Semigroup applications everywhere

Abstract

Most dynamical systems arise from partial differential equations (PDEs) that can be represented as an abstract evolution equation on a suitable state space complemented by an initial or final condition. Thus, the system can be written as a Cauchy problem on an abstract function space with appropriate topological structures. To study the qualitative and quantitative properties of the solutions, the theory of one-parameter operator semigroups is a most powerful tool. This approach has been used by many authors and applied to quite different fields, e.g. ordinary and PDEs, nonlinear dynamical systems, control theory, functional differential and Volterra equations, mathematical physics, mathematical biology, stochastic processes.

The present special issue of Philosophical Transactions includes papers on semigroups and their applications.

This article is part of the theme issue ‘Semigroup applications everywhere’.

1. Introduction

The exponential function, as one of the most important objects in pure and applied mathematics, has a long and fascinating history. Leonhard Euler’s original mid-eighteenth-century scalar-valued function, which arose in his treatment of elementary functions in 1748 and his study of calculus of differences and differentials, power series and summation formulae in 1755, was extended by Giuseppe Peano to the matrix-valued case by the end of the nineteenth century. After David Hilbert and Stefan Banach in the early twentieth century had established the theory of infinite-dimensional Hilbert and Banach spaces and the operators thereon, Marshall Stone in 1930 and Israel Gelfand in 1939 made the first steps towards an infinite-dimensional exponential function. The final step was taken in 1948 by Einar Hille and K$\overline{\text{o}}$saku Yosida, independently and in distinct contexts. They constructed and characterized, in what is now called the Hille–Yosida theorem, operator-valued exponential functions on general Banach spaces.

Such exponential functions

$${\mathbb{R}}_{+}\ni t\mapsto T(t)\in \mathcal{L}(X),X\text{a Banach space, }$$

are called one-parameter operator semigroups or semigroups for short. We refer to Engel–Nagel [1] for a brief history. These semigroups are one-parameter families (T(t))_t≥0 of bounded linear operators on a Banach space X satisfying the functional equation (also known as the semigroup law)

$$T(t+s)=T(t)T(s)\hspace{0.17em}\text{ for }s,\hspace{0.17em}t\ge 0\hspace{0.17em}\text{ and }T(0)=Id,$$

and such that the maps

$$t\mapsto T(t))x$$

are continuous for each x ∈ X.

In Peano’s matrix-valued case, for example if the Banach space is ${\mathbb{C}}^n$, the above two properties already imply that there exists an n × n matrix A such that

$$T(t)=exp(tA)\text{for }t\in \mathbb{R},$$

defined in the classical sense by the exponential series. Therefore, the map $t\mapsto x(t):=T(t)x$ is even differentiable and yields the unique solution of the initial-value problem

$$\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}(t)=Ax(t),t\ge 0,\\ x(0)=x.\end{array}$$

It seems to be a miracle that even in the infinite-dimensional case the above continuity combined with the semigroup law still yields the differentiability of the function $t\mapsto T(t)x$, at least for x in a dense subset of the Banach space X. This allows one to define an operator A as the derivative of this function at t = 0, namely

$$Ax:=\frac{\text{d}}{\text{d}t}T(t)x{|}_{t=0},$$

which is then called the generator of the semigroup (T(t)). This operator gives rise to an (abstract) initial-value problem of the form

$$\{\begin{array}{l}\frac{\text{d}x}{\text{d}t}(t)=Ax(t),t\ge 0,\\ x(0)=x,\end{array}$$

whose solution is obtained as x(t) = T(t)x.

So the semigroup (T(t))_t≥0, defined by an algebraic and a continuity property, solves an abstract differential equation. In order to solve such initial-value problems for a concrete (differential) operator, it suffices to show that the operator A is the generator of a semigroup. The Hille–Yosida theorem achieves exactly this. Based on the rich theory of these semigroups it is now possible to solve initial value problems for quite different types of evolution equations and to discuss properties of their solutions.

The basic theory for these semigroups was already established in Einar Hille’s monumental monograph Semigroups and Functional Analysis [2] from 1948, with a second and extended edition written jointly with Ralph Phillips [3] in 1957. It took the interested community a couple of years to digest this magnum opus.

Only around 1980 did several more accessible books appear, which served to introduce many readers to the basic theory and some applications of semigroups. We mention in chronological order Belleni-Morante (1979) [4], reviving the ideas of Peano in Italy, Davies (1980) [5], Pazy (1983) [6] and Goldstein (1985) [7]. In particular Pazy’s book [6], with his discussion of partial differential operators, contributed to the rising popularity of semigroup theory. In the following years, parts of the theory were completed (e.g. spectral theory, Nagel [8]), new aspects were studied (e.g. asymptotic behaviour, van Neerven [9], Eisner [10]), or special classes of semigroups were investigated in detail (e.g. positive semigroups, Nagel (ed.) [8], Clément et al. [11]). These more recent developments as well as a broad range of applications to linear evolution equations are collected in the 2000 volume by Engel–Nagel [1]. For other monographs on semigroup theory and applications, we refer to Lunardi [12], Batkai-Piazzera [13], Banasiak-Arlotti [14], Batkai-Kramar-Rhandi [15], Sinha-Srivastava [16], Applebaum [17] and Lorenzi-Rhandi [18].

Again 20 years later the time seems to be ripe to present with this volume a panorama of some of the many areas to which this theory can be applied. We hope that this collection under the title Semigroup applications everywhere will arouse the interest of curious readers in the spirit of Einar Hille, who wrote in his 1948 volume:

I hail a semigroup whenever I see one and I seem to see them everywhere.

2. The content of this issue

It is our intention to present a panorama of various applications based on a single mathematical theory under the leitmotif Semigroup applications everywhere. In this issue, we bring together leading groups working on such applications. In particular, we touch the following topics: semigroups and PDEs, form methods and perturbations of semigroups, semigroups and stochastic differential equations, semigroups for control problems, semigroups and asymptotics, semigroups for biology, as well as semigroups for dynamical systems.

We now summarize all contributions presented in this issue.

(a). Semigroups and asymptotics

• —Positive irreducible semigroups and their long-time behaviour, by W. Arendt and J. Glück:

  The notion Perron-Frobenius theory refers to the interaction between three properties of operator semigroups: positivity, spectrum and long-time behaviour. By a brief walk-through of the field with many examples, the authors highlight two aspects of the subject, both related to the long-time behaviour of semigroups: (i) The classical question how positivity of a semigroup can be used to prove convergence to an equilibrium as t → ∞. (ii) The more recent phenomenon that positivity itself sometimes occurs only for large t, while being absent for smaller times.

• —Semi-uniform stability of operator semigroups and energy decay of damped waves, by R. Chill, D. Seifert and Y. Tomilov:

  The notion of semi-uniform stability lies between exponential stability and strong stability, and became part of the asymptotic theory of C₀-semigroups. After briefly reviewing the various notions of stability, the authors present an overview of abstract results on semi-uniform stability. They indicate how these results can be applied to obtain (sometimes optimal) rates of energy decay for certain damped second-order Cauchy problems.

(b). Semigroups for dynamical systems

• —Towards a Koopman theory for dynamical systems on completely regular spaces, by B. Farkas and H. Kreidler:

  The Koopman linearization of measure-preserving or topological dynamical systems has proven to be extremely useful. In their article the authors look at dynamics given by continuous semiflows on completely regular spaces which arise naturally from solutions of PDEs. They introduce Koopman semigroups for these semiflows on spaces of bounded continuous functions and study their continuity properties as well as their infinitesimal generators. Finally, they demonstrate how this Koopman approach can be used to examine attractors of dynamical systems.

(c). Semigroups for control problems

• —Feedback theory to the well-posedness of evolution equations, by S. Boulite, S. Hadd and L. Maniar:

  Crossing the boundary between semigroup and general infinite-dimensional systems theory, the authors give a history of the development of the semigroup approach for control theory. They use the feedback theory to prove the well-posedness of a class of dynamic boundary problems. The results are applied to the well-posedness of neutral equations with nonautonomous past. They also explain the strong connection between the feedback theory and perturbations of semigroups.

(d). Form methods and perturbations of semigroups

• —Form-perturbation theory for higher-order elliptic operators and systems by singular potentials, by M. Mokhtar-Kharroubi:

  The author develops a form-perturbation theory by singular potentials for scalar elliptic operators on $L^2({\mathbb{R}}^d)$ of order 2m with Hölder continuous coefficients. The form-bounds are obtained from an L¹ functional analytic approach which takes advantage of both the existence of m-Gaussian kernel estimates and the holomorphy of the semigroup in $L^1({\mathbb{R}}^d)$. The results are extended to elliptic systems and singular matrix potentials.

(e). Semigroups and partial differential equations

• —First-order evolution equations with dynamic boundary conditions, by T. Binz and K.J. Engel:

  A general framework is proposed to study linear first-order evolution equations on a Banach space X with dynamic boundary conditions, that is, with boundary conditions containing time derivatives. The method is based on the existence of an abstract Dirichlet operator and yields to two simpler independent equations. Their approach is illustrated by examples and various generalizations are indicated.

• —On operator semigroups arising in the study of incompressible viscous fluid flows, by M. Hieber:

  Various operator semigroups arising in the study of viscous and incompressible flows, such as the classical Stokes semigroup, the hydrostatic Stokes semigroup, the Oldroyd as well as the Ericksen–Leslie semigroup. The properties of these semigroups are investigated and applied to the associated nonlinear equations.

• —The Ornstein-Uhlenbeck semigroup in finite dimension, by A. Lunardi, G. Metafune and D. Pallara:

  The Ornstein-Uhlenbeck semigroup describes the stochastic Ornstein-Uhlenbeck process which is a stationary Gauss-Markov process. It was used first in physics as a model for the velocity of a massive Brownian particle under the influence of friction. Here, the authors gather the main results concerning the non-degenerate Ornstein-Uhlenbeck semigroup in finite dimension.

• —Ornstein-Uhlenbeck semigroups in infinite dimension, by A. Lunardi and D. Pallara:

  This is a survey about Ornstein-Uhlenbeck semigroups in infinite dimension and their generators. The authors start from the classical Ornstein-Uhlenbeck semigroup on Wiener spaces and then discuss the general case in Hilbert spaces. Finally, they present some results for Ornstein-Uhlenbeck semigroups on Banach spaces.

(f). Semigroups and stochastic differential equations

• —Maximal inequalities for stochastic convolutions in 2-smooth Banach spaces and applications to stochastic evolution equations, by J. van Neerven and M. Veraar:
  This paper presents a survey of maximal inequalities for the stochastic convolution

  $$u_t={\int }_0^{t}S(t,s)g_s\hspace{0.17em}\text{d}{W}_s,t\in [0,T],$$

  in 2-smooth Banach spaces and their applications to stochastic evolution equations. Here, (S(t, s))_{0≤s≤t≤T} is a strongly continuous evolution family acting on a Banach space X, (W_t)_0≤t≤T is a (cylindrical) Brownian motion defined on a probability space Ω, and (g_t)_0≤t≤T is a stochastic process taking values in X.

(g). Semigroups for biology

• —Growth-fragmentation-coagulation equations with unbounded coagulation kernels, by J. Banasiak and W. Lamb:

  The aim is to prove, using semigroups, the global in time solvability of the continuous growth-fragmentation-coagulation equation with unbounded coagulation kernels in spaces of functions having finite moments of sufficiently high order. The main tool is the recently established result on moment regularization of the linear growth-fragmentation semigroup that allows one to consider coagulation kernels whose growth for large clusters is controlled by how good the regularization is, in a similar manner to the case when the semigroup is analytic.

• —On convergence and asymptotic behaviour of semigroups of operators, by A. Bobrowski and R. Rudnicki:

  The classical and modern theorems on convergence, approximation and asymptotic stability of semigroups of operators are presented, and their applications to recent biological models are discussed.

• —Semigroups for dynamical processes on metric graphs, by M. Kramar Fijavz and A. Puchalska:

  The authors present the operator semigroups approach to first and second-order dynamical systems taking place on metric graphs. They briefly survey the existing results and focus on the well-posedness of the problems with standard vertex conditions. Finally, they give two applications to biological models.

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