Evolutionary Equations

This chapter is intended to give a brief introduction as well as a summary of the present text. We shall highlight some of the main ideas and methods behind the theory and will also aim to provide some background on the main concept in the manuscript: the notion of so-called dating back to Picard in the seminal paper (Picard, Math. Methods Appl. Sci. 32, 1768–1803 (2009)); see also (Picard and McGhee, Partial differential equations: A unified Hilbert space approach, Chapter 6, vol. 55. Expositions in Mathematics (DeGruyter, Berlin, 2011)).

We will gather some information on operators in Banach and Hilbert spaces. Throughout this chapter let X ₀, X ₁, and X ₂ be Banach spaces and H ₀, H ₁, and H ₂ be Hilbert spaces over the field \(\mathbb {K}\in \{\mathbb {R},\mathbb {C}\}\).

It is the aim of this chapter to define a derivative operator on a suitable L ₂-space, which will be used as the derivative with respect to the temporal variable in our applications. As we want to deal with Hilbert space-valued functions, we start by introducing the concept of Bochner–Lebesgue spaces, which generalises the classical scalar-valued L _p-spaces to the Banach space-valued case.

In this chapter, we discuss a first application of the time derivative operator constructed in the previous chapter. More precisely, we analyse well-posedness of ordinary differential equations and will at the same time provide a Hilbert space proof of the classical Picard–Lindelöf theorem (There are different notions for this theorem. It is also called existence and uniqueness theorem for initial value problems for ordinary differential equations as well as Cauchy–Lipschitz theorem). We shall furthermore see that the abstract theory developed here also allows for more general differential equations to be considered. In particular, we will have a look at so-called delay differential equations with finite or infinite delay; neutral differential equations are considered in the exercises section.

In this chapter we introduce the Fourier–Laplace transformation and use it to define operator-valued functions of ∂ _t,ν; the so-called material law operators. These operators will play a crucial role when we deal with partial differential equations. In the equations of classical mathematical physics, like the heat equation, wave equation or Maxwell’s equation, the involved material parameters, such as heat conductivity or permeability of the underlying medium, are incorporated within these operators. Hence, these operators are also called “material law operators”. We start our chapter by defining the Fourier transformation and proving Plancherel’s theorem in the Hilbert space-valued case, which states that the Fourier transformation defines a unitary operator on \(L_2(\mathbb {R};H)\).

In this chapter, we shall discuss and present the first major result of the manuscript: Picard’s theorem on the solution theory for evolutionary equations which is the main result of Picard (A structural observation for linear material laws in classical mathematical physics. In Mathematical Methods in the Applied Sciences, vol 32, 2009, pp 1768–1803). In order to stress the applicability of this theorem, we shall deal with applications first and provide a proof of the actual result afterwards. With an initial interest in applications in mind, we start off with the introduction of some operators related to vector calculus.

This chapter is devoted to a small tour through a variety of evolutionary equations. More precisely, we shall look into the equations of poro-elastic media, (time-)fractional elasticity, thermodynamic media with delay as well as visco-elastic media. The discussion of these examples will be similar to that of the examples in the previous chapter in the sense that we shall present the equations first, reformulate them suitably and then apply the solution theory to them. The study of visco-elastic media within the framework of partial integro-differential equations will be carried out in the exercises section.

In this chapter we turn our focus back to causal operators. In Chap. 5 we found out that material laws provide a class of causal and autonomous bounded operators. In this chapter we will present another proof of this fact, which rests on a result which characterises functions in \(L_2(\mathbb {R};H)\) with support contained in the non-negative reals; the celebrated Theorem of Paley and Wiener. With the help of this theorem, which is interesting in its own right, the proof of causality for material laws becomes very easy. At a first glance it seems that holomorphy of a material law is a rather strong assumption. In the second part of this chapter, however, we shall see that in designing autonomous and causal solution operators, there is no way of circumventing holomorphy.

Up until now we have dealt with evolutionary equations of the form for some given \(F\in L_{2,\nu }(\mathbb {R};H)\) for some Hilbert space H, a skew-selfadjoint operator A in H and a material law M defined on a suitable half-plane satisfying an appropriate positive definiteness condition with \(\nu \in \mathbb {R}\) chosen suitably large. Under these conditions, we established that the solution operator, , is eventually independent of ν and causal; that is, if F = 0 on \(\left (-\infty ,a\right ]\) for some \(a\in \mathbb {R}\), then so too is U.

Let H be a Hilbert space and \(\nu \in \mathbb {R}\). We saw in the previous chapter how initial value problems can be formulated within the framework of evolutionary equations. More precisely, we have studied problems of the form for U ₀ ∈ H, M ₀, M ₁ ∈ L(H) and \(A\colon \operatorname {dom}(A)\subseteq H\to H\) skew-selfadjoint; that is, we have considered material laws of the form

In this chapter we study the exponential stability of evolutionary equations. Roughly speaking, exponential stability of a well-posed evolutionary equation means that exponentially decaying right-hand sides F lead to exponentially decaying solutions U. The main problem in defining the notion of exponential decay for a solution of an evolutionary equation is the lack of continuity with respect to time, so a pointwise definition would not make sense in this framework. Instead, we will use our exponentially weighted spaces \(L_{2,\nu }(\mathbb {R};H)\), but this time for negative ν, and define the exponential stability by the invariance of these spaces under the solution operator associated with the evolutionary equation under consideration.

This chapter is devoted to the study of inhomogeneous boundary value problems. For this, we shall reformulate the boundary value problem again into a form which fits within the general framework of evolutionary equations. In order to have an idea of the type of boundary values which make sense to study, we start off with a section that deals with the boundary values of functions in the domain of the gradient operator defined on a half-space in \(\mathbb {R}^d\) (for d = 1 we have \(L_2(\mathbb {R}^{d-1})=\mathbb {K}\)).

The power of the functional analytic framework for evolutionary equations lies in its variety. In fact, as we have outlined in earlier chapters, it is possible to formulate many differential equations in the form

This chapter is concerned with the study of problems of the form for a suitable sequence of material laws \(\left (M_{n}\right )_{n}\) when A ≠ 0. The aim of this chapter will be to provide the conditions required for convergence of the material law sequence to imply the existence of a limit material law M such that the limit U =lim_n→∞ U _n exists and satisfies

In this chapter, we address the issue of maximal regularity. More precisely, we provide a criterion on the ‘structure’ of the evolutionary equation in question and the right-hand side F in order to obtain \(U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)\). If \(F\in L_{2,\nu }(\mathbb {R};H)\), \(U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)\) is the optimal regularity one could hope for. However, one cannot expect U to be as regular since \(\left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )\) is simply not closed in general. Hence, in all the cases where \(\left (\partial _{t,\nu }M(\partial _{t,\nu })+A\right )\) is not closed, the desired regularity property does not hold for \(F\in L_{2,\nu }(\mathbb {R};H)\). However, note that by Picard’s theorem, \(F\in \operatorname {dom}(\partial _{t,\nu })\) implies the desired regularity property for U given the positive definiteness condition for the material law is satisfied and A is skew-selfadjoint. In this case, one even has \(U\in \operatorname {dom}(\partial _{t,\nu })\cap \operatorname {dom}(A)\), which is more regular than expected. Thus, in the general case of an unbounded, skew-selfadjoint operator A neither the condition \(F\in \operatorname {dom}(\partial _{t,\nu })\) nor \(F\in L_{2,\nu }(\mathbb {R};H)\) yields precisely the regularity \(U\in \operatorname {dom}(\partial _{t,\nu }M(\partial _{t,\nu }))\cap \operatorname {dom}(A)\) since where the inclusions are proper in general. It is the aim of this chapter to provide an example case, where less regularity of F actually yields more regularity for U. If one focusses on time-regularity only, this improvement of regularity is in stark contrast to the general theory developed in the previous chapters. Indeed, in this regard, one can coin the (time) regularity asserted in Picard’s theorem as “U is as regular as F”. For a more detailed account on the usual perspective of maximal regularity (predominantly) for parabolic equations, we refer to the Comments section of this chapter.

Previously, we focussed on evolutionary equations of the form In this chapter, where we turn back to well-posedness issues, we replace the material law operator M(∂ _t,ν), which is invariant under translations in time, by an operator of the form where both \(\mathcal {M}\) and \(\mathcal {N}\) are bounded linear operators in \(L_{2,\nu }(\mathbb {R};H)\). Thus, it is the aim in the following to provide criteria on \(\mathcal {M}\) and \(\mathcal {N}\) under which the operator is closable with continuous invertible closure in \(L_{2,\nu }(\mathbb {R};H)\). In passing, we shall also replace the skew-selfadjointness of A by a suitable real part condition. Under additional conditions on \(\mathcal {M}\) and \(\mathcal {N}\), we will also see that the solution operator is causal. Finally, we will put the autonomous version of Picard’s theorem into perspective of the non-autonomous variant developed here.

This chapter is devoted to the study of evolutionary inclusions. In contrast to evolutionary equations, we will replace the skew-selfadjoint operator A by a so-called maximal monotone relation A ⊆ H × H in the Hilbert space H. The resulting problem is then no longer an equation, but just an inclusion; that is, we consider problems of the form where \(f\in L_{2,\nu }(\mathbb {R};H)\) is given and \(u\in L_{2,\nu }(\mathbb {R};H)\) is to be determined. This generalisation allows the treatment of certain non-linear problems, since we will not require any linearity for the relation A. Moreover, the property that A is just a relation and not neccessarily an operator can be used to treat hysteresis phenomena, which for instance occur in the theory of elasticity and electro-magnetism.