An introduction to evolution PDEs September 24, 2018 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP

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1 An introuction to evolution PDEs September 24, 218 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP In this chapter we make the link between the existence theory for evolution PDEs we have presente in the two first chapters an the theory of continuous semigroup of linear an boune operators. In that unifie framework we may establish the Duhamel formula an the extension of the existence theory by perturbation argument. We also briefly present the Hille-Yosia-Lumer-Phillips existence theory for m-issipative operators. Contents 1. From linear evolution equation to semigroup Semigroup From well-poseness to semigroup 2 2. Semigroup an generator 3 3. Duhamel formula an mil solution 6 4. Dual semigroup an weak solution 7 5. Coming back to the well-poseness issue for evolution equations A perturbation trick Semilinear evolution equation Dissipativity an extension trick Semigroup Hille-Yosia-Lumer-Phillips existence theory Complements Continuity Nonautonomous semigroup Transport equation in measures an L frameworks 2 8. Discussion Several way to buil solutions From Hille-Yosia theory to variational solutions Very weak solution Bibliographic iscussion From linear evolution equation to semigroup 1.1. Semigroup. We state the efinition of a continuous semigroup of linear an boune operators. Definition 1.1. We say that (S t ) t is a continuous semigroup of linear an boune operators on a Banach space X, or we just say that S t is C -semigroup (or a semigroup) on X, we also write S(t) = S t, if the following conitions are fulfille: (i) one parameter family of operators: t, f S t f is linear an continuous on X; (ii) continuity of trajectories: f X, t S t f C([, ), X); (iii) semigroup property: S = I; s, t, S t+s = S t S s ; (iv) growth estimate: b R, M 1, (1.1) S t B(X) M e bt t. We then efine the growth boun ω(s) by ω(s) := lim sup t 1 log S(t) = inf{b R; (1.1) hols}. t 1

2 2 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP We say that (S t ) is a semigroup of contractions if (1.1) hols with b = an M = 1. Remark 1.2. The two continuity properties (i) an (ii) can be unerstoo both in the same sense of - the strong topology of X, an we will say that S t is a strongly continuous semigroup; - the weak topology σ(x, Y ) with X = Y, Y a (separable) Banach space, an we will say that S t is a weakly continuous semigroup. In the sequel, the semigroups we consier are strongly continuous except when we specify it. Anyway, many of the results are also true for weakly continuous semigroups. Remark 1.3. It is worth mentioning (an we refer to the sections 7.1 & 7.3 for more etails) that - For a given one parameter family (S t), the growth property (iv) is automatically satisfie when (i), (ii) an (iii) hol. - The continuity property (ii) can be replace by the following (right) continuity assumption in t = : (ii ) S(t)f f when t, for any f X. - When the continuity properties (i) & (ii) hol for the weak topology σ(x, X ) then they also hol for the strong topology in X: we o not nee to make any ifference between strongly an weakly continuous semigroups. - There o exist weakly continuous semigroups which are not strongly continuous because the continuity property (ii) for the weak topology σ(x, Y ) oes not implies the analogous strong continuity. Classical examples are the heat semigroup an the translation semigroup S tf = γ t f, γ t = (2πt) 1/2 e x 2 2t, an (S tf)(x) = f(φ 1 t (x)), Φ 1 t (x) := x t, in the Lebesgue space L (R) an in the space M 1 (R) := (C (R)) of boune Raon measures. - For a given semigroup (S t) on X, one may efine the new semigroup (T t) an the new norm on X by (1.2) T (t) := e ω t S(t) an f := sup T (t) f, t an then show that is equivalent to an T (t) is a semigroup of contractions for that new norm. However, that trick is not so useful because the new norm oes not satisfy the same nice regularity properties as the initial norm often satisfies. Exercise 1.4. (1) Prove that (i), (ii ) an (iii) imply (iv). (Hint. Use a contraiction argument an the Banach- Steinhaus Theorem or see Proposition 7.1). (2) Prove that (i)-(ii )-(iii) implies (i)-(ii)-(iii). (Hint. Use (1) or see Proposition 7.2). (3) Prove that in Remark 1.3, the two norms are equivalent an that T (t) is a semigroup of contractions for the new norm. (Hint. See Proposition 7.3). (4) Prove that if S t satisfies (iii) as well as the continuity properties (i) & (ii) in the sense of the weak topology σ(x, X ), then S t is a strongly continuous semigroup. (Hint. See Theorem 7.4) From well-poseness to semigroup. Given a linear operator Λ acting on a Banach space X (or on a subspace of X) an a initial atum g belonging to X (or to a subspace of X), we consier the (abstract) linear evolution equation (1.3) t g = Λg in (, ) X, g() = g in X. We explain how we may associate a C -semigroup to the evolution equation as a mere consequence of the linearity of the equation an of the existence an uniqueness result. Definition 1.5. Consier three Banach spaces X, Y, Z such that Z X Y, with continuous an ense embeing, an a linear an boune operator Λ : Z Y. We enote Λ : Y Y Z the ajoint operator. We say that a function g = g(t) E T := C([, T ); X ) L r (, T ; Z), 1 r, with X = X or X = X σ(x, Y ), is a weak solution to the evolution equation (1.3) associate to the initial atum g X if (1.4) [ g, ϕ X,Y ] T g, t ϕ X,Y t = g, Λ ϕ Z,Z t, for any test function ϕ Y T := C 1 ([, T ]; Y ). In the case X = X we can take Y = Y σ(y, X), while in the case X = X σ(x, Y ) we must take Y = Y. We o insist on that ψ C([, T ); X σ(x, Y)), with X Y or Y Y, means that the mapping t ψ(t), ϕ X,Y is continuous for any ϕ Y. We note E := {g : R + X; g [,T ] E T, T > }.

3 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 3 We give two examples. For a variational solution to an abstract parabolic equation, we take X = X = H, Z = Y = V an r = 2, with the notations of Chapter 1. For a weak (an thus renormalize) solution to a transport equation, we take X = X = Z = L p, 1 p <, Y = W 1,p an r =. Definition 1.6. We say that the evolution equation (1.3) is well-pose in the sense of Definition 1.5 of weak solutions, if for any g X there exists a unique function g E which satisfies (1.4), an for any R, T > there exists R T := C(T, R ) > such that (1.5) g X R implies sup g(t) X R T. [,T ] Proposition 1.7. To an evolution equation (1.3) which is well-pose in the sense of Definition 1.6, we may associate a continuous semigroup of linear an boune operators (S t ) in the following way. For any g X an any t, we set S(t)g := g(t), where g E is the unique weak solution to the evolution equation (1.3) with initial atum g. Corollary 1.8. To the time autonomous parabolic equation consiere in Chaper 1 an to the time autonomous transport equation consiere in Chapter 2, we can associate a strongly continuous semigroup of linear an boune operators. Proof of Proposition 1.7. We just check that S as efine in the statement of the Proposition fulfille the conitions (i), (ii) an (iii) in Definition 1.1. S satisfies (i). By linearity of the equation an uniqueness of the solution, we clearly have S t (g + λf ) = g(t) + λf(t) = S t g + λs t f for any g, f X, λ R an t. Thanks to estimate (1.5) we also have S t g C(t, 1) g for any g X an t. As a consequence, S t B(X) for any t. S satisfies (ii) since by efinition t S t g C(R + ; X ) for any g X. S satisfies (iii). For g X an t 1, t 2 enote g(t) = S t g an g(t) := g(t + t 1 ). Making the ifference of the two equations (1.4) written for t = t 1 an t = t 1 + t 2, we see that g satisfies g(t 2 ), ϕ(t 2 ) = g(t 1 + t 2 ), ϕ(t 1 + t 2 ) 1+t 2 { = g(t 1 ), ϕ(t 1 ) + Λg(s), ϕ(s) + ϕ (s), g(s) } s = g(), ϕ() + t 1 2 { Λ g(s), ϕ(s) + ϕ (s), g(s) } s, for any ϕ Y t1+t 2 with the notation ϕ(t) := ϕ(t + t 1 ) Y t2. We thus obtain S t1+t 2 g = g(t 1 + t 2 ) = g(t 2 ) = S t2 g() = S t2 g(t 1 ) = S t2 S t1 g, where for the thir equality we have use that the equation on the functions g an ϕ is nothing but the weak formulation associate to the equation (1.3) for the initial atum g(). 2. Semigroup an generator On the other way roun, in this section, we explain how we can associate a generator an then a solution to a ifferential linear equation to a given semigroup. Definition 2.1. An unboune operator Λ on X is a linear mapping efine on a linear submanifol calle the omain of Λ an enote by D(Λ) or om(λ) X; Λ : D(Λ) X. The graph of Λ is G(Λ) = graph(λ) := {(f, Λ f); f D(Λ)} X X. We say that Λ is close if the graph G(Λ) is a close set in X X: for any sequence (f k ) such that f k D(Λ), k, f k f in X an Λ f k g in X then f D(Λ) an g = Λ f. We enote C (X) the set of unboune operators with close graph an C D (X) the set of unboune operators which omain is ense an graph is close.

4 4 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP Definition 2.2. For a given semigroup (S t ) on X, we efine D(Λ) := { S(t) f f f X; lim exists in X }, t t S(t) f f Λ f := lim for any f D(Λ). t t Clearly D(Λ) is a linear submanifol an Λ is linear: Λ is an unboune operator on X. We call Λ : D(Λ) X the (infinitesimal) generator of the semigroup (S t ), an we sometimes write S t = S Λ (t). We enote G (X) the set of operators which are the generator of a semigroup. We present some funamental properties of a semigroup S an its generator Λ that one can obtain by simple ifferential calculus arguments from the very efinitions of S an Λ. Proposition 2.3. (Differentiability property of a semigroup). Let f D(Λ). (i) S(t)f D(Λ) an Λ S(t)f = S(t) Λsf for any t, so that the mapping t S(t) f is C([, ); D(Λ)). (ii) The mapping t S(t)f is C 1 ([, ); X), S(t)f = ΛS(t)f for any t >, an then t S(t)f S(s)f = s S(τ) Λf τ = Sketch of the proof of Proposition 2.3. Let f D(Λ). Proof of (i). We fix t an we compute S(s)S(t)f S(t)f lim s + s which implies S(t)f D(Λ) an ΛS(t)f = S(t)Λf. Proof of (ii). s = lim s + S(t)S(s)f f s Λ S(τ)f τ, t > s. = S(t)Λf, We fix t > an we compute (now) the left ifferential { S(t + s)f S(t)f } lim S(t)Λf = s s { ( S( s)f f ) ( )} = lim S(t + s) Λf + S(t + s)λf S(t)Λf =, s s using that the two terms within parenthesis converge to an that S(t + s) M e ωt for any s. Together with step 1, we euce that t S(t)f is ifferentiable for any t >, with erivative ΛS(t)f. We conclue to the C 1 regularity by observing that t S(t)Λf is continuous. Last, we have an in particular for any t > s. S(t)f S(s)f = s [S(τ) f] τ = τ s S(τ) Λf τ = S(t)f S(s)f (t s) M e bt Λf, s Λ S(τ)f τ Definition 2.4. Consier a Banach space X an an (unboune) operator Λ on X. We say that g C([, ); X) is a classical (or Hille-Yosia) solution to the evolution equation (1.3) if g C((, ); D(Λ)) C 1 ((, ); X) so that (1.3) hols pointwise. In it worth emphasizing that Proposition 2.3 provies a classical solution to the evolution equation (1.3) for any initial atum f D(Λ) by the mean of t S Λ (t)f. Lemma 2.5. For any f X an t, there hol an (ii) (i) 1 lim h h +h t S(s) f s = S(t)f, ( ) S(s)f s D(Λ), (iii) Λ S(s)f s = S(t)f f.

5 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 5 Sketch of the proof of Lemma 2.5. The first point is just a consequence of the fact that s S(s)f is a continuous function. We then euce 1 { ) S(h) S(s)f s S(s)f s = 1 { +h } S(s) f s S(s)f s h h h = 1 { +h h } S(s) f s S(s)f s S(t)f f, h h which implies the two last points. t In the next result we prove that G (X) C D (X). Definition 2.6. We say that C X is a core for the generator Λ of a semigroup S if C D(Λ), C is ense in X an S(t) C C, t. Proposition 2.7. (Properties of the generator) Let Λ G (X). (i) The omain D(Λ) is ense in X. In particular, D(Λ) is a core. (ii) Λ is a close operator. (iii) The mapping which associates to a semigroup its generator is injective. More precisely, if S 1 an S 2 are two semigroups with generators Λ 1 an Λ 2 an there exists a core C D(Λ 1 ) D(Λ 2 ) such that Λ 1 C = Λ 2 C, then S 1 = S 2. In other wors, S 1 S 2 implies Λ 1 Λ 2. Sketch of the proof of Proposition 2.7. For any f X an t >, we efine f t := t 1 t S(s)f s. Thanks to Lemma 2.5-(i) & (ii), we see that f t D(Λ) an f t f as t. In other wors, D(Λ) is ense in X. We prove (ii). Consier a sequence (f k ) of D(Λ) such that f k f an Λf k g in X. For t >, we write an passing to the limit k, we get S(t)f k f k = S(s)Λf k s, t 1 (S(t)f f) = t 1 S(s)g s. We may now pass to the limit t in the RHS term, an we obtain That proves f D(Λ) an Λf = g. t 1 (S(t)f f) g. We prove (iii). We observe that the mapping t S i (f)f, i = 1, 2, are C 1 for any f C, thanks to Proposition 2.3, an s S 1(s) S 2 (t s)f = S 1(s) S 2 (t s) f + S 1 (s) S 2(t s) f s s = S 1 (s) Λ 1 S 2 (t s) f + S 1 (s) Λ 2 S 2 (t s) f =. That implies S 2 (t)f = S 1 ()S 2 (t )f = S 1 (t)s 2 (t t)f = S 1 (t) f for any f C, an then S 2 S 1. Exercise 2.8. We efine recursively for n 2. Prove that D(Λ n ) is a core. D(Λ n ) := {f D(Λ n 1 ), Λf D(Λ n 1 )},

6 6 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP Consier the evolution equation (3.1) 3. Duhamel formula an mil solution t g = Λg + G on (, T ), g() = g, for an unboune operator Λ on X, an initial atum g X an a source term G : (, T ) X, T (, ). For G C((, T ); X), a classical solution g is a function (3.2) g X T := C([, T ); X) C 1 ((, T ); X) C((, T ); D(Λ)) which satisfies (3.1) pointwise. For U L 1 (, T ; B(X 1, X 2 )) an V L 1 (, T ; B(X 2, X 3 )), we efine the time convolution V U L 1 (, T ; B(X 1, X 3 )) by setting (V U)(t) := V (t s) U(s) s = V (s) U(t s) s, for a.e. t (, T ). Lemma 3.1 (Variation of parameters formula). Consier the generator Λ of a semigroup S Λ on X. For G C((, T ); X) L 1 (, T ; X), T >, there exists at most one classical solution g X T to (3.1) an this one is given by (3.3) g = S Λ g + S Λ G. Proof of Lemma 3.1. Assume that g X T satisfies (3.1). For any fiex t >, we efine s u(s) := S Λ (t s)g(s) C 1 ((, t); X) C([, t]; X). On the one han, we have On the other han, we compute g(t) S Λ (t)g = u(t) u() = u (s) s. u (s) = ΛS Λ (t s)g(s) + S Λ (t s)g (s) = S Λ (t s)g(s), for any s (, t). We conclue by putting together the two ientities. When G C((, T ); X) L 1 (, T ; D(Λ)) an g D(Λ), we observe that ḡ := S Λ g + S Λ G belongs to X T an tḡ(t) = ΛS Λ(t)g + Λ(S Λ G)(t) + S Λ ()G(t) = Λḡ(t) + G(t), so that ḡ is a classical solution to the evolution equation (3.1). When G L 1 (, T ; X) an g X, we observe that ḡ C([, T ]; X), ḡ() = g an it is the limit of classical solutions by a ensity argument. We say that ḡ is a mil solution to the evolution equation (3.1). Lemma 3.2 (Duhamel formula). Consier two semigroups S Λ an S B on the same Banach space X, assume that D(Λ) = D(B) an efine A := Λ B. If AS B, S B A L 1 (, T ; B(X)) for any T (, ), then S Λ = S B + S Λ AS B = S B + S B A S Λ in B(X). Proof of Lemma 3.1. Take f D(Λ) = D(B), t >, an efine s u(s) := S Λ (s)s B (t s)f C 1 ([, t]; X) C([, t]; D(Λ)). We observe that for any s (, t), from which we euce u (s) = S Λ (s)λs B (t s)f S Λ (s)bs B (t s)f S Λ (t)f S B (t)f = = S Λ (t s)as B (s)f, u (s) s = S Λ (t s)as B (s)f s. By ensity an continuity, we euce that the same hols for any f X, an that establishes the first version of the Duhamel formula. The secon version follows by reversing the role of S Λ an S B.

7 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 7 From the above secon version of Duhamel formula, we observe that for any g D(Λ), the function ḡ(t) := S Λ (t)g X T is a classical solution to the evolution equation (3.1) an satisfies the following functional equation (3.4) g = S B g + S B A g. On the other way roun, we observe that if g X T is a solution to the functional equation (3.4), then g (t) = BS B (t)g + B(S B A g)(t) + S B ()Ag(t) = Bg(t) + Ag(t) = Λg(t), so that g is a classical solution to the evolution equation (3.1). More generally, when S B A L 1 (, T ; B(X)), we say that g C([, T ]; X) is a mil solution to the evolution equation (3.1) if g() = an g is a solution to the functional equation (3.4). 4. Dual semigroup an weak solution Consier a Banach space X an an operator A C D (X), with X enowe with the topology norm, an we enote Y = X in that case, or with X = Y enowe with the weak topology σ(x, Y ) for a separable Banach space Y. We efine the subspace D(A ) := { ϕ Y ; C, f D(A), ϕ, Af C f X } an next the ajoint operator A on Y by A ϕ, f = ϕ, Af, ϕ D(A ), f D(A). Because D(A) X is ense, the operator A is well an uniquely efine an it is obviously linear. Because A has a close graph, the operator A has also a close graph. When A is a boune operator, then A is also a boune operator. When X is reflexive, then the omain D(A ) is always ense into X, so that A C D (X ). For a general Banach space X an a general operator A, then D(A ) is ense into X for the weak σ(x, X) topology, but it happens that D(A ) is not ense into X for the strong topology. Consier now a semigroup S with generator Λ an f D(Λ). Multiplying by ϕ Cc 1 ([, T ); D(Λ )) the equation (1.3) satisfie by g(t) := S(t)f an integrating in time, we get f, ϕ() X,X + S(t)f, t ϕ(t) + Λ ϕ(t) X,X t =. Because the mapping f S(t)f is continuous in X an the inclusion D(Λ) X is ense from Proposition 2.7, we see that the above formula is also true for any f X. In other wors, the semigroup S(t) provies a weak solution (in the above sense) to the evolution equation (1.3) for any f X. We aim to show now that the semigroup theory provies an answer to the well-poseness issue of weak solutions to that equation for any generator Λ. More precisely, given a semigroup, we introuce its ual semigroup an we then establish that the initial semigroup provies the unique weak solution to the associate homogeneous an inhomogeneous evolution equations. Proposition 4.1. Consier a strongly continuous semigroup S = S Λ on a Banach space X with generator Λ an the ual semigroup S as the one-parameter family S (t) := S(t) for any t. Then the following hol: (1) S is a weakly continuous semigroup on X with same growth boun as S. (2) The generator of S is Λ. In other wors, (S Λ ) = S Λ. (3) The mapping t S (t)ϕ is C([, ); X ) (for the strong topology) for any ϕ D(Λ ). Similarly, t S (t)ϕ is C 1 ([, ); X ) C([, ); D(Λ )) for any ϕ D(Λ 2 ). Proof of Proposition 4.1. (1) We just write S (t)ϕ, f = ϕ, S(t)f =: T f (t, ϕ) t, f X, ϕ X, an we see that (t, ϕ) T f (t, ϕ) is continuous for any f X.

8 8 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP (2) Denoting by D(L) an L the omain an generator of S as efine as in section 2, for any ϕ D(L) an f D(Λ) we have 1 Lϕ, f := lim t t (S(t) ϕ ϕ), f = lim ϕ, 1 t t (S(t)f f) = ϕ, Λf, from which we immeiately euce that D(L) D(Λ ) an L = Λ D(L). To conclue, we use that L is close. More precisely, for a given ϕ D(Λ ), we associate the sequence (ϕ ε ) efine through ϕ ε := 1 ε ε S(t) ϕ t. We have ϕ ε ϕ in the weak σ(x, X) sense, ϕ ε D(L) an, for any f D(Λ), Lϕ ε, f = Λ ϕ ε, f = ϕ ε, Λf = ϕ, 1 ε ε S(t)Λf t ϕ, Λf, so that Lϕ ε Λ ϕ in the weak σ(x, X) sense. The graph G(L) of L being close, we have (ϕ, Λ ϕ) G(L), which in turns implies ϕ D(L) an finaly L = Λ. (3) From Proposition 2.3, we have S (t)ϕ S (s)ϕ X = S (τ)λ ϕ τ Me bt (t s) Λ ϕ X X s for any t > s an ϕ D(Λ ), so that t S (t)ϕ is Lipschitz continuous from [, ) into X enowe with the strong topology. Proposition 4.2. Consier a weakly continuous semigroup T = S L on a Banach space X = Y with generator L, an the ual semigroup T as the one-parameter family T (t) := T (t) of boune operator on Y for any t. Then the following hol: (1) S = T is a strongly continuous semigroup on Y with same growth boun as T. (2) The generator Λ of S satisfies L = Λ. Proof of Proposition 4.2. Just as in the proof of Proposition 4.1, we have (t, f) ϕ, S(t)f is continuous for any ϕ X. That means that S(t) is a weakly σ(x, X ) continuous semigroup in X an therefore a strongly continuous semigroup in X thanks to Theorem 7.4. The rest of the proof is unchange with respect to the proof of Proposition 4.1. For any g X an G L 1 (, T ; X), we say that g C([, T ]; X) is a weak solution to the inhomogeneous initial value problem (3.1) if (4.1) ϕ(t ), g(t ) ϕ(), g = for any ϕ C 1 ([, T ]; X ) C([, T ]; D(Λ )). { ϕ + Λ ϕ, g + ϕ, G } t, Proposition 4.3. Assume that Λ generates a semigroup S on X. For any g X an G L 1 (, T ; X), there exists a unique weak solution to equation (3.1), which is nothing but the mil solution (4.2) ḡ = S Λ g + S Λ G. Proof of Proposition 4.3. We efine ḡ(t) = ḡ t := S(t)g + For any ϕ = ϕ t C 1 ([, T ]; X ) C([, T ]; D(Λ )), we have ϕ t, ḡ t = S t ϕ t, g + S(t s)g(s) s C([, T ]; X). S t sϕ t, G s s C 1 ([, T ])

9 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 9 an then t ϕ t, ḡ t = S t (Λ ϕ t + ϕ t), g + = Λ ϕ t + ϕ t, ḡ t + ϕ t, G t, S t s(λ ϕ t + ϕ t), G s s + G t, ϕ t from which we euce that ḡ is a weak solution to the inhomogeneous initial value problem (3.1) in the weak sense of equation (4.1). Now, if g is another weak solution, the function f := g ḡ is then a weak solution to the homogeneous initial value problem with vanishing initial atum, namely ϕ(t ), f(t ) = A first way to conclue is to efine ϕ + Λ ϕ, f t, ϕ C 1 ([, T ]; X ) C([, T ]; D(Λ )). ϕ(s) := s S (τ s) ψ(τ) τ, for any given ψ C 1 c ((, T ); D(Λ )), an to observe that ϕ C 1 ([, T ]; X ) C([, T ]; D(Λ )) is a (backwar) solution to the ual problem For that choice of test function, we get = ϕ = Λ ϕ + ψ on (, T ), ϕ(t ) =. ψ, f t, ψ C 1 c ((, T ); D(Λ )), an thus g = ḡ. An alternative way to get the uniqueness result is to efine ϕ(t) := S (T t)ψ for a given ψ D(Λ ). Observing that ϕ is a (backwar) solution to the ual problem (4.3) ϕ = Λ ϕ, ϕ(t ) = ψ, that choice of test function leas to ψ, f(t ) = ψ D(Λ ), T >, an thus again g = ḡ. Exercise 4.4. Consier a Banach space X an an unboune operator Λ on X. We assume that X = Y for a Banach space Y an that the ual opeartor Λ generates a strongly continuous semigroup T on Y. (1) Prove that S := T is a (at least) weakly σ(x, Y ) continuous semigroup on X with generator Λ an that it provies the unique weak solution to the associate evolution equation. (2) Prove that for any smooth functions a = a(x) an c = c(x), one can efine a weakly continuous semigroup S = S Λ on L = L (R ) associate to the transport operator (Λf)(x) := a(x) f(x) c(x) f(x), as the ual semigroup associate to the ual operator Λ efine on L 1 (R ). (3) Prove similarly that one can efine a weakly continuous semigroup on M 1 (R ) := (C (R )), the space of Raon measures, associate to the transport operator Λ. 5. Coming back to the well-poseness issue for evolution equations Using mainly the Duhamel formula an uality arguments, we present several ways for proving the well-poseness of evolution equation an builing the associate semigroup.

10 1 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 5.1. A perturbation trick. We give a very efficient result for proving the existence of a semigroup associate to a generator which is a mil perturbation of the generator of a semigroup. Theorem 5.5. Consier S B a semigroup satisfying the growth estimate S B (t) B(X) M e bt an A a boune operator. Then, Λ := A + B is the generator of a semigroup which satisfies the growth estimate S Λ (t) B(X) M e b t, with b = b + M A. Proof of Theorem 5.5. Step 1. Existence. We efine E := C([, T ]; B(X)) the space of family of operators (U(t)) t [,T ] such that U(t) B(X) for any t [, T ], U() = I an t U(t)g C([, T ]; X) for any g X. For any U E, we efine so that V E. More precisely, efining we observe that Φ : K K because V(t) = Φ[U](t) := S B (t) + (S B A U)(t), V E S B E + K := {U E ; U E e b T M}, S B (s)a B(X) s U E 2 e bt M, for T > small enough. Moreover, for U 1, U 2 K, we write U := U 2 U 1 an we have an then V(t) := V 2 (t) V 1 (t) = (S B A U)(t), Φ(U 2 ) Φ(U 1 ) E 1 2 U 2 U 1 E. From the Banach contraction Theorem, there exists a fixe point U K to the mapping Φ, so that U(t) = S B (t) + (S B AU)(t) = S B (t) + S B (t s)au(s) s, t [, T ]. We exten the function U to [, ) by iterating the construction. We may be a bit more accurate in the estimate of U(t). By writing U(t) B(X) M e bt + M A B(X) e b(t s) U(s) B(X) s an then applying the Gronwall lemma to the function u(t) := U(t) B(X) e bt, we inee obtain the announce growth rate. The very same kin of arguments tells us that there exists a unique solution g C(R + ; X) to the functional equation (5.4) g(t) = S B (t)g + (S B Ag)(t) in X, t, an g(t) = U(t)g. Step 2. Weak solution. We claim that U (t) = BS B (t) + B S B (t s)au(s) s + S B ()AU(t) = (B + A) U(t), on a weak sense, what we obtain by repeating the computation we have performe just after the proof of Lemma 3.2 about the Duhamel formula. More precisely, we fix g X, ϕ = ϕ t C 1 (R + ; X) C(R + ; D(B )), an enoting g t := U t g, St := S B (t), we efine We clearly have λ C 1 (R + ) an λ(t) = ϕ t, g t = S t ϕ t, g + λ (t) = S t B ϕ t + S t ϕ t, g + ϕ t, Ag t + = = ( B ϕ t + ϕ t, S B (t) + Λ ϕ t + ϕ t, g t. S t sϕ t, Ag s s. S t sb ϕ t + S t sϕ t, Ag s s. S t s AU s ) g + A ϕ t, g t

11 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 11 We conclue by writing ϕ t, g t ϕ, g = λ (s) s, an observing that this nothing but the weak formulation of the evolution equation associate to Λ. We use here that D(Λ) = D(B) because Λ : D(B) X is a boune operator an thus D(Λ ) = D(B ). Step 3. Semigroup property. We claim that Ug Y T := C 1 ([, T ]; X) for any g D(Λ) an T >. The proof follows by aapting the construction we have mae in Step 1. For f Y T an g D(Λ), we enote g t := ϕ t [g, f] = S B (t)g + (S B Af)(t), an we observe that 1 h (g t+h g t ) = 1 h [S B(t + h)g S B (t)g ] + 1 h + 1 h S B (s)a[f t+h s f t s ] s S B (t)bg + S B (t)ag + +h t S B (s)af t+h s s S B (s)af t s s =: g t, as h, where the limit term belongs to C([, T ]; X). We introuce the norms f YT := sup t [,T ] { ft X + f t } X, g D(Λ) := g X + Λg X on Y T an D(Λ). From the computations mae in Step 1 an the one mae just above, we see that g YT Me bt { g X + A T f E } + Me bt { Λg X + A T f E } Me bt ( g D(Λ) + A T f YT ), an for two images g i := Φ(U)f i, f i X T, i = 1, 2, that g 2 g 1 YT A MT e bt f 2 f 1 YT. We straightforwaly aapt the proof presente in Step 1 an we obtain the existence of a fix point g = ϕ[g, g] Y T. This one satisfies the functional equation (5.4) an that proves the claim Ug = g Y T. Also observing that 1 h (S B(h)g t g t ) = 1 h (g t+h g t ) 1 h g t S B (t)ag, +h t S B (s)af t+h s s as h, we have Ug C([, T ]; D(B)). Performing the same analyse for the ual operator, we get that for any ψ D(Λ ), there exists a solution ϕ C 1 ([, T ]; X ) C([, T ]; D(Λ )) to the (backwar) ual problem (4.3). Arguing as in the proof of Proposition 4.3, we euce that g := Ug C(R + ; X), g X, is the unique solution to the weak equation ϕ t, g t ϕ, g = Λ ϕ s + ϕ s, g s s, for any ϕ C 1 (R +, X ) C(R +, D(Λ )). We conclue that U satisfies the semigroup property by using Proposition Semilinear evolution equation. With very similar arguments as in the previous section, we present a possible extension of the existence theory for semilinear evolution equation. We consier the generator Λ of a semigroup of contractions S Λ an a function Q : X X which is Lipschitz continuous on boune sets: for any R > there exists C such that (5.1) f, g B(, R) Q(f) Q(g) C f g. We efine C(R) := inf{c >, such that (5.1) hols}. The mapping R C(R) is increasing. We consier the semilinear equation (5.2) t f = Λf + Q(f), f() = f

12 12 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP in classical form with f C([, T ]; D(Λ)) C 1 ([, T ]; X) an f D(Λ), as well as the associate mil formulation (5.3) f(t) = S Λ (t)f + with f C([, T ]; X) an f X. S Λ (t s)q(f(s)) s Proposition 5.1 (uniqueness). If f, g C([, T ]; X) are two solutions of (5.3) associate to the same initial atum f X, then f = g. Proof of Proposition 5.1. For any given T >, we efine R := max max( f(t), g(t) ), t T an we get that h := f g satisfies h(t) e C(R)T h() thanks to the Gronwall lemma. Proposition 5.2 (local existence). For R >, we efine M := 2R + Q(), T R := M R Q() + MC(M). For any f B(, R), there exists a unique function f C([, T R ]; X) solution to (5.3). Proof of Proposition 5.2. For any f E, we efine We efine so that Φ : E E. Inee, we observe that E := {f C([, T R ]; X); f(t) M, t [, T R ]}. Φ(f)(t) R + Φ(f) := S Λ f + S Λ Q(f), Q(f(τ)) τ R + T R [Q() + C(M)M] M, for any t T R. On the other han, for any f, g E an any we have t T R := (Φ(f) Φ(g))(t) M R Q() + MC(M) < 1 C(M), Q(f(τ)) Q(g(τ)) τ T R C(M) max f g. [,T R ] Thanks to the Banach fixe point theorem for contractions, we conclue to the existence of a unique fixe point for the function Φ, an that provies a solution to the semilinear equation (5.3). We set T max (f ) := sup{t > ; f C([, T ]; X) solution to (5.3)}. Theorem 5.3 (maximal solution). For any given f X, there exists a unique maximal solution f C([, T max (f )); X) to (5.3), for which the following alternative hols: (i) T max (f ) = +, we say that f is a global solution; (ii) T max (f ) < an f(t) as t T max (f ), we say that f blows up in finite time. Proof of Theorem 5.3. ( (5.4) 1 + f(t) Q() + f(t) The result is a straightforwar consequence of the estimate )( ) 1 + C( Q() + 2 f(t) ) 1 T max (f ) t, t [, T max(f )), that we prove now. We may assume T max (f ) <, an we assume by contraiction that there exists t [, T max (f )) such that ( )( ) C( Q() + 2 f(t ) ) < f(t ) Q() + f(t ) 1 T max (f ) t.

13 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 13 We efine R := f(t ), M := Q() + 2R, so that the above assumption writes M R T max (f ) t < M(1 + C(M)). On the other han, we have T R := M R Q() + MC(M) > M R M(1 + C(M)). Thanks to Proposition 5.2, there exists a unique g C([, T R ]; X) which satisfies g(t) = S Λ (t)f(t ) + S Λ (t τ)q(g(τ)) τ. We then set h(t) := f(t), t [, t ], h(t) := g(t t ), t [t, t + T R ], an we observe that h is a solution to (5.3) on the interval [, t + T R ], with t + T R > T max (f ), what is not possible. A straightforwar application of Theorem 5.3 is the following global existence result. Proposition 5.4 (global existence). Any solution is global when Q is globally Lipschitz. Exercise 5.5. Exten all the above results to the case when S Λ is a general (not necessarily of contractions) C -semigroup Dissipativity an extension trick. For f X, we efine its ual set F (f) := { f X, f, f = f 2 X = f 2 X }. That set is never empty thanks to the Hahn-Banach theorem (Exercise). Observe that when X is an Hilbert space F (f) = {f} an when X = L p, 1 p <, F (f) = {f f p 2 f 2 p L }. For a p general Banach space, one can show that F (f) thanks to the Hahn-Banach Theorem. We say that an (unboune) operator Λ is issipative if f D(Λ), f F (f), f, Λf. When X is an Hilbert space the issipativity conition writes (5.5) f D(Λ), (f, Λf), an we also say that Λ is coercive. We say that a Banach space X is regular if F (f) is a singleton {f }, f X, for any f X, the mapping ϕ : X R, f ϕ(f) := f 2 /2 is ifferentiable an Dϕ(f) h = f, h, f, h X. Examples of regular spaces are the Hilbert spaces an the Lebesgues spaces L p, 1 < p <. Theorem 5.6. Consier a Banach space X an an unboune opeartor L on X. We assume that (i) X is a regular space an L is issipative; (ii) there exists a ense Banach space X X an an operator L on X such that L = L X an L is the generator of a semigroup S L on X. Then L is the generator of a semigroup (of contractions) S L on X such that S L X = S L. Proof of Theorem 5.6. For any f D(L), the mapping t f t := S L (t)f is C 1 on X, so that also is the mapping t f t 2 X. By the chain rule an the issipativity property, we fin 1 2 t f t 2 X = Dϕ(f t ) t f t = ft, Lf t. Thanks to the Gronwall lemma, we euce S L (t)f X f X, t. We conclue by a extension argument using the above uniform continuity estimate an the ensity property D(L) X.

14 14 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP We often just say that L is a issipative operator in a (complex) Hilbert space H when there exists a real number b R such that (5.6) f D(L), f F (f), Re f, Lf b f 2. For any f D(L) an enoting f t := S t f, we have from (5.6) Thanks to the Gronwall lemma, we euce t f t 2 H := t ( f t, f t ) H = 2Re f t, Lf t 2b f t 2. (5.7) S t f e bt f t. It is then classical to show that the issipative growth rate ω (L) satisfies (5.8) ω (L) := inf{b R; (5.7) hols} = inf{b R; (5.6) hols}. 6. Semigroup Hille-Yosia-Lumer-Phillips existence theory We say that an (unboune) operator Λ is maximal if there exists x > such that (6.9) R(x Λ) = X. We say that Λ is m-issipative if Λ is issipative an maximal. We present now the Lumer-Phillips version of the Hille-Yosia Theorem which establishes the link between semigroup of contractions an issipative operator. Theorem 6.7 (Hille-Yosia, Lumer-Phillips). Consier Λ C D (X). The two following assertions are equivalent: (a) Λ is the generator of a semigroup of contractions; (b) Λ is issipative an maximal. For the sake of brevity an simplicity, we only present the proof of the implication (b) (a) in the case of an Hilbert framework. The reverse implication is let as an exercise. Exercise 6.8. Consier an (unboune) operator Λ on a Hilbert space X which is issipative an maximal. Prove that Λ C D (X). Prove the same result in the case when X is reflexive. Exercise 6.9. Consier a semigroup S = S Λ on an regular Banach space X (as efine in section 5.3). Prove that the generator Λ is issipative iff S is a semigroup of contractions. (Hint. Argue similarly as in the proof of Theorem 5.6). Exercise 6.1. Consier a semigroup S = S Λ on a Banach space X which satisfies the growth boun (1.1). Prove that Λ z is invertible for any z b := {z C; Re z > b}. (Hint. Prove that the operator U(z) := is well efine in B(X) for any z b an that S Λ (t) e zt t U(z) (Λ z) = I D(Λ), (Λ z) U(z) = I X ). Exercise Prove (a) (b) in Theorem 6.7. (Hint. Use Excercise 6.9 an Excercise 6.1). Lemma Uner conition (b), the operator Λ satisfies (6.1) ε >, R(I ελ) = X. Moreover, for any ε >, I ελ is invertible from D(Λ) into X, an (6.11) (I ελ) 1 B(X) 1 Finally, D(Λ n ) is ense in D(Λ n 1 ) for any n 1.

15 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 15 Proof of Lemma Step 1. We know that (6.1) hols with ε = ε := 1/x an we prove that it also hols for any ε > ε /2. First, we observe that for any g X there exists f D(Λ) such that f ε Λf = g. That solution f D(Λ) is unique because for any other solution h D(Λ), the ifference u := h f D(Λ) satisfies u ε Λu =, so that u 2 u, u ε u, Λu =, an u =. In other wors, I ε Λ is invertible. Moreover, we also have f 2 f, f ε f, Λf = f, g f g, so that f g. In other wors, (I ε Λ) 1 B(X) an (I ε Λ) 1 B(X) 1, which is nothing but (6.11) for ε = ε. Next, for a given ε > an a given g X, we want to solve the equation f D(Λ), f ελf = g. We write that equation as f ε Λf = (1 ε ε ) f + ε ε g, an then [ f = Φ(f) := (I ε Λ) 1 (1 ε ε ) f + ε ] ε g. Finally, when 1 ε /ε < 1, which means ε > ε /2, we euce from the Banach contraction fixe point Theorem that there exists a unique f X such that f = Φ(f) D(Λ). That conclues the proof of (6.1) for any ε > ε /2. Repeating the argument, we then get (6.1) an (6.11). Step 2. We alreay know (it is one of our assumptions) that D(Λ) is ense in X. We efine V ε := (1 ελ) 1, an for f X, we efine f ε = V ε f. We claim that (6.12) f ε f in X, as ε. First, we observe that V ε I = V ε (I (I ελ)) = ε V ε Λ. For any f X, we may introuce a sequence f n f in X such that f n D(Λ). We then have an f ε f = V ε f V ε f n + V ε f n f n + f n f, f ε f V ε f V ε f n + ε V ε Λf n + f n f 2 f n f + ε Λf n, from what (6.12) immeeiately follows. Consier f D(Λ) an f ε := V ε f D(Λ). From (I + ελ)f ε = f, we euce that ελf ε = f f ε D(Λ), which means that f ε D(Λ 2 ). Moreover, we have f ε f in X as well as f ε f an Λf ε = V ε Λf Λf in X as ε thanks to (6.12). We have prove that D(Λ 2 ) is ense in D(Λ). We prove that D(Λ n ) is ense in D(Λ n 1 ) for any n 2 in a similar way. Proof of (b) (a) in Theorem 6.7. Step 1. For fixe ε >, we may buil by inuction an thanks to Lemma 6.12, the sequence (g k ) k 1 in D(Λ) efine by the family of equations (6.13) k g k+1 g k = Λ g k+1. ε Observe that from the ientity we euce (g k+1, g k+1 ) ε Λg k+1, g k+1 = (g k, g k+1 ), g k g k.

16 16 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP We fix T >, n N an we efine an g ε (t) := t k+1 t ε The previous estimate writes then ε := T/n, t k = k ε, g ε (t) := g k on [t k, t k+1 ). g k + t t k ε (6.14) sup g ε g, [,T ] g k+1 on [t k, t k+1 ). sup g ε g. [,T ] Step 2. We next establish that g ε is equi-uniformly continuous in C([, T ); X) when g D(Λ). With the above notation, we write Observing that g k = (1 ελ) 1 g k 1 = V k ε g, V ε := (1 ελ) 1. k 1 ε I = (V ε I) Vε l, V k V ε commutes with Λ an V ε 1, we get l= V ε I = V ε ελ, k 1 g k g V ε l (V ε I)g k ε Λg. l= We see then that g ε (t) g t Λg for any t, ε >, an by construction, we also have g ε (t) g ε (s) (t s) Λg for any t > s > an ε >. Step 3. We finally improve the boun (6.14) by showing that g ε is a Cauchy sequence in C([, T ); X) for any T >, when ε := 2 n an g D(Λ 2 ). We fix t (, T ) yaic, that means t2 nt N for some n t N, an for any n n t, we write h n := g 2 n(t) = V 2n t n g, V n := (1 2 n Λ) 1 an h n+1 := g 2 n+1(t) = U 2n t n g, U n := (1 2 n 1 Λ) 2. Now, we observe that U 2n t n 2 n t 1 V 2n t n = (U n V n ) U 2n l n Vn l l= 2 n t 1 = U n [(1 2 n 1 Λ) 2 (1 2 n Λ)] V n U 2n l n Vn l 2 n t 1 = 2 2n 2 U 2n l+1 n Vn l+1 Λ 2, so that h n+1 h n 2 n 2 t Λ 2 g. As a consequence, for any m > n n t, we have l= h m h n 2 n 1 t Λ 2 g, an (h n ) is a Cauchy sequence in X. Thanks to Step 2, we conclue that g 2 n is a Cauchy sequence in C([, T ); X) for any T >. Step 4. Consier now a test function ϕ Cc 1 ([, T ); D(Λ )) an efine ϕ k := ϕ(t k ), so that ϕ n = ϕ(t ) =. Multiplying the equation (6.13) by ϕ k an summing up from k = to k = n, we get n n (ϕ, g ) ϕ k ϕ k 1, g k = ε Λg k+1, ϕ k. k=1 Introucing the two functions ϕ ε, ϕ ε : [, T ) X efine by ϕ ε (t) := ϕ k 1 an ϕ ε (t) := t k+1 t ϕ k + t t k ϕ k+1 ε ε for t [t k, t k+1 ), k= l=

17 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 17 in such a way that ϕ ε(t) = ϕ k+1 ϕ k ε the above equation also writes (6.15) ϕ(), g ε ϕ ε, g ε t = for t (t k, t k+1 ), Λg ε, ϕ ε t. On the one han, from Step 3, we know that there exists g C([, T ]; X), for any T >, such that g ε g in C([, T ]; X) an we then euce g ε g in L (, T ; X). On the other han, from the above construction, we have ϕ ε ϕ an ϕ ε ϕ both strongly in L (, T ; X ). We may then pass to the limit as ε in (6.15) an we get that (6.16) g, ϕ() + ϕ (s) + Λ ϕ(s), g(s) X,X s =. Step 5. All together, for g D(Λ 2 ), we have prove that there exists a function g C([, ); X) which satisfies the evolution equation in the weak form (6.16) an g(t) X g X for any t. Repeating the same argument as in steps 1, 2 an 3, we fin Λg C([, ); X) an Λg(t) X Λg X, at least when g D(Λ 3 ). By a ensity argument an using the two above contraction estimates, we get that the same hols for any g D(Λ). From (6.16) an that regularity estimate, we get g C 1 ([, ); X) an thus g is a classical solution to the evolution equation (1.3). In a Hilbert space, we have the uniquness of solution in such a class of functions by proving that g = implies g thanks to a stanar Gronwall argument. Inee, if g C 1 ([, ); X) C([, ); D(Λ)) satisfies (6.16) for g an thus (1.3), we compute t g(t) 2 = 2(Λg, g), g() 2 =, so that g =. In a general Banach space, we have the same existence theory for the backwar ual problem for any ϕ T D(Λ ). As a consequence, if ϕ is such a backwar solution associate to ϕ T, we have ϕ T, g(t ) = = ϕ(s), g(s) s + ϕ(), g() s [ Λ ϕ(s), g(s) + Λ ϕ(s), g(s) ] s =, an we conclue again that g(t ) = because ϕ T is arbitrary an D(Λ ) is ense (for the weak topology) into X. We conclue thanks to Proposition Complements 7.1. Continuity. For the sake of completeness an in orer to make the chapter as self-containe as possible, we establish the results we let as exercises in Excercise 1.4. Proposition 7.1. Let (S t ) satisfy (i), (ii ) an (iii) in Definition 1.1 an Remark 1.3. Then (S t ) also satisfies the growth estimate (iv) in Definition 1.1. Proof of Proposition 7.1. We first claim that δ >, C 1, such that S(t) C t [, δ]. On the contrary, there exists a sequence (t n) such that t n an S(t n). On the other han, we know that S(t n)f f for any f X which implies sup S(t n)f < for any f X (that is a consequence of the Banach- Steinhaus Theorem for a weak continuous semigroup). Using the Banach-Steinhaus Theorem (again) that implies sup S(t n) < an a contraiction. We then obtain the growth estimate (1.1) with M := C an b := (log C)/δ thanks to an eucliian ivision argument. Proposition 7.2. Let (S t ) satisfy (i), (ii ) an (iii) in Definition 1.1 an Remark 1.3. Then (S t ) also satisfies the continuity of trajectories conition (ii) in Definition 1.1.

18 18 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP Proof of Proposition 7.2. when h >, For t >, we write S(t + h) S(t) = S(t)(S(h) I) S(t + h) S(t) = S(t + h)(i S( h)) when h <. We conclue using the conition (ii ) together with the growth estimate (iv) establishe in Proposition 7.1. Proposition 7.3. For a given semigroup (S t ) on X, the new norm efine in (1.2) is equivalent to the initial norm an the new semigroup (T t ) efine in (1.2) is a semigroup of contractions for that new norm. Proof of Proposition 7.3. Moreover, for any t, The two norms are equivalent because f = T ()f f = sup e ωt S(t) f M f. t T (t)f = sup T (s) T (t)f s = sup e ω(s+t) S(t + s)f s sup e ωτ S(τ)f = f, τ which proves that T (t) is a semigroup of contractions for that norm. Theorem 7.4. Let (S t ) be a semigroup in the sense of Definition 1.1 in which conitions (i) an (ii) are unerstoo in the sense of the weak topology σ(x, X ). Then (S t ) is a strongly continuous semigroup. Proof of Theorem 7.4. For any fixe f X, we efine (7.1) f ε := 1 ε S tft. ε Using the growth estimate, we then compute S h f ε f ε 1 ε ε = sup S t+h f, ϕ t S tf, ϕ t ϕ 1 ε 1 ε+h h = sup S tf, ϕ t S tf, ϕ t ϕ 1 ε ε as h. We efine now h ε f 2Me b (ε+h), X := {f X; S tf f in norm as t }. It is a norm close linear subspace so that it is weakly close. Because f ε f an f ε X for any f X, it is also weakly ense. That proves that X = X Nonautonomous semigroup. We briefly present a possible extension of the semigroup theory to the nautonomous evolution equation framework. Definition 7.5. Let fix T (, ). We say that a two parameters family (U t,s ) T t s is a continuous nonautonomous semigroup of linear an boune operators on X (or we just say a nonautonomous semigroup on X), if the following conitions are fulfille: (i) family of operators: s, t, f U t,s f is linear an continuous on X; (ii) continuity of trajectories: f X, {(t, s); s t} (t, s) U t,s f is continuous; (iii) semigroup property: t r s, U s,s = I an U t,r U r,s = U t,s ; (iv) growth estimate: b R, M 1, (7.2) U t,s B(X) M e b(t s) t s. of (un- To a nonautonomous semigroup, we may associate a one parameter family (Λ(t)) t T boune) operators on X an the forwar nonautonomous evolution equation (7.3) in the following way. t f = Λ(t)f on (, T ), f() = f,

19 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP 19 Theorem 7.6. Consier a nonautonomous semigroup (U t,s ) T t s on a Banach space X. For any given t [, T ), we efine the (linear unboune) operator Λ(t) by D(Λ(t)) := { U t+h,t f f f X; lim exists in X }, h h U t+h,t f f Λ(t) f := lim for any f D(Λ(t)). h h Assume that there exists X 1 X ense such that U t,s is a nonautonomous semigroup on X 1 an X 1 D(Λ(t)) for any t [, T ). Then (7.4) t U t,s f = Λ(t) U t,s f f X 1, s t T ; (7.5) s U t,s f = U t,s Λ(s)f f X 1, s t T. In particular, for any f X 1, the function t f(t) := U t, f provies a solution to the evolution equation (7.3). Proof of Theorem 7.6. t U 1 t,s = lim h h s U 1 t,s = lim h h We (at least formally) compute ( ) 1 ( Ut+h,s U t,s = lim Ut+h,t I ) U t,s = Λ(t) U t,s, h h ( ) 1 Ut,s U t,s+h = lim U t,s+h h h ( Us+h,s I ) = U t,s Λ(t), an we observe that these limits can be easily rigorously justify in the space B(X 1, X). Finally, efining f(t) := U t, f for any f X 1, we obserse that (7.3) immeiatley follows from (7.4). Corollary 7.7. Uner the asumptions of Theorem 7.6 an for any G C([, T ]; X), any solution to the non homogeneous equation satisfies t f = Λ(t)f + G on (, T ), f() = f, f(t) = U t, f + U t,s G(s) s. In the other way roun, by the mean of the J.-L. Lions theory or the characteristics metho one can show that we can associate a nonautonomous semigroup U t,s which provies solutions to the forwar nonautonomous evolution equation. That is let as an exercise an we refer to chapters 1 an 2 for etails. We also present a result (without proof) which is more in the spirit of the Hille-Yosia semigroup theory. Theorem 7.8. Consier a one parameter family (Λ(τ)) τ T of (unboune) operators on a Banach space X. We assume (1) Λ(τ) generates a semigroup S Λ(τ) on X for any τ [, T ] with growth boun inepenent of τ. (2) Λ(τ) generates a semigroup on another Banach space X 1 X for any τ [, T ], with growth boun inepenent of τ [, T ]. (3) The mapping [, T ] B(X 1, X), τ Λ(τ), is continuous an X 1 X is ense. Then there exists a unique nonautonomous semigroup (U t,s ) T t s in X with infinitesimal generators Λ(t) t T. Exercise 7.9. Establish Theorem 7.8 by first assuming that Λ(τ) is a piecewise constant family an next approximating a general family Λ(τ) by a sequence of piecewise constant families. Theorem 7.1. Consier a one parameter family (Λ(τ)) τ T of (unboune) operators on X = Y for a Banach space Y. We assume that there exists Y 1 Y ense an U t,s a nonautonomous semigroup on Y such that t U t,s = Ut,sΛ(t), on {t > s} B(Y 1, Y ), s U t,s = Λ(s) Ut,s, on {t > s} B(Y 1, Y ).

20 2 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP Then U t,s is a nonautonomous semigroup on X which satisfies (7.4) an (7.5) in the weak sense. It provies the unique weak solution to the equation to the evolution equation (7.3) for any f X. Proof of Theorem 7.1. an For f X an ϕ Y 1, we write t U t,s f, ϕ := f, t U t,sϕ = f, U t,sλ(t) ϕ s U t,s f, ϕ := f, s U t,sϕ = f, Λ(s) U t,sϕ, in which we recognize a weak formulation of equations (7.4) an (7.5). For f X, we efine f(t) := U t, f, an more precisely, we efine by uality From the first above ientity, we have f(t), ϕ := f, U t,ϕ, ϕ Y. t f(t), ϕ = f, U t,λ(t) ϕ = f(t), Λ(t) ϕ on (, T ), for any ϕ Y 1, which is a weak formulation of the evolution equation (7.3). On the other han, in orer to prove the uniqueness of the solution, we consier a weak solution f(t) to the evolution equation (7.3) associate to the initial atum f =. For any τ (, T ) an ϕ τ Y 1, we efine ϕ(s) := U τ,sϕ τ on [, τ], so that an ϕ(τ) = ϕ τ. We then compute As a consequence s ϕ(s) = s U τ,sϕ τ = Λ(s) U τ,sϕ τ = Λ(s) ϕ(s) on (, τ) t f(t), ϕ(t) = t f(t), ϕ(t) + f(t), t ϕ(t) = Λ(t)f(t), ϕ(t) + f(t), Λ(t) ϕ(t) =. f(τ), ϕ τ = f, ϕ = for any ϕ τ Y 1 an that implies f(τ) = for any τ (, T ) Transport equation in measures an L frameworks. We consier the transport equation (7.6) t f = Λf = a(t, x) x f c(t, x)f, with smooth cofficients a an c. Denoting by Φ t,s the characterics of the associate ODE, namely for any x s R, x(t) := Φ t,s x s is the solution to t x = a(t, x), x(s) = x s, we observe that any smooth solution f to (7.6) satisfies [ f(t, Φt,s ) e s c(τ,φτ,s(x))τ ] =. t As a consequence, for any smooth function f s, the function f(t, x) := f s (Φ 1 t,s ) e s c(τ,φ 1 τ,s (x))τ is the unique solution to the equation (7.6) corresponing to the initial conition f(s, x) = f s (x). We now consier the ual equation (7.7) t ϕ = Λ ϕ := a x ϕ + (iv a c)ϕ. That last equation generates a strongly continuous nonautonmous semigroup V t,s in C (R ) an in L 1 (R ) that one can buil by the above characteristics metho in Cc 1 (R ) an next by a ensity an continuity argument in L p (R ), p =, 1. Finally, by setting U t,s := Vt,s an using Theorem 7.1, we buil a weakly continuous nonautonmous semigroup in M 1 (R ) an in L (R ) which is associate to the initial transport equation (7.6).

21 CHAPTER 3 - EVOLUTION EQUATION AND SEMIGROUP Discussion 8.1. Several way to buil solutions. In the previous chapters, we have seen two ways to buil a solution to an evolution equation associate to an abstract or a PDE operator. More precisely, we have built (1) variational solutions for coercive operator, (2) weak (an in fact renormalize) solutions for transport operator. (3) There exist other classical ways to buil solutions in some particular situations. On the one han, we may use some explicit representation formula exactly as we i to solve the transport equation thanks to the characteristics metho. The most famous example concerns the Laplacian operator an the associate heat equation which can be solve in the all space by introucing the heat kernel. More precisly, one may observe that S(t)f := γ t f, γ t(x) = 1 t /2 γ ( x t ), γ(x) = 1 (2π) /2 exp( x 2 /2), which is meaningfull for f L 1 (R ) + L (R ), efines a semigroup (for instance in X = L p (R ), 1 p, or X = C (R )) an a solution to the heat equation That is immeiate from the explicit formulas an tf = 1 2 f in (, ) R, f(, ) = f in R. tγ t(z) = /2 t γ t(z) + z 2 2t 2 γt(z) zγ t(z) = z t γt(z), zγt(z) = z 2 γt(z) + t t γt(z). 2 On the other han, in some situation, we may buil (a bit less explicit) representation formula by introucing convenient basis. To give an example, we consier the operator Λ = in the space X = C([, 2π]). We observe that ϕ k (x) := e i kx is an eigenfunction associate to the eigenvalues problem ϕ k = k 2 e i kx = λ k ϕ k, ϕ k () = ϕ k (2π). We then efine the semigroup S(t) : C per([, 2π]) C per([, 2π]) which to any function f(x) = k Z c k ϕ k (x) associate the function (S tf)(x) := k Z c k e λ kt ϕ k (x), an we easily verify that S tf gives a solution to the heat equation in [, 2π] (with perioic bounary conitions). The same spectral ecomposition metho can be generalize to the case where Λ = (or even Λ is a general parabolic operator) is pose in a (smooth) boune open omain with Dirichlet, Neumann or Robin conitions at the bounary. Similarly, when (Λf)(x) := b(y, x) f(y) y Ω with b L 2 (Ω Ω), then Λ is an Hilbert-Schmit operator in L 2 (Ω). That means that the exists a Hilbert basis (φ k ) of L 2 (Ω) mae of eigenfunctions of Λ. As in the first example, in orer to solve the evolution equation associate to Λ, one just has to write the Hilbert expansion series of the initial atum on the basis (φ k ) an to solve (straightforwarly) the evolution equation for each coorinate. (4) In this chapter 3, we have presente several simple but powerful tricks, as extension arguments an uality arguments, in orer to buil a semigroup from one other. (5) Finally, we have briefly presente the Hille-Yosia theory (or more precisely its Lumer-Phillips version) which provies a clear link between the semigroup theory an its abstract evolution equation counterpart From Hille-Yosia theory to variational solutions. The Hille-Yosia-Lumer-Phillips Theorem 6.7 can be seen as a generalisation of the J.-L. Lions theorem presente in chapter 1, in the sense we explain now. We consier a Hilbert space H an an operator Λ : D(Λ) H H such that the Hille-Yosia theory applies: for any g D(Λ), there exists g C([, ); D(Λ)) C 1 ([, ); H) such that g = Λg in H. t We moreover assume that there exists a Hilbert space V such D(Λ) V H an realizes the hypothesizes (i) an (ii) of Theorem I.3.2. We claim that for any g H, the semigroup solution g(t) := e Λ t g, given by the Hille-Yosia theory (an thus obtaine by a uniform continuity principle from the solutions