with viscous, incompressible, and immiscible fluids with density $\rho_\pm$ and viscosity $\mu_\pm$, respectively. Here, $\rho_\pm$ are positive constants, while $\mu_\pm=\mu_\pm(x)$ are functions of $x\in{\mathbf{R}^N}$. On the boundaries $ \Gamma $, $ \Gamma _+$, and $ \Gamma _-$, we consider an interface condition, a free boundary condition, and the Dirichlet boundary condition, respectively. We also show, by using the $\mathcal{R}$-bounded solution operator families, some maximal $L_p{\theta}xt{-}L_q$ regularity as well as generation of analytic semigroup for a time-dependent problem associated with the two-phase Stokes resolvent equations. This kind of problems arises in the mathematical study of the motion of two viscous, incompressible, and immiscible fluids with free surfaces. The essential assumption of this paper is the unique solvability of a weak elliptic transmission problem for $\mathbf{f}\in L_q(\Omega )^N$, that is, it is assumed that the unique existence of solutions ${\theta}\in\mathcal{W}_q^1(\Omega )$ to the variational problem: $ (\rho^{-1}\nabla{\theta},\nabla{\varphi})_{\dot\Omega }=(\mathbf{f},\nabla{\varphi})_{\Omega } $ for any ${\varphi}\in\mathcal{W}_{q'}^1(\Omega )$ with $1 < q < \infty$ and $q'=q/(q-1)$, where $\rho$ is defined by $\rho=\rho_+$ ($x\in\Omega _+$), $\rho=\rho_-$ ($x\in\Omega _-$) and $\mathcal{W}_q^1(\Omega )$ is a suitable Banach space endowed with norm $\|\cdot\|_{\mathcal{W}_q^1(\Omega )}:=\|\nabla\cdot\|_{L_q(\Omega )}$. Our assumption covers e.g. the following domains as $\Omega $: ${\mathbf{R}^N}$, $\mathbf{R}_\pm^N$, perturbed $\mathbf{R}_\pm^N$, layers, perturbed layers, and bounded domains, where $\mathbf{R}_+^N$ and $\mathbf{R}_-^N$ are the open upper and lower half spaces, respectively.

On measuring unboundedness of the H∞-calculus for generators of analytic semigroups

We investigate the boundedness of the H∞-calculus by estimating the bound b(ε) of the mapping H∞→B(X): f↦f(A)T(ε) for ε near zero. Here, −A generates the analytic semigroup T and H∞ is the space of bounded analytic functions on a domain strictly containing the spectrum of A. We show that b(ε)=O(|log⁡ε|) in general, whereas b(ε)=O(1) for bounded calculi. This generalizes a result by Vitse and complements work by Haase and Rozendaal for non-analytic semigroups. We discuss the sharpness of our bounds and show that single square function estimates yield b(ε)=O(|log⁡ε|).

On some two phase problem for compressible and compressible viscous fluid flow separated by sharp interface

In this paper, we prove a local in time unique existence theorem for some two phase problem of compressible and compressible barotropic viscous fluid flow without surface tension in the $L_p$ in time and the $L_q$ in space framework with $2< p <\infty$ and $N< q <\infty$ under the assumption that the initial domain is a uniform $W^{2-1/q}_q$ domain in $\mathbb{R}^N (N\ge 2)$. After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve the problem by the contraction mapping principle with the maximal $L_p$-$L_q$ regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key step of our method is to prove the existence of $\mathcal{R}$-bounded solution operator to resolvent problem corresponding to linearized problem. The $\mathcal{R}$-boundedness combined with Weis's operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal $L_p$-$L_q$ regularity theorem.

Generation of Semigroups for Vector-Valued Pseudodifferential Operators on the Torus

We consider toroidal pseudodifferential operators with operator-valued symbols, their mapping properties and the generation of analytic semigroups on vector-valued Besov and Sobolev spaces. Here, we restrict ourselves to pseudodifferential operators with x-independent symbols (Fourier multipliers). We show that a parabolic toroidal pseudodifferential operator generates an analytic semigroup on the Besov space $$B_{pq}^s({\mathbb T}^n,E)$$ and on the Sobolev space $$W_p^k({\mathbb T}^n,E)$$ , where E is an arbitrary Banach space, $$1\le p,q\le \infty $$ , $$s\in {\mathbb R}$$ and $$k\in {\mathbb N}_0$$ . For the proof of the Sobolev space result, we establish a uniform estimate on the kernel which is given as an infinite parameter-dependent sum. An application to abstract non-autonomous periodic pseudodifferential Cauchy problems gives the existence and uniqueness of classical solutions for such problems.

Rs-bounded H∞-calculus for sectorial operators via generalized Gaussian estimates

We show that, for negative generators of analytic semigroups, a bounded -calculus self-improves to an -bounded -calculus in an appropriate scale of -spaces if the semigroup satisfies suitable generalized Gaussian estimates. As application of our result we obtain that large classes of differential operators have an -bounded -calculus.

Generation of analytic semigroups with generalized Wentzell boundary condition

In this paper we study second order ordinary and partial differential equations with generalized Wentzell boundary condition. We prove, by a perturbation method, that certain second order ordinary differential operators generate an analytic semigroup in \(W^{1,p}([0,1])\) and the same result has been extended for the degenerate operator \(Au(x):=x(1-x)u''(x)\). Finally, we prove that certain linear partial differential operators of the second order generate analytic semigroups in the space of continuous functions.

On the ℛ-boundedness for the two phase problem: compressible-incompressible model problem

The situation of this paper is that the Stokes equation for the compressible viscous fluid flow in the upper half-space is coupled via inhomogeneous interface conditions with the Stokes equations for the incompressible one in the lower half-space, which is the model problem for the evolution of compressible and incompressible viscous fluid flows with a sharp interface. We show the existence of ℛ-bounded solution operators to the corresponding generalized resolvent problem, which implies the generation of analytic semigroup and maximal L p - L q regularity for the corresponding time dependent problem with the help of the Weis’ operator valued Fourier multiplier theorem. The problem was studied by Denisova (Interfaces Free Bound. 2(3):283-312, 2000) under some restriction on the viscosity coefficients and one of our purposes is to eliminate the assumption in (Denisova in Interfaces Free Bound. 2(3):283-312, 2000).MSC: 35Q35, 76T10.

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.

On some free boundary problem for a compressible barotropic viscous fluid flow

In this paper, we prove a local in time unique existence theorem for the free boundary problem of a compressible barotropic viscous fluid flow without surface tension in the \(L_p\) in time and \(L_q\) in space framework with \(2 < p < \infty \) and \(N < q < \infty \) under the assumption that the initial domain is a uniform \(W^{2-1/q}_q\) one in \({\mathbb {R}}^{N}\, (N \ge 2\)). After transforming a unknown time dependent domain to the initial domain by the Lagrangian transformation, we solve problem by the Banach contraction mapping principle based on the maximal \(L_p\)–\(L_q\) regularity of the generalized Stokes operator for the compressible viscous fluid flow with free boundary condition. The key issue for the linear theorem is the existence of \({\mathcal {R}}\)-bounded solution operator in a sector, which combined with Weis’s operator valued Fourier multiplier theorem implies the generation of analytic semigroup and the maximal \(L_p\)–\(L_q\) regularity theorem. The nonlinear problem we studied here was already investigated by several authors (Denisova and Solonnikov, St. Petersburg Math J 14:1–22, 2003; J Math Sci 115:2753–2765, 2003; Secchi, Commun PDE 1:185–204, 1990; Math Method Appl Sci 13:391–404, 1990; Secchi and Valli, J Reine Angew Math 341:1–31, 1983; Solonnikov and Tani, Constantin caratheodory: an international tribute, vols 1, 2, pp 1270–1303, World Scientific Publishing, Teaneck, 1991; Lecture notes in mathematics, vol 1530, Springer, Berlin, 1992; Tani, J Math Kyoto Univ 21:839–859, 1981; Zajaczkowski, SIAM J Math Anal 25:1–84, 1994) in the \(L_2\) framework and Holder spaces, but our approach is different from them.

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.

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 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.

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.

On the $\mathcal{R}$-Sectoriality and the Initial Boundary Value Problem for the Viscous Compressible Fluid Flow

In this paper, we prove the $\mathcal{R}$-sectoriality of the resolvent problem for the boundary value problem of the Stokes operator for the compressible viscous fluids in a general domain, which implies the generation of analytic semigroup and the maximal Lp-Lq regularity for the initial boundary value problem of the Stokes operator.<! > Combining our linear theory with fixed point arguments in the Lagrangian coordinates, we have a local in time unique existence theorem in a general domain and a global in time unique existence theorem for some initial data close to a constant state in a bounded domain for the initial boundary value problem of the Navier-Stokes equations describing the motion of compressible viscous fluids. All the results obtained in this paper were announced in Enomoto-Shibata [12].