What is semigroup theory

Evolution equations

Transcript

1 Evolution equations Linear and semi-linear semigroup theory Ben Schweizer lecture in winter 29/1 at TU Dortmund

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3 Contents 1. Introduction and examples Basic terms Examples The heat conduction equation I. Linear semigroup theory Operators and semigroups Operator, spectrum and resolvents Semigroups Generators of the semigroup resolvents and semigroups Continuous and contractive semigroups Generators of contractive semigroups Contractive semigroups in applications Generators of strongly continuous semigroups Analytical semigroups Sectoral Operators and analytical semigroups Invariant subspaces, normal operators Broken powers of sectoral operators Regularization and interpolation II. Nonlinear semigroup theory Semilinear equations The inhomogeneous equation Existence theorem for semilinear equations Behavior for long periods Regularity theory Continuous and differentiable dependencies Improved regularity Applications and outlook Applications The semilinear environment of a stationary solution Quasilinear equations

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5 1. Introduction and examples 1.1. Basic concepts Aim of the lecture: We want to solve equations of the following form. t u = Au, u () = u. (1.1) For this we have to be given a Banach space X and an A. We are looking for a u: [,) X that solves (1.1). Different cases: 1. X = R n, AR nn 2. X = R n, A: XX Lipschitz, non-linear 3. X general Banach space, AL (X, X), 4. A: D (A) XX linear (name : A: XX is an unbounded linear operator) 5. A: D (A) XX non-linear 6. AXX non-linear and multi-valued Here, cases 1 and 2 are essentially known, case 3 will be a preparation for the general results, with case 4 things will be exciting. For 5 and 6 we will only deal with special cases in this lecture. Fundamental example: The heat conduction equation t u = u: = n x 2 i u, u (t = ,.) = u (1.2) i = 1 is to be solved on R n. This is an example of an operator as in 4. Choose X = L 2 (R n) and A =: L 2 (R n) L 2 (R n) is an unbounded operator with D (A) = H 2 ( R n). We will get to know solution methods for (1.2) in Section 1.3. After the first part of this lecture we will only say about solvability: is self-adjoint and in particular has a real spectrum. is dissipative and in particular fulfills the resolvent assessment. It follows that it is sectorial and that (1.2) creates an analytic semigroup. 5

6 The methods will be so general that, in addition to (1.2), many other equations from applications are also dealt with (see Section 1.2). Names: In applications u (t) describes the state of the system in the state space X at time t. The differential equation (1.1) describes the evolution of the system: For an initial value u, we can state the state at time t. If (1.1) has a unique solution, this defines a time-t-map S (t), S (t): X uu (t) X. (1.3) S (t) has the properties S (t) S (s) = S (t + s) and S () = id. We abstract this to the concept of the semigroup. Concept of semigroup. A semigroup on a Banach space X is a family S (t) L (X) for all t [,) with the semigroup property S (t) S (s) = S (t + s) t, s, S () = id. (1.4) With these terms our central questions are: Q1 Does equation (1.1) define a semigroup on X? Q2 Is a given semigroup S generated by a suitable equation (1.1)? The candidate for the operator A is the creator of the semigroup, which we define as follows. The producer. The generator of a semigroup S on X is S (h) x x A: x lim, (1.5) h h defined on D (A) X, D (A): = {x X lim h S (h) x x h} exists. (1.6) We also see from this definition that we should not limit ourselves to mappings A: X X. Rather, we should admit that A is not defined on the entire space X, but only on a subset D (A). If we want to solve equation (1.1) for such an operator, then we require example 1.1 (exponential function) for the solution u (t) D (A) for all t examples. For X = R n and a matrix A we consider the equation t u = A u, u () = u. (1.7) 6

7 In the case n = 1 and A = a the solution is u (t) = ue at. The semigroup S (t): RR is given by S (t) = e at. For general n, (1.7) is also solvable ( see Analysis II). The solution operator is S (t) = e At. This operator can be defined as S (t): u u (t), where u is a solution of the equation. Another possibility is to define e At by the series. One can prove in an elementary way that the series again provides a solution to the equation. We ask about the producer of the semigroup. The mapping t e At is differentiable and because of t e At = Ae At in t = S (h) x x lim h h = (t e At x) t = = (Ae At x) t = = Ax. So the matrix A: R n R n is the generator of the semigroup e At. Example 1.2 (nonlinear finite dimensional). An ordinary differential equation with globally Lipschitz continuous f: R n R n defines a semigroup S (t) on X = R n. We define S (t) u = u (t), where u is the solution of tu = f (u) , u () = u. (1.8) The semigroup is nonlinear, because S (t) (u + u 1) S (t) u + S (t) u 1 in general. The non-linear generator is f: X X. Example 1.3 (displacement dynamics). For X = L p (R, R) we define the right shift S (t) by (S (t) u) (x) = u (x t). The generator of the semigroup is A = x: W 1, p (R, R) L p (R, R). Proof: For u W 1, p we have S (h) u u lim h h = L p lim h u (. H) u (.) H = x u (.). (a property of the spaces L p: the difference quotient on the left is a convolution of f = x u with a special Dirac sequence; but the convolution of L p functions f with Dirac sequences converges in L p to f). Conversely, x u L p only holds for functions u W 1, p. Therefore the Limes only exists in this case. Two concepts of continuity. The semigroup is called strongly continuous if [,) t S (t) x X is continuous for all x X. (1.9) The semigroup is called uniformly continuous if for a continuous ρ: [,) R + with ρ () = : S (t) uu X ρ (t) u X u X. Exercise 1.1. Consider that 1. for p = the semigroup is not strongly continuous, 2. for all p the semigroup is not uniformly continuous. 7th

8 Example 1.4 (displacement dynamics on half space). On the space X = L p (R +, R), p> 1, we define S (t) by {u (xt) for xt>, (S (t) u) (x) = otherwise. The creator of the Semigroup is formally A = x again, but this time with a different domain, A = x: XD (A) = {u W 1, p (R +, R) u () =} L p (R +, R). Proof: We consider u W 1, p (R +, R) with u (). Then (S (h) u u) / h on the interval (, h) is of the order of 1 / h. Such a sequence of functions is unbounded in L p for p> 1. The limit does not exist. Conversely, for u W 1, p (R +) with u () = the trivial continuation in W 1, p (R). So the Limes exists for such u. We see: 1. D (A) {x X Ax can be defined} 2. D (A) contains central information, e.g. Boundary values ​​3. The initial values ​​do not have to be from D (A). Next we consider a variant of the displacement dynamics. Example 1.5 (transport at variable speed). Let b C 1 (R, R) with b (x) [b min, b max] (,) for all x R. We consider the equation tu (x, t) = b (x) xu (x, t) . The semigroup of this equation can be stated explicitly as follows. For every point x R n we solve the ordinary differential equation t ξ (x, t) = b (ξ (x, t)), ξ (x,) = x. ξ t: x ξ (x, t) is strictly monotonic and therefore invertible. The semigroup is then implicitly given by (S (t) u) (ξ (x, t)) = u (x), or by the explicit formula (S (t) u) (y) = u (ξt 1 (y) )). The proof is done by differentiating the implicit formula with the abbreviation u (x, t) = (S (t) u) (x). = tu (x) = t [(S (t) u) (ξ (x, t))] = (t S (t) u) (ξ (x, t)) + x (S (t) u) (ξ (x, t)) t ξ (x, t) = tu (ξ, t) + b (ξ) xu (ξ, t). 8th

9 Exercise 1.2 (Method of Characteristics). Consider the same procedure for transport in R n in order to solve the equation for b C 1 (R n, R n). t u (x, t) = b (x) x u (x, t). Example 1.6 (delay differential equation). For g, h: R n R n, we can consider equations of the type t u (t) = g (u (t)) + h (u (t 1)). If initial values ​​for u are given on the interval [1,], the equation can be solved (assuming g and h). At first it is not clear why this defines a semigroup. We can introduce: X = C ([1,]) and the operator A: XX, ff, D (A): = {f C 1 ([1,]) f () = g (f ()) + h (f (1))}. In these spaces the above equation is equivalent to t U = AU, U () = u [1,] X. Example 1.7 (infinite-dimensional exponential function). For X = l 2 (N) and u = (uk) k we define S (t) u = u (t) = (uk (t)) k as a solution of tuk (t) = k 2 uk (t), uk () = uk (1.1) In this example we can solve explicitly by decoupling the components (diagonal matrix). u k (t) = e k2t u k. We will see that with this example we have found a solution to the heat conduction equation. A generalization of the previous example is as follows. Example 1.8 (multiplication semigroup). We consider a space X = L 2 (Σ, µ), where Σ R n is open and µ is a Borel measure on Σ. Furthermore, let a measurable function q: Σ C be bounded with Re (q). Then we can consider the multiplication operator M q: X u q u with domain D (M q) = {u X q u X}. The semigroup generated by M q is S (t): X ue tq u X. We recognize Example 1.7 again if we set Σ = R and q (k) = k 2. With µ = k N δ k, X = L 2 (Σ, µ) = l 2 (N). The Laplace operator can be represented as a multiplication operator in the simple space l 2 (N) with q (k) = k 2. According to a general spectral theorem: For X Hilbert space, A: D (A) XX normal, one finds a representation in which A is diagonal, i.e., µ, q: X = L 2 (Σ, µ), A = M q. 9

10 Partial Differential Equations Some important linear partial differential equations in mathematical physics are as follows (we do not include initial and boundary conditions here). tuu = heat conduction equation (1.11) tu + bxu = simple transport equation (1.12) tu + buu = heat conduction with transport (1.13) 2 tuu = wave equation (1.14) 2 tu + 2 u = rod equation (1.15) itu + u + V (.) u = Schrödinger equation (1.16) We now give two important nonlinear examples. What they have in common is that the non-linearity is not of the highest order. t u u f (t, u) = reaction diffusion Eq. (1.17) t v + (v) v v + p = div (v) = Navier-Stokes equation (1.18) In the next equation, the highest derivatives occur non-linearly. t u ij a ij (u) ij u = nonlinear heat conduction (1.19) The variety of equations is almost limitless. Our goal must therefore be to develop a theory that is as general as possible and to find suitable categories for equations. Our ambitious goal is to develop methods that cover a large class of equations and, in particular, provide local solutions for all of the above equations. As an outlook, we give another equation here. It can no longer be treated with linearizations and leads to a free boundary value problem. t u (u γ) = porous media equation (1.2) 1.3. The heat conduction equation We want to remind you of three elementary methods with which the heat conduction equation can be solved. Each of the methods will be used in a generalized form in semigroup theory Development in Eigenfunctions First we solve the one-dimensional equation tu (x, t) = u (x, t), u (, t) = u (π, t) = t >, u (.,) = u (.) 1

11 for u L 2 ((, π), R). We expand u into a Fourier series, u (x) = u k sin (kx). k = 1 (Formally the odd continuation of u is expanded into sin and cos, and because of the symmetry no cos-terms appear.) We now also expand the solution u (., t) into a Fourier series, u (x, t ) = uk (t) sin (kx). (1.21) k = 1 u is the solution of the equation if for all ktuk (t) sin (kx) = uk (t) sin (kx) = uk (t) k 2 sin (kx), uk () = u k . The solutions u k (t) of t u k = k 2 u k are in particular exponentially decreasing for t> and k. This ensures the convergence of the series. The boundary condition u (, t) = u (π, t) = is also kept: The series (1.21) converges uniformly. In particular, the boundary values ​​of u agree with the formal boundary values ​​=. So in Example 1.7 we have solved the heat conduction equation on an interval. Exercise 1.3. Think about what happens if we a) expand u in the approach and u (., T) in cos; what equation do we solve? b) develop the Dirichlet solution in cos. The same method can be used in any field. For a smooth bounded domain Ω R n we will use the equation tu (x, t) = u (x, t), u = on Ω, u (x,) = u (x) for u L 2 (Ω, R) to solve. With the family of eigenfunctions v k H 1 (Ω, R) of the Laplace operator v k = λ k v k we can again expand, u (x, t) = u k (t) v k (x). (1.22) k = 1 As u k we choose the solutions of t u k = λ k u k, u k () = u k and find a solution to the heat conduction equation. Again we have the generator: H 1 (Ω) H 2 (Ω) L 2 (Ω) diagonalized. We will encounter this procedure (decomposition into eigenfunctions) in the definition of an analytic semigroup (Definition 4.5 and Theorem 4.6). 11

12 Galerkin Method In section we implemented the following idea for special shape functions vk: In the Hilbert space X = L 2 (Ω) we choose a basis (vk) k of X with vk H 1 (Ω) H 2 (Ω) and consider the finite-dimensional subspaces X n spanned by (v 1, ..., vn). We denote the L 2 orthogonal projection on X n by P n. We assume the following for the basis: P n ϕ ϕ in H 1 for n, for all ϕ H 1 (Ω). (1.23) We solve the ordinary differential equations tu (t) = P nu (t), u () = P n u. For these ordinary DGL s with Lipschitz continuous right side we find solutions u = un C 1 (R +, X n). A priori estimates. Multiplication of the equation by u n and integration over Ω yields 1 t u n (x, t) 2 dx + u n (x, t) 2 dx =. 2 Ω Here we have used the orthogonality of P n. An integration over the time interval [, T] gives 1 2 Ω u n (x, T) 2 dx + T Ω Ω u n (x, T) 2 dx dt = 1 2 Ω P n u (x) 2 dx. The right hand side is bounded for all n by 1 u 2 2 L. This means that the functions u n are bounded uniformly in L 2 ((, T), H 1 (Ω)). We can then choose a 2 subsequence that converges to a u in the same space. Equations for u. If we test the un equation with a test function ϕ, we find the weak form of the equation T Ω und ϕ + T Ω un (P n ϕ) = ϕ C 1 (Ω (, T), R) . Because of P n ϕ ϕ in C ((, T), H 1 (Ω)) (by assumption (1.23)) and u n u, we can move to the limit in both integrals. We find T Ω u t ϕ + T Ω u ϕ = ϕ C 1 (Ω (, T), R), so the weak form of the equation is fulfilled. With a little extra effort it can be shown that u C ([, T], L 2 (Ω)) and that the initial values ​​are also accepted. This method is used, for example, in finite element methods. Then the u n are chosen as piecewise polynomials over a triangulation of Ω. The method is therefore also referred to as a discretization in space. In semigroup theory we will do something similar in Hille-Yosida theorem 3.1: We approximate the operator A by bounded operators A n. The limit of the associated semigroups S n is then the semigroup S of A. 12

13 Rothe method In contrast to the previous method, the Rothe method is a discretization of time. The simplest procedure is as follows. We want to solve t u = u with u () = u on Ω and for a time interval [, T]. We choose a small t and subdivide [, T] into intervals [tk, t k + 1] with tk = kt and t N = T. The solution u should be approximated at the times tk by u (tk) uk X = L 2 (Ω). For this we define the family u k by the rule u k + 1 u k = u k + 1. t A priori estimates. We test the rule with u k + 1 and get 1 (uk + 1 2 uk, u k + 1) = u k + 1 2. t We sum over all k with weight t, and use uk, u k + 1 ( ukuk 2) / 2. t N uk 2 = k = 1 k = N 1 k = u k + 1 2 uk, u k + 1 N (u k + 1 2 uk 2) = 1 2 u N u 2. On the values ​​(uk) k = 1, ..., n one can define the linear interpolation. This gives a function u N: [, T] X, u N (t) = µu k + (1 µ) u k + 1 for t = µt k + (1 µ) t k + 1, µ [, 1 ]. For the functions u N we found estimates in the spaces L 2 ((t, T), H 1 (Ω)). A weak limit of the sequence is again a solution to the equation. Connection with semigroup theory. We set λ = 1 t u k + 1 equation as (λ) u k + 1 = λu k. If we can prove that (λ) 1 L (X, X) 1 λ and write that (1.24) holds for all λ R +, then u k X u X for all k, independent of t (i.e. of N). The linear interpolations u N are then a bounded family in C ([, T], X) and we can find a weak limit.In fact, relation (1.24) holds for the Laplace operator. We only have to test with u k + 1, as we did before for the a priori estimation. The result was u k + 1 2 u k, u k + 1 u k + 1 u k, so u k + 1 u k. But this is exactly (1.24). In semigroup theory we will see that under very general circumstances the following applies: An operator A (for us A =), which satisfies (1.24), is the generator of a continuous (even a contractive) semigroup on X. 13

14 Part I. Linear semigroup theory 14

15 2. Operators and semigroups 2.1. Operator, spectrum and resolvent definition 2.1 (linear operator). An (unbounded) linear operator on a Banach space X is given by a linear subspace D (A) X and a linear mapping A: D (A) X X. We usually require that A is closed, that is, for all uk D (A ) with uku X and Au kv X we have u D (A), Au = v. Definition 2.2 (spectrum and resolvent). For a closed linear operator we define the spectrum of A as σ (a): = {λ C λ A: D (A) X is not bijective} Resolvent set ρ (a): = C \ σ (a) Resolvent R (λ, A): = (λ A) 1 for λ ρ (a). Note: The spectrum and resolvent set are defined differently for operators that are not closed. One sets λ ρ (a) if λ A: D (A) image (λ A) has a continuous inverse and image (λ A) is dense in X. The spectrum is again the complement of the resolvent set. The definitions are the same for closed operators. The following comment shows one of the two directions. Note 2.3. For closed operators A, the resolvent is a bounded operator. This follows from the theorem about the closed graph. The linear operator R (λ, A): X X has a closed graph in X X. Then the operator is bounded (see e.g. [1]). Elementary properties of resolvents. Lemma 2.4 (resolvent identity). For λ, µ ρ (a) In particular, resolvents, R (λ, A) commute R (µ, A) = (µ λ) r (λ, A) R (µ, A). (2.1) R (λ, A) R (µ, A) = R (µ, A) R (λ, A). (2.2) 15

16 proof. We start with the observation that id (λ A) R (µ, A) = (µ λ) r (µ, A), which follows by inserting = A A on the right-hand side. Applying (λ A) 1 = R (λ, A) yields (2.1). Lemma 2.5. Let A: D (A) X X be a closed operator. Then ρ (a) is open and σ (a) is closed. More precisely applies to µ ρ (a): For r = R (µ, A) 1 the open sphere B r (µ) C is contained in ρ (a). In the sphere, R (λ, A) depends analytically on λ and the representation R (λ, A) = (µ λ) n R (µ, A) n + 1 applies. (2.3) Proof. For λ C we write n = λ A = µ A + λ µ = [id (µ λ) r (µ, A)] (µ A). If the operator id (µ λ) r (µ, A) L (X) is invertible, so is λ A. This is the case for λ µ

17 Comment 2.8. A linear semigroup S (t) from Definition 2.9 is strongly continuous if and only if initial values ​​are assumed, i.e. if lim S (t) x = x for all x X. (2.6) t Proof. The strong continuity particularly implies the Limes property. 1. We first show that for a linear semigroup S (t) with property (2.6) the following restriction applies: There is a δ> such that sup S (t) <. (2.7) t [, δ] For a contradiction proof we assume the opposite, that for t k we have S (t k). Now we apply the principle of uniform limitedness (see e.g. [1]). According to this, for the family of operators S (t k) k N there is also a fixed x X with S (t k) x. But this contradicts the continuity in. 2. Let S (t) with (2.6) again. We first show that S (t) x is then right-continuous. In fact, S (t + h) x S (t) x S (t) S (h) x x for h + because of the Limes property, so S (t) is right-continuous. For left continuity we first notice that, because of the semigroup property (2.7), improves to sup t [, t1] S (t) . Now we write S (t h) x S (t) x S (t h) x S (h) x. Since the family S (t) L (X), t [, t 1] is uniformly bounded, the continuity to the left follows from the Limes property. Definition 2.9 (continuity properties). A family of mappings S (t) L (X, X), t [,) with the semigroup property (2.4) is called uniformly continuous if t S (t) L (X, X) is continuous if S (t) 1 ω-contractive, if S (t) e ωt analytical, if the family t S (t) can be analytically extended into the complex plane (see Definition 4.5). Semi-groups can grow exponentially at most. Throughout the lecture we will often use constants M and ω from the following statement. Comment 2.1. Every strongly continuous semigroup S (t) satisfies an estimate S (t) Me ωt t for constants M 1 and ω R. (2.8) 17

18 evidence. For fixed x X, t S (t) x is continuous and therefore sup t S (t) x <. The principle of uniform boundedness yields the boundedness of the family (S (t)) t [, 1], that is, a bound M with S (t) M t [, 1]. According to the semigroup property, for any t = n + τ, n N, τ [, 1] for ω = log M. S (t) S (1) n S (τ) M n + 1 Me n log M Me ωt In According to the theory, ω-contractive and contractive semigroups can be converted into one another. This is done with the following lemma. Remark 2.11 (rescaled semigroup). Let S (t) be an ω-contractive semigroup on X. Then T (t) u: = e ωt S (t) u defines a contractive semigroup on X. In particular we see that in inequality (2.8) the number ω for the Theory doesn't matter; we can always assume ω = by scaling. The prefactor M makes the important difference between an only strongly continuous (M> 1) and a contractive (M = 1) semigroup. Remark 2.12 (strongly continuous, but not contractive). The transition to an equivalent norm on X can turn contractive semigroups into ones that are only strongly continuous. We want to construct such an example. We consider X = l 2 (N, R 2) l 2 (N, C) as a Banach space over R and for u = (u k) k N we define Au = v = (v k) k, v k = iku k. This is a multiplication operator and, as in Example 1.8, the generated semigroup can be specified explicitly, S (t) u = u (t) = (u (t) k) k, u (t) k = e ict u k. In each component, the semigroup is a rotation; in components with large indices, it is rotated as quickly as required. We can define a norm on X by u 2 1: = k Re u k Im u k 2. This norm is equivalent to the natural norm. But: Regarding the norm. 1 the semigroup is not contractive. The M in (2.8) must be set to at least 2. 18th

19 2.3. Creator of the semigroup Definition 2.13 (Creator of a semigroup). The generator of a linear semigroup S on X is the operator S (h) x x A: x lim. (2.9) h h It is defined on D (A) X, {D (A): = x X lim h S (h) x x h} exists. (2.1) In order to match, we determine the producer of the rescaled half-group where S has producer A. We find T (h) u u lim h h = lim h So T has the producer B = A ω. T (t) u: = e ωt S (t) u, (ωh S (h) u u e h) + e ωh 1 u = Au ωu. h Proposition 2.14 (solution properties). Let S (t) be a strongly continuous linear semigroup and (A, D (A)) its generator. Then for u D (A): 1. S (t) u D (A) for all t 2. AS (t) u = S (t) Au for all t 3. The map t S (t) u is differentiable to (,), and 4. there the equation t S (t) u = AS (t) u applies. (2.11) Proof. We ask ourselves: What would AS (t) u be? To do this, we calculate with the semigroup property S (h) S (t) u S (t) u lim h + h = S (t) lim h + S (h) u u h = S (t) Au. So we have S (t) u D (A) and the interchangeability AS (t) u = S (t) Au. At the same time we see that the right derivative of S (t) u is given by AS (t) u. It remains to calculate the difference quotients to the left. {S (t) u S (th) u lim = lim S (th) S (h) uu} h + hh + h {()} S (h) uu = lim S (th) Au + S (th) au = S (t) Au. h + h We have used that the family S (t h) L (X) is bounded for h [, h] (see proof of Remark 2.8) and the strong continuity of S (t). Right and left limits exist and both agree with AS (t) u (especially both with each other). Hence the semigroup (for u D (A)) is differentiable and (2.11) applies. 19th

20 Theorem 2.15 (properties of the producer). The generator of a strongly continuous linear semigroup is a closed and densely defined linear operator. The semi-group is clearly determined by the producer. Proof. For continuous linear S (t), A is by definition a linear operator. 1. We set V (t): = S (τ) dτ. Since the mapping τ S (τ) u is continuous for u X, 1 V (t) u S () u = u t for t. We claim that V (t) u D (A) for all t>. This then proves the tightness of D (A). In fact, S (h) V (t) u V (t) u = 1 hh = 1 h Our result is + ht S (τ) u 1 hh S (h + τ) u S (τ) u dτ S (τ ) u S (t) u u. AV (t) u = S (t) u u. (2.12) Formally, this follows immediately from the main theorem of differential calculus. 2. Seclusion. Let D (A) uku and Au kv be in X. Because of (2.12) and Proposition 2.14 we have S (t) ukuk = We pass to the limit and find S (t) uu = AS (τ) uk dτ = We form the difference quotient and find S (t) uut = 1 t S (τ) v dτ. S (τ) v dτ v. S (τ) Au k dτ. This proves u D (A) and Au = v. 3. Let T (t) be another semigroup with producer A. For u D (A) we consider [, t] sv (s): = T (ts) s (s) u X. Our goal is to show that v is constant. For h> we form the difference quotient v (s + h) v (s) h = T (tsh) 1 (S (s + h) u T (h) s (s) u) h = T (ts) 1 ( S (s + h) u S (s) u) h + (T (tsh) T (ts)) 1 (S (s + h) u S (s) u) h + 1 (T (tsh) T ( ts)) S (s) u. h 2

21 We form the Limes h +. According to Proposition 2.14, the first and third terms converge to T (t s) as (s) u and AT (t s) s (s) u, respectively. They agree except for the sign and cancel each other out. We write the second term as B (h) xh: = (T (tsh) T (ts)) xh with xh = 1 h (S (s + h) u S (s) u) x = AS (s) u . We estimate and form the limit h, B (h) xh XB (h) x X + B (h) L (X) xhx X. We have the convergences B (h) x (strong continuity) and the uniform restriction of the operators B (h) are used. The above calculation can also be carried out for negative h. We conclude that the function s v (s) is differentiable and that the derivative vanishes. So v is constant. We compare the final values ​​and find T (t) u = v () = v (t) = S (t) u. The semigroups on D (A) agree with this. Since D (A) is dense and the maps u T (t) u and u S (t) u are continuous, equality follows all of X. Uniformly continuous semigroups With the next theorem we characterize the producers of uniformly continuous semigroups. Theorem In the Banach space X applies: 1. A L (X) generates a uniformly continuous semigroup. 2. The generator of a uniformly continuous semigroup is an operator A L (X). 3. If A: D (A) = X X creates a strongly continuous semigroup, then A L (X) holds. Proof. 1. For A L (X) we define S (t): = e At: = t k A k. (2.13) k! k = S (t) is well defined, because the partial sums N t k A k S N (t): = k! are Cauchy sequences in the Banach space L (X). For the differentiability we argue as follows: The partial sums S N: t S N (t) are Cauchy sequences in C 1 ([, 1], L (X)). In fact, for m

22 M A k k! k = m M k A k + k! k = m for m. So S N has a limit in C 1 ([, 1], L (X)), and in particular t S N t S in C ([, 1], L (X)). We can now show that S (t) u solves the equation t u = Au on [, 1] (one-sided derivatives are carried out at boundary points). In particular, A is then the producer of S (t). AS (t) (1) AS N (t) = = N + 1 k = 1 t t k A k k! N t k A k + 1 = k! k = N + 1 k = 1 = t S N + 1 t S (t). kt k 1 A k In (1) we use the boundedness of A. We write values ​​t (T ε, T + ε) for T> 1/2 as t = T + τ for T: = T ε and τ (, 1). Then S (t) = S (τ) S (T) t S (t) = t S (τ) S (T) = AS (t), where in particular the corresponding functions are differentiable. The equation t S (t) u = AS (t) u therefore applies to all of R +. The uniform continuity on R + also follows from S C ([, 1], L (X)). 2. For the semigroup S (t) and for t> we define V (t): = The uniform continuity of S implies S (τ) dτ. 1 V (t) S () = id L (X) t for t. Therefore the operator V (t) is invertible for small t>. The semigroup property implies S (t) = V (t) 1 V (t) S (t) = V (t) 1 S (t + τ) dτ + t = V (t) 1 S (τ) dτ, t and therefore the semigroup S (t) is even differentiable on (,) and on the right differentiable in. The derivative in is A = t S () = V (t) 1 (S (t) S ()). In particular, A is a bounded operator. 3. As the generator of a strongly continuous semigroup, A is closed by Theorem 2.15. By the theorem of the closed graph, A is bounded (see [1], 6.8). k! 22nd

23 Theorem 2.16 says above all: Uniformly continuous semigroups belong to bounded operators. They will therefore not appear in our applications. On the other hand, we will now be able to proceed as follows. We will approximate unbounded operators A by bounded operators A n. Every A n generates a semigroup S n according to theorem A Limes which defines S n. As an illustration of the results of this section we consider the displacement semigroup S (t) from Example 1.3, (S (t) u) (x) = u (xt). This defines a strongly continuous semigroup on X = L 2 (R) u. By Theorem 2.15 the generator A = x is defined on a dense subspace of X (namely H 1 (R)). Furthermore, x: H 1 L 2 L 2 is a closed operator (in fact u k u in L 2 (R) and x u k v in L 2 (R) implies that u k u in H 1 and x u = v). The semigroup to t u = x u is uniquely determined. The generator is not restricted, so, according to Theorem 2.16, the semigroup cannot be uniformly continuous (compare exercise 1.1) Resolvents and semigroups The connection between semigroup S (t) and generator A lies in the spectral properties of A. We have already seen that a bounded spectrum of A belongs to a uniformly continuous semigroup. We will complete this with the following picture. Spectrum of A semigroup σ (a) limits S (t) uniformly continuous σ (a) in half-plane S (t) strongly continuous / contractive σ (a) in sector S (t) analytical However, not only the position of the spectrum is important , but also the resolvent mapping, or, for short, the resolvent. Theorem 2.17 (resolvents). Let S (t) be a strongly continuous semigroup with M and ω from equation (2.8), and let (A, D (A)) be the generator. Then: 1. If for λ C the improper integral R (λ) u: = e λτ S (τ) u dτ (2.14) exists for all u X, then λ ρ (a) and R (λ, A) = R (λ). 2. For all λ in the half-plane Re λ> ω applies λ ρ (a) and the integral representation for R (λ, A) = R (λ). In addition, the resolvent estimate R (λ, A) M Re λ ω applies. (2.15) 23

24 evidence. To 1. It is sufficient to consider the case λ =. Otherwise we rescale the semigroup as in Remark 2.11 and consider à = A λ with S (t) = e λt S (t). We want to show that the existence of the improper integral implies the invertibility of A: D (A) X (with the corresponding formula). To do this, we apply (an approximation of) A to the claimed inverse: S (h) id h (= 1 h = 1 hh S (h) id R () u = hh S (τ) u dτ + S (τ) u dτ u. S (τ) u dτ) S (τ) u dτ We see that R () u D (A) for all u X and that AR () = id. Finally, to show that A is also the right inverse, we fix u D (A) and use successively: R () right inverse, definition of R (), closure of A, interchangeability of A and S (t), definition. u = AR () u = A lim = lim t t S (τ) u dτ = lim S (τ) Au dτ = R () Au. t AS (τ) u dτ We found an inverse to A and thus showed that λ = ρ (a). The formula is also proven. To 2. The convergence of the improper integral is clear for Re λ> ω. Therefore, all statements follow from 1 except for the resolvent estimate. For this we calculate R (λ, A) = R (λ) = e λτ S (τ) dτ = M e λτ S (τ) dτ e τre λ Me ωτ dτ e τ (ω Re λ) dτ M Re λ ω , and thus have the estimate for the resolvent.On the basis of the theorem, we can already state the following: Information about the position of the spectrum is not sufficient to find an associated (solution) semigroup for an operator. Exercise 2.1 (displacement operators). We consider the three (closed) right-shift operators A i = x, A i: D (A i) X i X i, i = 1, 2, 3, with the spaces X 1 = L 2 (R), D (A 1) = H 1 (R), X 2 = L 2 ((, 1), R), D (A 2) = {u H 1 ((, 1), R) u () =}, X 3 = L 2 ((, 1), R), D (A 3) = H 1 ((, 1), R). 24

25 Consider: σ (a 1) = ir, σ (a 2) =, σ (a 3) = C. Hint: (λ A i) u = f is an ordinary differential equation. The operators A 1 and A 2 create a semigroup (right shift). The operators A 2 and A 3 do not. According to Theorem 2.17: The spectra of A 1 and A 2 lie in a left half-plane, in a right half-plane the resolvent estimate is fulfilled. The operators A 2 and A 3 do not satisfy the resolvent estimation in any right half-plane. The operator A 2 shows: The limitedness of the operator does not follow from the limitedness of the spectrum. The operator A 2 shows: From the limitedness of the spectrum it does not follow that the operator creates a semigroup. Another example is the following. It is similar to A 2 in the exercise, but here the problem is not a non-adjustable boundary condition. Example 2.18 (resolvent estimate not met). In the space X: = {u L 2 (R +) u (, 1) H 1 ((, 1))} with the natural norm, we consider the left-shift operator S (t) from Example 1.4. The associated producer is (as before) A = x, this time defined on D (A): = {u: R + R x u X}. A is densely defined, closed, and λ A is invertible for all Re λ>: The solution of (λ x) u = f is unique and given by u (x) = x e λ (τ x) f (τ) dτ. Although A has all these properties, S (t) is still not a semigroup. In fact, S (t) L (X) does not hold (because S (t) does not map to X at all). Exercise 2.2. Show directly that the unbounded operator x does not satisfy the resolvent estimate. : X X the 25th

26 3. Continuous and contractive semigroups 3.1. Generators of contractive semigroups In Section 2.2 we collected the necessary properties of generators A: D (A) X of ω-contractive semigroups (M = 1). A is closed D (A) is dense σ (a) {λ Re λ <ω} R (λ, A) Re λ ω 1 λ, Re λ> ω. It is now our aim to show that these properties are also sufficient. If this is shown, we have found a method to decide for an operator A whether the equation t u = Au has a (ω-contractive) solution. The case of strongly continuous solutions (M> 1) is a little more complicated, we will give the corresponding results in Section 3.3. First remark: If S (t) = e At is the semigroup of a bounded operator A and B interchanged with A, then BS (t) = S (t) B also applies. This can be shown, for example, using the series display. Theorem 3.1 (Hille-Yosida (1948), contractive case). Let ω R. For a linear operator (A, D (A)) on a Banach space X, the following are equivalent: 1. (A, D (A)) generates a strongly continuous ω-contractive semigroup. 2. (A, D (A)) is closed, densely defined, and for all λ C with Re λ> ω we have λ ρ (a) and 1 R (λ, A) Re λ ω. For it is sufficient to assume that the resolvent estimate holds for λ (ω,) ρ (a) and that a λ (ω,) exists with λ ρ (a). Proof. The theorems 2.15 and 2.17 already show the direction. For 2.1 .: Because of Remark 2.11 we can assume ω = without restriction. If there is only one λ (,) in the resolvent set, we use the principle from Remark 2.6: The estimate implies that all (,) is in the resolvent set. 26th

27 We now want to construct a semigroup. The basic idea is the following: We approximate the operator A by bounded operators A n. According to Theorem 2.16, the operators A n generate semigroups S n (t) on X. If we can carry out the limit n, we have the semigroup we are looking for with S ( t) = lim n S n (t) found. The suitable choice of A n are the Yosida approximations A n: = AR (1, 1n) A = nar (n, A) = n 2 R (n, A) nid. We will show that for S n (t) = e Ant the following applies: a) A nu Au for u D (A), b) S (t) u = lim n S n (t) u exists for all u X, c) S (t) is a strongly continuous semigroup on X ist, d) A is the generator of S (t) ist. With c) and d) the theorem is proven. a) We first show nr (n, A) id. For u D (A) we have nr (n, A) uu R (n, A) Au 1 Au, n The operators nr (n, A) are uniformly bounded (by 1) and D (A) is dense, so we have nr (n, A) uu for all u X. For u D (A) then A nu = nr (n, A) Au Au holds as claimed. b) We use the representation A n = n + n 2 R (n, A), the estimates for the exponential formula and the resolvent estimate, S n (t) e nt e n2 R (n, A) te nt e nt = 1 So the semigroups S n are contractive semigroups. According to our preliminary remarks, the resolvents R (n, A) and R (m, A), thus also the operators A n and A m, and then these commute with the semigroups S m (t) and S n (t). We can therefore calculate S n (t) u S m (t) u = = = d ds (S m (ts) sn (s) u) ds (A m S m (ts) sn (s) u + S m (ts) an S n (s) u) ds S ​​m (ts) sn (s) (A n A m) u ds. Because of the convergence A n A and the boundedness of S (τ) it follows that S n (t) u is a Cauchy sequence, i.e. lim S n (t) u =: S (t) un exists for u D (A) . 27

28 Because of the boundedness of S n (t) and tightness of D (A), the limit even applies to all u X. c) The norm estimate for S n (t) carries over and therefore S (t) is contractive. The convergence S n (t) u S (t) u is uniform in t on compact intervals. Hence, from the continuity of S n (t) u the continuity of t S (t) u follows. d) We now show that A is the generator of S (t). Let (B, D (B)) be the generator of S (t). Because of the solution properties, we also have for u D (A) S n (t) u u = S n (τ) a n u S (τ) Au S n (τ) a n u dτ. S n (τ) A n u Au + (S n (τ) S (τ)) Au. So we can go over to the Limes above and find for u D (A) we can form S (t) u u = S (t) u u lim t + t S (τ) Au dτ. = Au. This implies D (A) D (B) and Bu = Au for u D (A). For λ>, (λ B) (D (A)) = (λ A) (D (A)) = X, because A = B on D (A) and λ A is invertible as operator D (A) X. The operator B creates a contractive semigroup, which is why (,) ρ (b), that is, λ B: D (B) X is invertible. Then id = (λ B) 1 (λ B): D (A) D (B) surjective, i.e. D (B) D (A). We conclude D (B) = D (A). The two operators also agree in their domain of definition. Dissipative operators On a Hilbert space X a linear operator A: D (A) X X is called dissipative if Re Au, u u D (A). Note 3.2. Dissipative operators A satisfy the resolvent estimate R (λ, A) 1 Re (λ) Testing (λ A) u = f with u yields We take the real part and find λ ρ (a), Re λ>. λ u 2 Au, u = f, u. Re λ u 2 f, u f u with Cauchy black. Shortening u thus provides the resolvent estimate. Re λ u f, 28

29 Theorem 3.3 (Lumer-Phillips (1961), simplified). If A is densely defined and dissipative with (1 A) (D (A)) = X, then A creates a contractive semigroup. Proof. The resolvent estimate has already been recalculated. It remains to show that A is closed. Then Hille-Yosida's theorem can be applied. According to the assumption, Figure 1 A is surjective, but also injective because of the calculation in Note 3.2. So 1 A is invertible, the inverse is bounded according to the resolvent estimate. According to the theorem of the closed graph, the graph of (1 A) 1 is closed, and thus A is also closed. Contractive semigroups in applications In this section we consider Hilbert spaces X self-adjoint operators For every operator A: D (A) XX we can adjudicate Define operator A: X = XD (A) by (A x) (u): = x (au). For x D (A) the form is represented with the scalar product, and Au, x = x (au) = (A x) (u) = u, A x applies. The operator A: D (A) XX is called self-adjoint if D (A): = {µ XA µ bounded in X} = D (A) and A = A. An immediate consequence is for u D (A) Au, u = u, A u = u, Au = Au, u, where the conjugate denotes complexes. We conclude Au, u R. Remark 3.4. Let A be densely defined, self-adjoint and dissipative. Then A creates a contractive semigroup. We first find out that adjoint operators are always closed (exercise), so A is closed. We will show that C: = (1 A) (D (A)) = X. (3.1) As soon as this is shown, (1 A) (D (A)) = X also holds, because limits of picture elements are due Isolation of 1 A again images of 1 A. We can apply the Lumer-Phillips theorem. We now show C = X. If C is not quite X, then by Hahn-Banach's theorem there is an element µ X with µ C =, so (1 A) u, µ = u D (A). Formally, then (1 A) µ =, the estimate for solutions yields µ =, the desired contradiction. It remains to show that µ D (A). The figure (1 A) µ: u (1 A) u, µ is defined on D (A) and restricted with regard to the X-norm (since it is identical), so it can be continued to (1 A) µ X. Thus µ D ( A) according to the definition of D (A), and thus µ D (A) = D (A) according to assumption. 29

30 Parabolic Equation Example 3.5 (Parabolic Equation). Let Ω R n be bounded by Lipschitz's boundary and Ω C 2. We consider the unbounded operator A on X, defined with C coefficients by A: D (A) = H 2 (Ω) H 1 (Ω) XX = L 2 (Ω), Au (x) = i, j xi (ai, j (x) xj u (x)) + ibi (x) xi u (x) + c (x) u (x). (3.2) A is elliptical, i.e. for η> we have a i, j ξ i ξ j η ξ 2 ξ R n. Then A creates an ω-contractive semigroup. i, j A is densely defined. For large λ R the bilinear form to λ A is coercive (see calculation below) and thus (λ A) u = f uniquely solvable for all f X according to Lax-Milgram with solution u H 1 (Ω). The regularity theory yields the estimate u H 2 C f X for H 1 -solutions u of (λ A) u = f. In particular, λ is in the resolvent set. We continue to conclude that A is complete. Reason: It is sufficient to show that (λ A) is closed. So let u n u solutions to f n f. U n u m H 2 C f n f m X. So u n u in H 2, in particular u is in the domain and the equation holds. It remains to show the resolvent estimate. We show that for a suitable ω R the operator A + ω is dissipative. Re Au, u = ai, j (x) xj u (x) xi ū (x) dx + Ω i, j Ω bi (x) xi u (x) ū (x) dx + i η 2 u 2 L 2 (Ω) + C u 2 L 2 (Ω). Ω c (x) u (x) ū (x) dx With ω = C the following applies: A generates an ω-contractive semigroup. The above calculation also shows the coercivity of the bilinear form misadjointed operators. Our next application concerns skewed adjoint operators. A: D (A) XX is called skew adjoint if D (A): = {µ XA µ bounded in X} = D (A) and A = A. An immediate consequence is for u D (A) Au, u = u , A u = u, Au = Au, u, thus Au, u ir. We see: Skewed operators are automatically dissipative. 3

31 Comment 3.6. Let X be a Hilbert space over C, A densely defined and skew adjoint. Then A creates a contractive semigroup. The proof is the same as for dissipative self-adjoint operators. This time the dissipativity already follows from the skewed adjointness, as before the closure follows from the fact that adjoint operators are always closed. From the dissipativity the estimate for solutions follows, u Re λ 1 f, if (λ A) u = f. (3.3) We now want to show C \ iR ρ (a). For this, let λ C \ iR be arbitrary. We have ker (λ A) = {}, because solutions u of (λ A) u = satisfy u = according to (3.3). As with the self-adjoint operators one shows C: = (1 A) (D (A)) = X. Formally: If C X, then there is µ X that vanishes on C. We have (λ + a) µ =, i.e. due to the resolvent estimate µ =. Example 3.7 (Schrödinger equation). For VC c (R n, R) we define A = i (+ V (.)): D (A): = H 2 (R n, C) L 2 (R n, C) =: X. ​​(3.4 ) The operator A is a tightly defined, skew adjoint operator on the Hilbert space X, in particular A creates a contractive semigroup. We see immediately that X is a Hilbert space and that A is densely defined. So it suffices to show that A is skew adjoint. The formal equality A = A follows immediately through partial integration, Au, v L 2 = = R ni (u (x) + V (x) u (x)) v (x) dx R nu (x) i (v ( x) + V (x) v (x)) dx = u, Av L 2, because ī = i. The fact that the form A µ is bounded for µ X means that X = L 2 u µ, Au = R n i (ū + V (.) Ū) µ is bounded (for all u D (A)). This means that the distribution µ as a form is restricted to L 2, so that µ L 2. Together with µ X = L 2 this means µ H 2. This shows that D (A) = D (A). The equality A = A applies rigorously to D (A). We conclude that the Schrödinger equation creates a contractive semigroup. Wave equations We consider again an elliptic differential operator as in (3.2), albeit symmetrically and with a sign, A u (x) = i, j xi (ai, j (x) xj u (x) )) c (x) u (x). Let A be elliptical and symmetrical with c. 31

32 Example 3.8 (wave equation, symmetrical). Let Ω R n be bounded and Lipschitz, we investigate the wave equation 2 t u A u =, in Ω (,) u = u (.,) = G (.), T u (.,) = H (.), On Ω. on Ω [,), To put this equation in the form tw = Bw, we define v: = tu, w: = (u, v), X: = H 1 (Ω) L 2 (Ω), B ( u, v): = (v, A u), Then (B, D (B)) creates a contractive semigroup. D (B): = (H 2 (Ω) H 1 (Ω)) H 1 (Ω). Proof. We want to apply the Lumer-Phillips theorem. We set X = L 2 (Ω). On H 1 (Ω) we choose an equivalent norm by u 1: = A u, u L 2, and denote the associated Hilbert space by X 1. We claim that B is dissipative. In fact, for w = (u, v) Bw, w X1 X = v, u X1 + A u, v L 2 =. B is tightly defined. It remains to show that λ B is invertible for λ = 1. For w = (u, v) the equation is equivalent to v = λu f and (λ B) w = λ (u, v) (v, A u) = (f, g) λ 2 u A u = λf + G. This can be solved because λ 2 A is invertible (see parabolic equation). Two remarks on the wave equation. 1. You can also solve backwards. If we set s = t and solve 2 su A u =, we also find a solution for negative t. The equation not only creates a semigroup, but even a group. 2. The conservation of energy applies to the time-dependent solution (testing the equation with t u): (1 t 2 tu L) 2 A u, u =. We now want to consider general elliptic operators as in (3.2), that is, not necessarily symmetric and not necessarily negative. A: D (A) = H 2 (Ω) H 1 (Ω) XX = L 2 (Ω), defined with elliptic C coefficients by Au (x) = i, j xi (ai, j (x) xj u (x)) + ibi (x) xi u (x) + c (x) u (x). 32

33 Example 3.9 (wave equation, non-symmetrical). We examine the equation from Example 3.8, t 2 ū Aū =, in Ω (,). not necessarily symmetric with the operator A, but with a i, j = a j, i. This equation creates a ω-contractive semigroup. We want to rewrite the equation and set ū (x, t) = u (x, t) e αt. Then the equation for u t 2 u + 2α t u + α 2 u Au =. To put this equation in the form t w = Bw, we define v: = t u + α 2 u, w: = (u, v), X and D (B) as before. We calculate t v = 2 t u + α 2 tu = 3 2 α tu + Au α 2 u = 3 α2 αv 2 4 u + Au. So we put the main part B (u v) (= A u (x) = i, j αu + v 2 (α2 + A) u 3αv 4 2 xi (a i, j (x) xj u (x))). is symmetrical according to the assumption. We choose X: = L 2 (Ω) and on X 1: = H (Ω) 1 the scalar product () α 2 u, v 1: = 4 A u, v. L 2 We claim that on X 1 X the operator B is dissipative (for α> large). Bw, w X1 X = α 2 uv, u 1 + α 2 u C u H 1 v L α v 2 L 2.) (α2 4 + A u, v 3 L 2 α v 2 L 2 2 for α> large if this is negative. So B is dissipative. Furthermore, B is densely defined and (1 B) (u, v) = (f 1, f 2) is solvable for all (f 1, f 2) X, because (λ A) u = f is solvable for all f L 2 (Ω), with solution u H 2. Because of the regularity estimate u H 2 C f L 2, (1 B) 1 is bounded, so B creates a contractive semigroup α-contractive semigroup 33