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Some Elements of Semigroup Theory

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Some Elements of Semigroup Theory Some elements of semigroup theory J´erˆome Le Rousseau January 8, 2019 Contents 1 Strongly continuous semigroups 2 1.1 Definition and basic properties ...................... 2 1.2 The Hille-Yosida theorem ........................ 5 1.3 The Lumer-Phillips theorem ....................... 6 2 Differentiable and analytic semigroups 9 3 Mild solution of the inhomogeneous Cauchy problem 10 4 The case of a Hilbert space. 12 Semigroup theory is at the heart of the understanding of many evolution equa- tions that can be put in the form d x(t)+ Ax(t)= f(t), t> 0, x(t)= x , (0.1) dt 0 with x(t) and x0 in a proper function space, usually a Banach space, denoted by X below, if not a Hilbert space, with A an unbounded operator on X, with dense do- main, and f a function of the time variable t taking values in X. First, in Sections 1 and 2, we review the case of a homogeneous equation, that is f ≡ 0. Second, in Section 3 we consider the more general case of an inhomogeneous equation. In par- ticular, we provide the necessary form of the solution, given by the so-called mild solution based on the Duhamel formula. Third, in Section 4 we show how some results improve in the case the function space X is a Hilbert space. 1 1 Strongly continuous semigroups Consider the homogeneous equation associated with the evolution problem (0.1), that is, d x(t)+ Ax(t)=0, t> 0, x(t)= x , (1.1) dt 0 Under proper assumptions on A we can write the solution in the form x(t)= S(t)x0, where S(t): X → X is a bounded operator. As some sort of differentiation with respect to time is expected in (1.1), a minimal assumption is then that S(0)x = x and t 7→ S(t)x be continuous for all x ∈ X. (1.2) With t 7→ x(t) solution to (1.1), if the evolution problem is well posed, we expect from uniqueness that solving the following problem, for some t0 ≥ 0, d y(t)+ Ay(t)=0, t>t , y(t )= x(t ), (1.3) dt 0 0 0 yield a solution that satisfies y(t) = x(t) for t ≥ t0. In particular, this implies the following property S(t + t′)= S(t) ◦ S(t′), for t, t′ ∈ [0, +∞). (1.4) Properties (1.2) and (1.4) are precisely the starting point of semigroup theory in Banach spaces. 1.1 Definition and basic properties Let X be a Banach space. Definition 1. A family S(t) of bounded operators on X, indexed by t ∈ [0, +∞) is called a semigroup if: ′ ′ ′ S(0) = IdX and S(t + t )= S(t) ◦ S(t ) for t, t ∈ [0, +∞). (1.5) The semigroup is called strongly continuous if, moreover, for all x ∈ X we have limt→0+ S(t)x = x. One says that S(t) is a C0-semigroup for short. For each C0-semigroup there exist M ≥ 1 and ω ∈ R such that S ωt k (t)kL (X) ≤ Me , (1.6) by the uniform boundedness principle [3, Theorem 1.2.2]. It follows that the map (t, x) 7→ S(t)x is continuous from [0, +∞) × X into X. A C0-semigroup S(t) is said 2 S to be bounded if there exists M ≥ 1 such that k (t)kL (X) ≤ M, for t ≥ 0. In the case M = 1 one says that the C0-semigroup is of contraction. We define the unbounded linear operator A from X to X, with domain D(A)= {x ∈ X; lim t−1(x − S(t)x) exists}, (1.7) t→0+ and given by Ax = lim t−1(x − S(t)x), x ∈ D(A). (1.8) t→0+ For basic aspects of unbounded operators we refer to [1]. This operator (A, D(A)) is called the generator of the C0-semigroup. One can prove that the generator of a C0-semigroup has a dense domain in X and is a closed operator [3, Corollary 1.2.5] (see [1] for the notion of closed operators). The domain D(A) can be equipped with the graph norm kxkD(A) = kxkX + kA(x)kX . Since A is closed one finds that (D(A), k.kD(A)) is complete. Note that the map S(t) 7→ A (1.9) is injective [3, Theorem 1.2.6]. The following proposition shows that computing S(t)x yields a solution of an evolution equation. Proposition 2. Let T ∈ R+ ∪{+∞} and let x ∈ D(A). We have u(t) = S(t)x ∈ C 0([0, T ],D(A)) ∩ C 1([0, T ],X) and d u(t)+ Au(t)=0, 0 ≤ t ≤ T, u(0) = x. (1.10) dt Moreover, S(t)x is the unique solution to (1.10) in C 0([0, T ],D(A)) ∩ C 1([0, T ],X). In addition, we have AS(t)x = S(t)Ax. Here, [0, T ] means [0, +∞) if T =+∞. Proof. Let x ∈ D(A). We write, for h> 0, −1 −1 h IdX −S(h)S(t)x = S(t)h IdX −S(h)x →h→0+ S(t)Ax, as S(t) is bounded on X. Hence, we have S(t)x ∈ D(A) and AS(t)x = S(t)Ax by d+ S S (1.7)–(1.8). Moreover, we have the following right derivative dt (t)x = −A (t)x. To compute the left derivative we write, for h> 0, h−1(S(t)x − S(t − h)x)= S(t − h)h−1(S(h)x − x). 3 Since h−1(S(h)x − x) → −Ax as h → 0+ and since (t, x) 7→ S(t)x is continuous on [0, T ] × X we obtain h−1(S(t)x − S(t − h)x) → −S(t)Ax. h→0+ d S S S S We thus obtain dt (t)x + A (t)x = 0. From A (t)x = (t)Ax and the continuity of t 7→ S(t)y for all y ∈ X we conclude that S(t)x ∈ C 0([0, T ],D(A)) and finally using the equation we have S(t)x ∈ C 1([0, T ],X). Uniqueness. Let u ∈ C 0([0, T ],D(A)) ∩ C 1([0, T ],X) be a solution to (1.10) satisfying u(0) = x. Let 0 < s ≤ T . For t ∈ [0,s] we set v(t)= S(s − t)u(t). With the first part we have d d v(t)= S(s − t) u(t)+ S(s − t)Au(t)=0. dt dt We thus find S(s)x = S(s)u(0) = v(0) = v(s)= u(s). We provide also the following result that shows that integration with respect to time yields a gain of regularity. T S Lemma 3. For x ∈ X and T > 0, we have ∫0 (t)x dt ∈ D(A) and T S(T )x − x + A ∫ S(t)x dt =0. 0 Proof. For h> 0 we compute T T −1 −1 Fh = h (IdX −S(h)) ∫ S(t)x dt = h ∫ S(t) − S(t + h)x dt 0 0 h = h−1 ∫ S(t)x − S(t)S(T )x dt. 0 With the continuity of t 7→ S(t)x and t 7→ S(t)S(T )x, the fundamental theorem of calculus yields the result, by (1.7)–(1.8). We shall use the following version in which a smooth window function is intro- duced. C ∞ Lemma 4. For x ∈ X and χ ∈ c (0, ∞) we have ∞ ∞ ∞ ∫ χ(t)S(t)x dt ∈ D(A) and A ∫ χ(t)S(t)x dt = ∫ χ′(t)S(t)x dt. 0 0 0 We see that the result of Lemma 3 formally coincides with that of Lemma 4 in the case χ = 1(0,T ). 4 Proof. For h> 0 we compute ∞ ∞ −1 −1 Fh = h (IdX −S(h)) ∫ χ(t)S(t)x dt = h ∫ χ(t)S(t) − S(t + h)x dt. 0 0 Observe that we have ∞ ∞ ∞ ∫ χ(t)S(t + h)x dt = ∫ χ(t − h)S(t)x dt = ∫ χ(t − h)S(t)x dt, 0 h 0 because of the support of χ. We thus obtain ∞ −1 Fh = h ∫ χ(t) − χ(t − h)S(t)x dt. 0 With the continuity of t 7→ S(t)x, the Lebesgue dominated-convergence theorem yields ∞ ′ lim Fh = ∫ χ (t)S(t)x dt. h→0+ 0 Consequently, by (1.7)–(1.8), we obtain the result. zt Observe that if S(t) is a C0-semigroup and z ∈ C then e S(t) satisfies (1.5). The following proposition is then clear from what precedes. zt Proposition 5. Let S(t) be a C0-semigroup and z ∈ C. Then e S(t) is also a C0-semigroup and its generator is A − z IdX . Note that, because of the uniqueness of the generator of a C0-semigroup [3, Theorem 1.2.6], conversely, if A generates a C0-semigroup, then A − z IdX is the zt generator of a C0-semigroup, namely e S(t). 1.2 The Hille-Yosida theorem The next natural question is to wonder if an unbounded operator on X is the generator of a C0-semigroup. The Hille-Yosida theorem is central in the semigroup theory, providing a clear answer to this question. We refer to [3, Theorem 1.3.1] for a proof. Theorem 6. Let (A, D(A)) be a linear unbounded operator on a Banach space X.
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