Sharp rates of energy decay for damped waves - Oxford Mathematics Research

Differential equations arising in physics and elsewhere often describe the evolution in time of quantities which also depend on other (typically spatial) variables. Well known examples of such evolution equations include the heat equation and the wave equation. A rigorous, functional analytic approach to the study of linear autonomous evolution equations begins by considering the associated abstract Cauchy problem, \begin{equation}\label{eq:ACP} \hspace{100pt}\left\{\begin{aligned} \dot{u}(t)&=Au(t),\quad t\ge0,\hspace{200pt} (1)\\ u(0)&=x\in X. \end{aligned}\right. \end{equation} Here $A$ is a linear operator (typically unbounded) acting on a suitably chosen Banach space $X$, which is usually a space of functions or a product of such spaces. For instance, in the case of the heat equation on a domain $\Omega$ we might choose $X=L^2(\Omega)$ and let $A$ be the Laplace operator with suitable boundary conditions, and for the wave equation we would take $X$ to be a product of two function spaces corresponding, respectively, to the displacement and the velocity of the wave. Assuming the abstract Cauchy problem to be well posed, there exists a family $(T(t))_{t\ge0}$ of bounded linear operators (a so-called $C_0$-semigroup) acting on (1) such that the solution of (1) is given by $u(t)=T(t)x$, $t\ge0$. Of course, we cannot normally hope to solve (1) exactly, so the operators $T(t)$, $t\ge0$, are in general unknown. The main task is to deduce useful information about the semigroup $(T(t))_{t\ge0}$ from what is known about $A$, in particular its spectral properties.

In concrete applications, the norm on the space $X$ often admits a physical interpretation. An important example of this kind is the wave equation, where $X$ is a Hilbert space with the property that the induced norm of the solution $u(t)=T(t)x$ is related in a very natural way to the energy of the solution at time $t\ge0$. Thus we may study energy decay of waves, a fundamental problem in mathematical physics, by investigating the asymptotic behaviour of the norms $\|u(t)\|$ as $t\to\infty$. In the classical (undamped) wave equation the operators $T(t)$, $t\ge0$, are isometries (even unitary operators), so energy is conserved. On the other hand, as soon as there is some sort of damping, for instance due to air resistance or other frictional forces, we expect the energy of any solution to decay over time. But at what rate? As it turns out, we may associate with any damped wave equation an increasing continuous function $M\colon[0,\infty)\to(0,\infty)$, which captures important spectral properties of the operator $A$ and in all cases of interest will satisfy $M(s)\to\infty$ as $s\to\infty$. In practice, obtaining good estimates on the function $M$ may itself be a non-trivial problem (the precise behaviour of $M$ is determined by the nature of the damping), but at least in principle the function $M$ is part of what one knows about the problem at hand. The question becomes: given the function $M$, what can we say about the rate of energy decay of (sufficiently regular) solutions of our damped wave equation?

It is known that the best result one may hope for is an estimate of the form \begin{equation}\label{eq:opt}\hspace{100pt} \|u(t)\|\le \frac{C}{M^{-1}(ct)} \hspace{200pt} (2)\end{equation} for all sufficiently large values of $t>0$, where $C,c$ are positive constants. It is also known that this best possible rate does not hold in all cases, and that sometimes a certain correction factor is needed. On the other hand, a celebrated result from 2010 due to A. Borichev and Y. Tomilov shows that if we consider the damped wave equation (or more generally any abstract Cauchy problem in which $X$ is a Hilbert space) and if $M(s)$ is proportional to $s^\alpha$ for some $\alpha>0$ and all sufficiently large $s>0$, then we do obtain the best possible rate given by (2). This result has been applied extensively throughout the recent literature on energy decay of damped waves and similar systems. A natural question, then, is whether the best possible estimate in (2) holds only for functions $M$ of this special polynomial type or for other functions as well.

In a recent paper (to appear in Advances in Mathematics) Oxford Mathematician David Seifert and his collaborators proved that one in fact obtains the optimal estimate in (2) for a much larger class of functions $M$, known in the literature as functions with positive increase. This class includes all functions of sufficiently rapid and regular growth, and in particular it includes functions $M(s)$ which are eventually proportional to $s^\alpha\log(s)^\beta$, where $C,\alpha >0$ and $\beta\in\mathbb{R}$. Such functions arise naturally in models of sound waves subject to viscoelastic damping at the boundary. Furthermore, the class of functions with positive increase is in a certain sense the largest possible class for which one could hope to obtain the estimate in (2), as is also shown in the paper. The proofs of these results combine techniques from operator theory and Fourier analysis. One particularly important ingredient is the famous Plancherel theorem, which states that the Fourier transform (suitably scaled) is a unitary operator on the space of square-integrable functions taking values in a Hilbert space. In future work, David and his collaborators hope to extend their results to the setting of more general Banach spaces. In such cases, however, the Plancherel theorem is known not to hold, so new ideas based on the finer geometric properties of Banach spaces are likely to be needed.