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Lecture Notes Evolution Equations Lecture Notes Evolution Equations Roland Schnaubelt These lecture notes are based on my course from summer semester 2020, though there are minor corrections and improvements as well as small changes in the numbering of equations. Typically, the proofs and calculations in the notes are a bit shorter than those given in the course. The drawings and many additional oral remarks from the lectures are omitted here. On the other hand, the notes contain very few proofs (of peripheral statements) not presented during the course. Occasionally I use the notation and definitions of my lecture notes Analysis 1{4 and Functional Analysis without further notice. I want to thank Heiko Hoffmann for his support in the preparation of an earlier version of these notes. Karlsruhe, July 23, 2020 Roland Schnaubelt Contents Chapter 1. Strongly continuous semigroups and their generators1 1.1. Basic concepts and properties1 1.2. Characterization of generators 15 1.3. Dissipative operators 22 1.4. Examples with the Laplacian 35 Chapter 2. The evolution equation and regularity 46 2.1. Wellposedness and the inhomogeneous problem 46 2.2. Mild solution and extrapolation 51 2.3. Analytic semigroups and sectorial operators 56 Chapter 3. Perturbation and approximation 72 3.1. Perturbation of generators 72 3.2. The Trotter-Kato theorems 80 3.3. Approximation formulas 86 Chapter 4. Long-term behavior 90 4.1. Exponential stability and dichotomy 90 4.2. Spectral mapping theorems 99 Chapter 5. Stability of positive semigroups 106 Bibliography 112 ii CHAPTER 1 Strongly continuous semigroups and their generators Throughout, X and Y are non-zero complex Banach spaces, where we mostly write k · k instead of k · kX etc. for their norms. The space of all bounded linear maps T : X ! Y is denoted by B(X; Y ) and endowed with the operator norm kT kB(X;Y ) = kT k = supx6=0 kT xk=kxk: We abbreviate B(X) = B(X; X). Further, X∗ is the dual space of X acting as hx; x∗i, and I is the identity map on X. For ! 2 R, we denote R≥0 = [0; 1); R+ = (0; 1); R≤0 = (−∞; 0]; R− = (−∞; 0); C! = fλ 2 C j Re λ > !g; C+ = C0; C− = fλ 2 C j Re λ < 0g: In this course we study linear evolution equations such as 0 u (t) = Au(t); t ≥ 0; u(0) = u0; (EE) on a state space X for given linear operators A and initial values u0 2 D(A). (For a moment we assume that A is closed and densely defined.) We are looking for the state u(t) 2 X describing the system governed by A at time t ≥ 0. A reasonable description of the system requires a unique solution u of (EE) that continuously depends on u0. In this case (EE) is called wellposed, cf. Definitions 1.10 and 2.1. We will show in Section 2.1 that wellposedness is equivalent to the fact that A generates a C0-semigroup T (·) which yields the solutions via u(t) = T (t)u0. In the next section we will define and investigate these concepts, before we characterize generators in Sections 1.2 and 1.3. In the final section the theory is then applied to the Laplacian. Three intermezzi present basic notions and facts from the lecture notes [ST] on spectral theory, mostly without proofs. These are not needed later on. 1.1. Basic concepts and properties We introduce the fundamental notions of these lectures. Definition 1.1. A map T (·): R≥0 !B(X) is called a strongly continuous operator semigroup or just C0-semigroup if it satisfies (a) T (0) = I and T (t + s) = T (t)T (s) for all t; s 2 R≥0, (b) for each x 2 X the orbit T (·)x : R≥0 ! X; t 7! T (t)x, is continuous. Here, (a) is the semigroup property and (b) the strong continuity of T (·). The generator A of T (·) is given by n 1 o D(A) = x 2 X j the limit lim t (T (t)x − x) exists ; t!0; t2R≥0nf0g 1 Ax = lim t (T (t)x − x) for x 2 D(A): t!0; t2R≥0nf0g 1 1.1. Basic concepts and properties 2 If one replaces throughout R≥0 by R, one obtains the concept of a C0-group with generator A. Observe that the domain of the generator is defined in a `maximal' way, in the sense that it contains all elements for which the orbit is differentiable at t = 0. In view of the introductory remarks, usually the generator is the given object and T (·) describes the unknown solution. We will first study basic properties of C0-semigroups, starting with simple observations. Remark 1.2. a) Let A generate a C0-semigroup or a C0-group. Then its domain D(A) is a linear subspace and A is a linear map. b) Let (T (t))t2R be a C0-group with generator A. Then its restriction (T (t))t≥0 is a C0-semigroup whose generator extends A. (Actually these two operators coincide by Theorem 1.30.) c) Let T (·): R≥0 !B(X) be a semigroup. We then have T (t)T (s) = T (t + s) = T (s + t) = T (s)T (t); Yn T (nt) = T Pn t = T (t) = T (t)n j=1 j=1 for all t; s ≥ 0 and n 2 N. If T (·) is even a group, these properties are valid for all s; t 2 R and hence T (t)T (−t) = T (0) = I = T (−t)T (t): −1 There thus exists the inverse T (t) = T (−t) for every t 2 R. ♦ We next construct a C0-group with a bounded generator, which is actually dif- ferentiable in operator norm. Conversely, an exercise shows that a C0-semigroup with T (t) ! I in B(X) as t ! 0+ must have a bounded generator. Example 1.3. Let A 2 B(X). For t 2 C with jtj ≤ b for some b > 0, the numbers tn (b kAk)n An ≤ n! n! are summable in n 2 N0. As in Lemma 4.23 of [FA], the series 1 X tn T (t) = etA := An; t 2 ; n! C n=0 thus converges in B(X) uniformly for jtj ≤ b. In the same way one sees that N N N−1 d X tn X tn−1 X tk An = An = A Ak dt n! (n − 1)! k! n=0 n=1 k=0 tA tends to Ae in B(X) as N ! 1 locally uniformly in t 2 C. As in Analysis 1 tA one then shows that the map C !B(X); t 7! e , is continuously differentiable tA tA with derivative Ae . Moreover, (e )t2C is a group (where one replaces R≥0 by C in Definition 1.1(a)). m The case of a matrix A on X = C was treated in Section 4.4 of [A4]. ♦ 1.1. Basic concepts and properties 3 For a semigroup a mild extra assumption implies its exponential boundedness. This assumption is satisfied if kT (t)k is uniformly bounded on an interval [0; b] with b > 0 or if T (·) is strongly continuous. (We need both cases below.) We set !+ = maxf!; 0g and !− = max{−!; 0g for ! 2 R. Lemma 1.4. Let T (·): R≥0 !B(X) satisfy condition (a) in Definition 1.1 as well as lim supt!0 kT (t)xk < 1 for all x 2 X. Then there are constants !t M ≥ 1 and ! 2 R such that kT (t)k ≤ Me for all t ≥ 0. Proof. 1) We first claim that there are constants c ≥ 1 and t0 > 0 with kT (t)k ≤ c for all t 2 [0; t0]. To show this claim, we suppose that there is a null sequence (tn) in R≥0 such that limn!1 kT (tn)k = 1: The principle of uniform boundedness (Theorem 4.4 in [FA]) then yields a vector x 2 X with supn kT (tn)xk = 1. There thus exists a subsequence satisfying kT (tnj )xk ! 1 as j ! 1. This fact contradicts the assumption, and so the claim is true. 2) Let t ≥ 0. Then there are numbers n 2 N0 and τ 2 [0; t0) such that −1 t = nt0 + τ. Take ! = t0 ln kT (t0)k if T (t0) 6= 0 and any ! < 0 otherwise. Set M = ce!−t0 . We estimate n n nt0! t! −τ! !t kT (t)k = kT (τ)T (t0) k ≤ c kT (t0)k ≤ ce = ce e ≤ Me ; using Remark 1.2. The above considerations lead to the following concept, which is discussed below and will be explored more thoroughly in Section 4.1. Definition 1.5. Let T (·) be a C0-semigroup with generator A. The quantity !t !0(T ) = !0(A) := inff! 2 R j 9 M! ≥ 1 8 t ≥ 0 : kT (t)k ≤ M!e g 2 [−∞; 1) is called its (exponential) growth bound. If supt≥0 kT (t)k < 1, then T (·) is bounded. (Similarly one defines !0(f) 2 [−∞; +1] for any map f : R≥0 ! Y .) Remark 1.6. Let T (·) be a C0-semigroup. a) Lemma 1.4 implies that !0(T ) < 1. b) There are C0-semigroups with !0(T ) = −∞, see Example 1.9. c) In general the infimum in Definition 1.5 is not a minimum. For instance, 2 0 1 let X = C be endowed with the 1-norm j · j1 and A = 0 0 . We then have tA 1 t T (t) = e = 0 1 and kT (t)k = 1 + t for t ≥ 0.
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