NOTES ON LINEAR SEMIGROUPS AND GRADIENT FLOWS

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1 NOTES ON LINEAR SEMIGROUPS AND GRADIENT FLOWS F. MAGGI Tese notes ave been written in occasion of te course Partial Differential Equations II eld by te autor at te University of Texas at Austin. Tey are mostly based on Evans s PDE book, and on Brezis Functional Analysis book. Wit respect to tese classical references, te autor as tried to supply additional insigts and explanations meant to facilitate te student s understanding. 1. Linear semigroups Te idea explored ere is te possibility of seeing te Caucy problem for a linear PDE like te eat equation, u t u = on (, ), u t= = u on, (1.1) u = t. as a linear ODE on some infinite dimensional normed vector space X, U (t) = AU(t), t, U() = U. (1.2) In tis ODE, A denotes a linear operator from X to X, and we ave a continuous function U : [, T ] X suc tat te limit in X of te incremental ratios 1 (U(t + ) U(t)) as exists for every t, and is denoted by U (t). Te interpretation of (2.1) as an ODE is obtained by identifying u(x, t) wit U(t) = u(, t) X, were X is a space of functions of te space variable x R n. Te linear operator A will of course consist in taking te spatial Laplacian of U X. We sall set X = L 2 (). Te domain of A = will ten be te subspace D(A) = H 1 () H2 () of X. So typically A will not be defined on te wole X, but just on a dense subspace, and A will be unbounded in te norm of X Linear ODE in Banac spaces. Let us consider te general linear ODE (1.2) in a given normed vector space X, and wit A L(X, X), tat is A is a linear bounded operator from X to X. If X is a Banac space ten we can solve (1.2) by te exponential construction. Teorem 1.1. If X is a Banac space and A L(X, X), ten for every t R te limit exists in L(X; X) and is denoted by lim Id + N t k A k N k! k=1 e t A. It commutes wit A, it is equal to Id for t =, and as te property tat U(t) = e t A U solves (1.2), tat is d dt et A U = A e t A U = e t A AU U X. (1.3) 1

2 Proof. Fix t R, and denote by S N (t) te sequence of operators in te statement. For N < M we ave, in te norm of L(X, X), S N (t) S M (t) M k=n t k A k k! as N, M. If X is a Banac space ten so is L(X, X), tus te limit S(t) = e t A exists in L(X, X). Given U X, t E N (t)u is a smoot function, and as N, E N (t)u S(t)U locally uniformly in t. Te derivatives of every order of E N (t)u also converge locally uniformly, so tat S(t)U is smoot in t, and its derivatives are easily computed. For example d dt E N(t) = N k=1 implies tat (d/dt)s(t)u = AS(t)U, as claimed. t k 1 A k (k 1)! = AE N 1(t) 1.2. Semigroups arise from generators. We single out te following properties of te map S : [, ) L(X, X), S(t) = e t A, constructed in Teorem 1.1 starting from A L(X, X): (i) S() = Id ; (ii) S(t + s) = S(t)S(s) = S(s)S(t) for every t, s ; (iii) S( )U is continuous on [, ); (iv) tere exists ω R suc tat S(t) e ω t for every t. In general, a family of linear operators S(t) t L(X, X) is an ω-contractive semigroup on X is properties (i) (iv) old. One may naively ask if every semigroup arises from an operator A L(X, X) by means of te exponential construction. Tis is morally true, but it cannot be exactly true, as someting like te eat flow on L 2 () as to arise from an operator like A =, wic is defined as a linear map taking values in L 2 () only if we restrict its domain to H 2 (), wic is a proper dense subspace of L 2 (). But wit tis caveat in mind, namely tat te generating operator A does not need to be bounded on X, and may be actually defined only on a dense subspace of X, it is essentially true tat every semigroup is generated by a linear operator defined on a dense subspace. Teorem 1.2. Let S(t) t be an ω-contractive semigroup. Consider te linear subspace D of X and te linear operator A : D X defined by D = U X : te limit lim S(t)U U t + t exists in X and set D = D(A). Ten AU = lim t + S(t)U U t U D, (i) D(A) is dense in X and A is closed (tat is, u k k N D, u k u and Au k v imply u D(A) and v = Au); (ii) for every U D(A) and t >, S(t)U D(A), and t S(t)U is differentiable on (, ), wit d S(t)U = AS(t)U = S(t)AU. (1.4) dt Proof. Proof of (ii): We sow tat S(t)U D(A) for eac U D(A) and t >. Indeed S()S(t)U S(t)U = S(t) S()U U S(t)AU as + 2

3 as (1/)(S()U U) AU for +, and since S(t) L(X, X). Tis sows tat S(t)U D(A) wit AS(t)U = S(t)AU. Concerning te differentiability of S(t)U in t for t >, we notice tat S(t + )U S(t)U = S(t) S()U U To compute te limit as set k =, ten S(t + )U S(t)U S(t) AU wen +. S(t)U S(t k)u = = S(t k) S(k)U U ( k ) k S(k)U U = S(t k) AU + S(t k)au. k Considering tat S(t k) e ω t for every k > and tat S(t k) S(t) in L(X, X) as k, we find tat as tat is (1.4). Proof of (i): Let U X, and set S(t + )U S(t)U lim U (t) = 1 t t = S(t)AU, S(s)U ds, t >, so tat, as t +, by continuity of S(s)U U for s [, ) and by S() = Id, U (t) U 1 t t S(s)U U ds. We claim tat U (t) D(A): indeed, S()U (t) U (t) = 1 t t = 1 t+ S(s)U ds t = 1 t+ S(s)U ds t t t S( + s)u ds t S(s)U ds S(s)U ds S(s)U ds so tat as +, S()U (t) U (t) S(t)U U. t Tis sows tat U (t) D(A), tus tat D(A) is dense in X. Now consider u k k N D(A) wit u k u and Au k v in X. Integrating (1.4) we find tat S()u k u k = A S(s) u k ds = S(s) A u k ds. Now S(s)Au k S(s)v uniformly as k, wile S()u k S()u, so tat we find S()u u = 1 S(s) v ds. Te limit as + of te left-and side exists, and it is equal to S()v = v, tat is u D(A) and Au = v. Tis proves tat A is a closed operator. 3

4 1.3. From generators to semigroups, general strategy. We now discuss te converse construction to te one of Teorem 1.2, namely, given a closed operator A densely defined on X we wis to construct a semigroup aving A as its generator, and tus solving te ODE (1.2). Notice tat wen D(A) = X, closedness implies boundedness, i.e. A L(X, X), and tus te exponential construction of Teorem 1.1 applies. Tus we are really concerned wit te case wen D(A) is a proper dense subspace of X. Te idea is approximating A wit operators A λ L(X, X) and considering te semigroups S λ (t) = e t A λ defined from A λ by means of te exponential construction (Teorem 1.1). Ten te semigroup S generated by A will be obtained as te limit of te approximating semigroups S λ. Te construction of te approximating operators will be as follows. Having in mind te case wen A = and D(A) = H 1() H2 (), we expect tat for µ > sufficiently small, te linear operator Id µa will be bounded and invertible from D(A) to X. Indeed tere exists no u H 1 () \ wit u = µ u and µ >, as a reflection of te fact tat is positive definite on H 1 () (Poincaré inequality). Tus, for µ small, we sall be able to apply te exponential construction to (Id µa) 1 L(X, X). (1.5) An even better idea is applying te exponential construction to (Id µa) 1 Id µ Indeed a formal expansion in µ + gives (Id µa) 1 Id µ L(X, X). (1.6) = A + O(µ) so tat we expect (1/µ) [(Id µa) 1 Id ] to converge to A as µ +. Te induced exponential semigroups will ten converge to a limit semigroup S(t) aving A as its generator. Tis strategy works and leads to te Hille-Yosida teorem. Te statement uses te following terminology. If A is a linear operator defined from some subspace D(A) of X wit values in X, we denote by ρ(a) = λ R : (λ Id A) is injective and surjective from D(A) to X te resolvent of A. For λ ρ(a), te resolvent operator R λ is defined as R λ = (λ Id A) 1 L(X, X). (1.7) Notice tat (1.5) corresponds to (1.7) wit µ = 1/λ. In particular te limit as µ + will become a limit as λ +, and te approximating operators defined in (1.6) will take te form A λ = λ Id + λ 2 R λ. (1.8) A formal expansion in λ + gives of course A λ = A + O(1/λ). Te operators A λ are called te Hille-Yosida approximating operators of A. For example, let us consider te above definitions in te case of te Laplacian wit omogeneous Diriclet condition on a bounded open set, tat is, wen A =, D(A) = H 1() H2 (), and X = L 2 (). Let < λ k () < λ k+1 () denote te sequence of te eigenvalues of on H 1 (). If λ R is suc tat (λ Id A) is injective, ten tere cannot be u H 1() \ suc tat u = λu, tat is λ λ k() : k 1. Tus if 4

5 λ > λ 1 () we definitely ave tat (λ Id A) 1 is injective. About surjectivity, pick any f L 2 () and consider te minimization of E(u) = u 2 + λ u 2 fu on H 1 (). By te Poincaré inequality, for every ε >, u 2 + λ u 2 fu ε u 2 + ( λ 1 () ε + λ ) u 2 f 2 ε ε u2 so tat if ε is small enoug wit respect to λ 1 () + λ >, te energy E(u) is coercive on H 1() and tere exists u H1 () suc tat (λ Id A) 1 f = u. By te incremental ratios metod we immediately see tat u Hloc 2 (), and if as Lipscitz boundary, tat u H 2 (). Tis sows tat for a bounded open set wit Lipscitz boundary one always as ( λ 1 (), ) ρ( ). In particular, it definitely makes sense to consider te Hille-Yosida approximating operators ( ) λ = λid + λ 2 (λid ) 1 in te limit as λ. It will turn out tat for every λ > λ 1 (), ( ) λ is indeed an element of L(L 2 (), L 2 ()), and tat its exponential map converge to te eat flow. All tese facts can be proved in an abstract framework according to te Hille-Yosida teorem. Teorem 1.3 (Hille-Yosida teorem). Let X be a Banac space and let ω R. Part one: If S(t) t is an ω-contractive semigroup, and A is te generator of S(t), ten (ω, ) ρ(a), R λ 1 λ > ω, λ ω and R λ is te Laplace transform of S(t), R λ U = e λ t S(t) U dt, U X. Part two: If A is a closed operator defined on a dense subspace D(A) of X suc tat (ω, ) ρ(a), R λ 1 λ > ω, (1.9) λ ω ten te Hille-Yosida approximating operators A λ L(X, X) defined by A λ = λ Id + λ 2 R λ λ > ω, are suc tat: (i) for every U D(A), A λ U AU as λ ; (ii) S λ (t) = e t A λ t is a semigroup wit S λ ωλ λ ω, λ > ω (iii) te limit S(t) of S λ (t) as λ exists and defines an ω-contractive semigroup wose generator is A. Solving te eat equation To exemplify let us ceck ow Teorem 1.3 can be applied to construct solutions to te eat equation. Set X = L 2 (), A =, D(A) = H 1() H2 (). If u k u and u k v in L 2 () for some u k k N H 1() H2 (), ten v η = lim η u k = lim u k η = u η η Cc (). k k By te L 2 -regularity teory for we also ave tat u k L 2 () 2 u k L 2 () 5

6 so tat u H 2 () and u k u in H 2 () wit u = v. Tus is closed in L 2 (). We ave already seen te argument sowing tat ( λ 1 (), ) ρ( ). Let us tus consider te map R λ = (λid ) 1. For f L 2 () set u = R λ f, so tat u η λuη = fη η H 1 (). Testing wit η = u gives f L 2 () u L 2 () tat is exactly fu = u 2 + λ u 2 (λ 1 () + λ) u 2 L 2 () R λ f L 2 () = u L 2 () f L 2 () λ + λ 1 (), as required. Tus te Hille-Yosida teorem provides a weak solution to te weak equation in te form of a ( λ 1 ())-contractive semigroup on L 2 (). Te solution u(t) satisfies te exponential convergence-to-equilibrium estimate u(t) 2 e λ 1()t u 2. Solving te wave equation: Given u H 1 () and v L 2 () we want to solve u tt u = in wit u = on and u t= = u, (u t ) t= = v. To tis end consider te space X = H 1 () L 2 () and denote by U = (u, v) te generic element of X, endowed wit te norm U = u H 1 () + v L 2 (). Next define te operator AU = (v, u) wit domain D(A) = (H 2 () H 1 ()) H 1 (), wic is dense in X. Solving te ODE U (t) = AU(t) wit U() = (u, v ) amounts in providing a weak solution to te wave equation. Let λ >. For every F = (f, g) X, solving λu AU = F means tat λu v = f λv u = g so tat u = g λv = g + λf λ 2 u. Since (λ 2 Id ) is injective and surjective from H 1() H2 () to L 2 (), we find tat tere exists a unique u suc tat u = g+λf λ 2 u, and ten by setting v = λu f we ave found tat for every F X tere exists a unique U D(A) suc tat λu AU = F. Set U = R λ F. Notice tat by λv u = g, multiplying by v we find λv 2 v u = gv and ten inserting v = λu f Integrating λ v 2 + u 2 = tat is λ ( v 2 u u) + f u = gv. gv + f u ( g L 2 () + f H 1 ()) ( ) v L 2 () + u L 2 () R λ F = U = v L 2 () + u L 2 () g L 2 () + f H 1 () λ = F λ. 6

7 Tus A generates a contractive semigroup on X, wic provides a notion of weak solution for te wave equation. Notice tat te estimate S(t)U U now means tat u(t) L 2 () + u t (t) L 2 () u L 2 () + v L 2 (), were we actually know (by differentiating te total energy of te wave) tat tis quantity is conserved! 1.4. Proving part one of te Hille-Yosida teorem. It is a simple exercise to prove te following lemma. Lemma 1.4. Let A be a closed operator defined on a dense subset D(A) of a Banac space X. Ten for every λ >, R λ is a closed operator, and tus belongs to L(X, X). Moreover, for every λ, µ ρ(a), A R λ = R λ A on D(A), (1.1) R λ R µ = R µ R λ on X, (1.11) R λ R µ = (µ λ) R λ R µ on X. (1.12) Proof. If u k u and R λ u k v in X, ten λ u k Au k v implies Au k λu v. Since A is closed, u D(A) and Au = λ u v, tat is v = R λ u. Tus R λ is closed, and by te closed grap teorem, R λ L(X, X). To ceck (1.1) just notice tat Id = (λid A)R λ = λ R λ AR λ so tat A R λ = R λ A on D(A). Next, by (1.1) we find Id = R λ (λid A) = λ R λ R λ A R λ = R λ R µ (µid A) = R λ (µid A)R µ = R λ (λid A)R µ + R λ (µ λ)id R µ = R µ + (µ λ)r λ R µ, tat is (1.12). As a consequence we find (1.11) R λ R µ = R λ R µ µ λ = R µ R λ λ µ = R µr λ. Proof of Teorem 1.3, Part one. For U X and λ > ω let us set Rλ U = e λ t S(t)U dt. Notice tat te integral is convergent as e λ t S(t)U e (ω λ) t U and λ > ω. particular, Rλ 1 l ω. We sall now prove tat Rλ = R λ on X. First of all, for U X and λ > ω we ave S()Rλ U R λ U = 1 e λ t S(t + )U dt e λ t S(t)U dt = 1 e λ t+λ S(t)U dt e λ t S(t)U dt = 1 wic as +, converges to e λ t+λ S(t)U dt + eλ 1 ARλ U = S()U + λ e λt S(t)U dt = U + λ Rλ U. e λt S(t)U dt In 7

8 Tis identity means tat (λ Id A)R λ U = U U X. On te oter and ARλ = R λa on D(A) as a consequence of AS(t) = S(t)A on D(A), so tat we also ave Rλ (λ Id A)U = U U D(A). Tese two identities imply tat (λ Id A) is injective and surjective from D(A) to X and tat Rλ = (λ Id A) 1 for every λ > ω Proving part two of te Hille-Yosida teorem. We start proving (i). We first sow tat as λ λ R λ U U U X. (1.13) Indeed, let us first pick U D(A) and set V = λ R λ U for λ > ω Ten λ U = λv AV so tat (1.1) and (1.9) imply U λr λ U = U V AV = AR λ U = R λ AU R λ AU AU λ λ ω, were AU <. In te general case wen U X we just pick U δ D(A) suc tat U U δ < δ and notice tat ( R λ U U R λ U δ U δ ) U U δ λ ω so tat (1.13) follows. By (1.13) and recalling tat Id = (λid A)R λ = λ R λ R λ A, we deduce tat for every U D(A) A λ U = λu + λ 2 R λ U = λ R λ AU AU. Tis proves assertion (i). Since R λ L(X, X) we ave A λ L(X, X) and tus S λ (t) = e t A λ defines a contraction semigroup tanks to Teorem 1.2. To estimate S λ, let us notice witout proof tat if L, M L(X, X) and LM = ML ten Hence, so tat e (L+M)t = e Lt e Mt = e Mt e Lt on X. e t A λ = e λ t e t λ2 R λ e t A λ U e λt t k λ 2k R λ k U k! k=1 e λt t k λ 2k U (λ ω) k k! k=1 = e λt e λ2 t/(λ ω) U e ωλ/(λ ω) t U Tis proves (ii). We now prove tat for every t and U D(A), S λ (t)u λ>ω as te Caucy property as λ. Indeed, let t >, U D(A), and define φ : [, t] X by setting φ(s) = e (t s)a λ e s A µ U s t. We ave φ(t) φ() = S λ (t)u S µ (t)u sup s t φ(s) e ωt. Moreover, on noticing tat A µ A λ = A λ A µ tanks to (1.11), and by using (1.3) we find φ (s) = e (t s)a λ e s Aµ (A µ U A λ U) s [, t], 8

9 were, as λ λω/(λ ω) is decreasing, Hence, Tus, e (t s)a λ e s A µ e λω λ ω (t s) e µω µ ω s e λω λ ω t, µ > λ. φ (s) e ωλt/(λ ω) A µ U A λ U s [, t], µ > λ > ω. S λ (t)u S µ (t)u t e ωλt/(λ ω) A µ U A λ U U D(A). (1.14) Since A λ U AU as λ, we conclude tat S λ (t)u λ>ω is a Caucy family, and tat for every t tere exists a linear map S(t) : D(A) X defined by S(t)U = lim λ S λ(t)u, U D(A). Sending µ in (1.14) wile keeping λ fixed we find S(t)U S λ (t)u t e λωt/(λ ω) AU A λ U U D(A). (1.15) By (1.15), S λ (t)u S(t)U uniformly over t [, T ] as λ, so tat t S(t)U is continuous on [, ) for U D(A). Since S µ (t)u e ωµ t/(µ ω) U for every U X, we immediately see tat S(t) L(X, X) wit S(t) e ωt. From tis we easily deduce tat S λ (t)u S(t) U for every U X, and tat t S(t)U is continuous on [, ) for every U X. Te properties S(t + s) = S(t)S(s) = S(s)S(t) and S() = Id are easily transferred by pointwise convergence, and so (iii) is proved up to sowing tat A is te generator of S(t) t. To tis end, let B denote te generator of S(t) t, so tat B is a closed operator wit domain D(B) = as sown in Teorem 1.2, wit Recall tat for λ > ω S λ (t)u U = U X : te limit BU = lim t + S(t)U U t t In particular, if U D(A), ten so tat (ω, ) ρ(b). A λ S λ (t) U dt = t exists in X. S λ (t) A λ U dt U X. S λ (t) A λ U S(t) AU S(t) A λ U AU + S λ (t) S(t) AU S(t)U U = As a consequence if U D(A), ten t S(t) AU dt U D(A). U D(B) wit BU = lim + S()U U = AU tat is B = A on D(A) D(B). Now (ω, ) ρ(a) ρ(b) and if λ > ω X = (λ Id A)(D(A)) = (λ Id B)(D(A)), so tat (λ Id B) is injective and surjective from D(A) to X. Since (λ Id B) is also injective and surjective from D(B) to X, it follows tat and tus A is te generator of S(t) t. D(A) = D(B) 9

10 2. Nonlinear semigroups as gradient flows We now consider a class of nonlinear semigroups arising as gradient flows of convex energies. Natural examples to keep in mind are nonlinear parabolic equations of te form u t u = f(u) f : R R decreasing and u t div ( L( u)) = L : R n R convex. Tese will be te gradient flows of te convex energies u 2 F (u) L( u), were, say, F (u) = u f(s) ds. We now illustrate (i) te notion of gradient flow of a convex energy in te model case of finite dimension and (ii) in wic sense a PDE on R n like te eat equation can be seen as a gradient flow of te Diriclet energy Gradient flows in R n. Gradient flows in R n are systems of ODE aving te form X (t) = E(X(t)), t, (2.1) X() = X, for a given scalar function E : R n R. Here E represents te total energy of a system expressed wit respect to te states variable (x 1,..., x n ) = X, and tese variables evolve according to an infinitesimal minimization principle, tat is, te systems sniffs around for te nearest accessible states of lower energy by moving in te direction E(X(t)). Wenever E is a Lipscitz map, te Caucy-Picard teorem implies te existence of a unique solution for every sufficiently small time. A global in time solution exists, for example, if E is globally bounded, or if E is just locally bounded one can prove tat te solution never leaves a compact sublevel set of E. Te latter is te case for convex gradient flows, see below. Te dynamics of a gradient flow can be quite ric. Every critical point X of E is stationary for te flow, in te sense tat X(t) X will be te unique solution corresponding to te initial data X. Among critical points, local maxima will be unstable (under small perturbations of te initial data, te solution will flow away); but all local minima will be possible asymptotic states. Tis last remark illustrate te great importance of understanding local minimizers, and not just global minimizers, in te study of variational problems. Te following simple teorem gives a good idea of te properties of convex gradient flows. Teorem 2.1. Let E : R n R be a smoot convex function suc tat lim E(x) = +. (2.2) x Ten a solution X(t) of (2.1) exists for every t. If E is strictly convex, ten E as a unique global minimizer X min on R n, and X(t) X min as t. If in addition E is uniformly convex, tat is, if tere exists λ > suc tat ten X(t) converges to X min exponentially, Y 2 E(X)Y λ Y 2 X, Y R n, (2.3) X(t) X min e λ t X X min, t. (2.4) 1

11 Proof. Notice tat E is only locally Lipscitz, so tat te existence for every time does not follow immediately from te Caucy-Picard teorem. Te flow exists for every time neverteless, because differentiating E(X(t)) in time we find tat E(X(t)) is decreasing along te flow, d dt E(X(t)) = X (t) E(X(t)) = E(X(t)) 2. (2.5) In particular, te flow never leaves te set E E(X ), wic is a bounded set by te coercivity assumption (2.2). One can tus apply te local existence and uniqueness teorem indefinitely and construct a unique solution for every time. By coercivity and strict convexity of E tere exists a unique minimizer X min of E on R n. We now prove tat X(t) X min as t. We first notice tat (2.7) implies te dissipation inequality E(X(t)) 2 dt E(X ). (2.6) Differentiating in turn E(X(t)) 2, and using te second order caracterization of convexity Y 2 E(X)Y X, Y R n, we deduce tat d dt E(X(t)) 2 = 2 E(X(t)) 2 E(X(t))[X (t)] (2.7) = 2 E(X(t)) 2 E(X(t))[ E(X(t))]. (2.8) Combining (2.6) and (2.7) we see tat E(X(t)) as t. Since E(X(t)) is bounded and E is coercive, for every t j tere exists a subsequence x j of X(t j ) suc tat x j X as j. It must be E(X ) =, and since E admits a unique global minimum and its convex, we find X = X min. By te arbitrariness of t j, we conclude tat lim X(t) = X min. (2.9) t To prove exponential convergence, we take te scalar product of E(X(t)) E(X min ) = 1 wit (X(t) X min ) and apply uniform convexity to find But tanks to (2.1) 2 E(sX(t) + (1 s)x min ) (X(t) X min ) ds λ X(t) X min 2 ( E(X(t)) E(X min ) ) (X(t) X min ). d dt X(t) X min 2 = 2 (X(t) X min ) X (t) = 2 (X(t) X min ) ( E(X(t)) E(X min )) 2 λ X(t) X min 2 so tat (2.4) follows Te eat equation as a gradient flow in L 2. We now informally discuss ow te following boundary value problem for te eat equation u t u = on (, ), u t= = u on, (2.1) u = ψ t. (were u and ψ are given initial conditions and boundary data) can be seen as a gradient flow in L 2 (). An important conclusion will be tat in order to see (2.1) as a gradient 11

12 flow we sall need to work wit convex functions tat are possibly infinite value and non-differentiable. Te energy functional will be te Diriclet energy E : L 2 () R +, defined as 1 u 2 if u ψ + H 1 E(u) = 2 () +, oterwise. Te differential of E at u is defined as de(u)[φ] = lim t E(u + tφ) E(u) t wenever φ L 2 () is suc tat te limit exists. Tis is equivalent in asking φ H 1(), and in tis case we find E(u + tφ) = E(u) + t u φ + t 2 E(φ), so tat de(u) : H 1 () R is te linear operator defined by de(u)[φ] = u φ φ H 1 (). Defining te gradient of E at u means representing te linear functional de(u) via a duality pairing. Considering tat de(u) is defined on H 1 (), any Hilbert space H containing H 1 () and suc tat sup de(u)[φ] : φ H 1 (), φ H 1 < (2.11) is an admissible coice. Indeed, by Riesz teorem, if (2.11) olds, ten tere exists a unique H E(u) H suc tat de(u)[φ] = H E(u) φ H, φ H. Wen u H 2 () an integration by parts gives de(u)[φ] = u φ = so tat (2.11) olds wit H = L 2 () and L2 E(u) = u L 2 (). φ u, Terefore te equation u t = u can be seen as a gradient flow U (t) = L 2E(U(t)). Let us review te argument of Teorem 2.1 in ligt of tese definitions. Te computation (2.5) takes te form, d dt E(u(t)) = u(t) u t (t) = ( u(t)) 2 = L 2E(u) 2 L 2 (), so tat te dissipation inequality is ( u(t)) 2 1 u 2. 2 Similarly (2.7) takes te form d dt E(u(t)) 2 L 2 () = 2 = 2 u(t) u t (t) = 2 ( u(t)) 2. u(t) ( u(t)) 12

13 Tere is exponential convergence to te unique minimum of E(u) over L 2 (), wic is te unique armonic function Ψ wit ψ as its boundary data. Te fastest way to see tis is differentiating d 1 u(t) Ψ 2 = (u(t) Ψ) u t (t) = (u(t) Ψ) ( u(t) Ψ) dt 2 = (u(t) Ψ) 2 λ 1 () u(t) Ψ 2 were we ave applied te Poincaré inequality to u(t) Ψ H 1(). dimensional case, we tus ave u(t) Ψ 2 e 2λ 1()t u Ψ 2. As in te finite 2.3. Non-smoot convex functions. Te considerations from te previous section sow tat studying te eat equation as a gradient flow requires understanding convex functions wic possibly take infinite values and are not everywere differentiable. Let X be a vector space. We say tat E : X R + is convex if E(tu+(1 T )v) t E(u) + (1 t)e(v) wenever t [, 1] and u, v X. Tis inequality implies in particular tat if E(tu + (1 T )v) = + for some t [, 1], ten eiter E(u) or E(v) equals +. Tus dom(e) = u X : E(u) < is a convex subset of te vector space X. We say tat E is proper if dom(e). Convexity and topology. Now assume tat E is a Banac space, and consider a proper convex function E on X. Te natural property tying te convexity of E to te metric properties of X is te lower semicontinuity inequality, namely E(u) lim inf E(u k) wenever u k u in X. (2.12) k Notice tat morally convex functions sould automatically be lower semicontinuous, because intuitively tey are upper envelopes of affine (tus, continuous) functions. However, tis latter properties may fail wen we allow for infinite values. For example, consider te proper convex function E : R R + defined by, for x < 1, E(x) = 1, for x = 1, (2.13) +, for x > 1. Ten E is strictly larger tan te supremum of its affine minorants, as E(1) = 1 > = supa + b : a + bx E(x) x R. Notice tat, indeed, E is not lower semicontinuous at x = ±1. So (2.12) is not automatically satisfied. An important remark is tat lower semicontinuity along strongly convergent sequences (2.12) implies lower semicontinuity along weakly convergent sequences. Indeed, pick u k u weakly in X, and up to extract a subsequence assume tat lim inf k By Mazur s lemma, for suitable λ (j) k N(k) j=k E(u k) = lim k E(u k). 1, j = k,..., N(k), wit N(k) j=k λ(j) k = 1 we ave λ (j) k u j u in X 13

14 so tat by (2.12) we find as claimed. E(u) lim inf k lim inf k ( N(k) E N(k) j=k j=k ) λ (j) k u j λ (j) k E(u j) lim inf k max E(u j) = lim E(u k), k j N(k) k Subdifferentials We now restrict our attention to te case wen X = H is a Hilbert space, and introduce te important notion of subdifferential of a convex proper function E on H. Te subdifferential of E at u H is te set E(u) = v H : E(w) E(u) + v w u w H, and dom( E) = u H : E(u). Te subdifferential is te set of slopes v suc tat te affine function w E(u)+ v w u of slope v and value E(u) at w = u lies below E everywere on H; te domain dom( E) is te set of points u of H suc tat tere exists at least one affine function passing troug (u, E(u)) and lying below E everywere on H. Clearly, dom( E) dom(e). Te inclusion may be strict: in example (2.13), 1 dom(e) but 1 dom( E), i.e., E(1) =. Important simple properties of sub-differentials are: (i) E(u) is a convex set for every u H; (ii) in finite dimension, E is differentiable at any u lying in te interior of dom(e) if and only if E(u) consists of a single point, te gradient of E at u; (iii) E(u) = min H E if and only if E(u); (iv) if u, v dom( E), ten v u v u v E(v), u E(u). Tis last property follows immediately by considering te inequalities E(w) E(u) + u w u w H, E(w) E(v) + v w u w H, testing te first one at w = v, te second one at w = u and ten adding up. Property (iv) is called monotonicity because in te case H = R n, n = 1, for a differentiable convex function E, it means (E (u) E (v))(u v), wic of course implies E (u) E (v) wenever u v Construction of gradient flows on Hilbert space. We now state te existence teorem for gradient flows in Hilbert spaces. Teorem 2.2. Let E : H R + be a proper, convex lower-semicontinuous function on a Hilbert space H. Assume tat dom( E) = H. Ten for every u H tere exists a unique function U(t) C ([, ); dom( E)) wit U() = u and U (t) L ((, ); H), suc tat U (t) E(U(t)) for a.e. t >. (2.14) 14

15 Remark 2.1. Notice tat uniqueness assertion may seem non-obvious, as te condition U (t) E(U(t)) does not uniquely identify U (t) for any given t suc tat E(U(t)) contains more tan one element. However, it U and U are two gradient flows wit initial condition u in te sense of Teorem 2.2, ten for a.e. t > we obtain d U(t) U (t) 2 = U(t) U (t) U (t) U dt 2 (t) tanks to te monotonicity of E. Tus for every t > and so U U. U(t) U (t) U() U () = u u =, Te approac to Teorem 2.2 is, on a formal level, very similar to te approac to te Hille-Yosida teorem. In analogy wit equations (1.5) and (1.6) we may try to define maps A µ approximating E by setting A µ = Id (Id µ E) 1 µ = E + O(µ). To make sense of tis we will first need to sow tat J µ = (Id µ E) 1 really defines a single valued map. From tis point of view, te second identity sign sould be interpreted as saying tat A µ (u) is O(µ) distant from E(u). All tese tings turn out to be true, and we will see tat A µ are Lipscitz maps, wose classical flow converges to a solution of (2.14) as µ +. One can develop a better geometric intuition on tis construction by considering te problem in finite dimension. Consider a piece-wise affine convex energy E : R R. Te subdifferential of E is multivalued at eac point were E jumps, but neverteless we can visualize E as a function wose grap contains a few vertical walls. Te idea is obtaining A µ as function wit Lipscitz constant of order 1/µ and smaller slopes tan E, in oter words, we approximate te walls from below wit very steep slopes. Notice tat, actually, A µ is te subdifferential of a C 1,1 convex function E µ (uniquely determined up to a constant). Te flows X µ (t) satisfying X µ(t) = A µ (X µ (t)) and X µ () = x are compact in µ > and converge to a limit flow, wic actually solves (uniquely!) X (t) E(X(t)) and X() = x. In te next teorem we rigorously describe te approximation procedure describe above. Teorem 2.3. Let E be a proper, lower-semicontinuous convex function on a Hilbert space H. Ten for every µ > and u H tere exists a unique J µ (u) dom( E) suc tat u J µ (u) µ E(J µ (u)). (2.15) Te resulting map J µ = (Id µ E) 1 : H H is 1-Lipscitz continuous and satisfies Moreover, if we define A µ : H H by setting lim J µ(u) = u u dom( E). (2.16) µ + A µ (u) = u J µ(u) µ u H, ten A µ is 2/µ-Lipscitz continuous, A µ is monotone on H, and A µ (u) E(J µ (u)) u U (2.17) A µ (u) inf z : z E(J µ (u)), u dom( E). (2.18) 15

16 Proof. Step one: To sow tat J µ is well-defined, given u H, consider te energy F (v) = µ E(v) + v 2 2 u v. We claim tat tere exists a global minimizer v of F on H. Notice tat suc a minimizer will be unique (F is strictly convex as v 2 is so). Moreover, ( 2 /2)(v) = v and ( u )(v) = u, so tat F (v) = µ w + v u : w E(v), will immediately imply tat v = J µ (u) satisfies (2.15). To prove te existence of a minimizer of F, we first notice tat tere exist a R and v H suc tat E(w) a + v w w H. (2.19) Tis is a consequence of te Han-Banac teorem, but a direct proof is also simple. Let us assume tat k N tere exists w k H s.t. E(w k ) < k k w k. (2.2) If w k is bounded, ten w k w and te lower-semicontinuity of E give E(w ) =, a contradiction. So it must be w k, and ten setting ŵ k = w k / w k w and using convexity we get tat, for every u H, E (( 1 1 w k ) u + w k w k ) tat is, by lower-semicontinuity and by (2.2) ( 1 1 ) E(u) + E(w k) w k w k k k w k E(w ) E(u) + lim =, k w k again a contradiction. Tus (2.2) cannot old, and tere exists C > suc tat E(w) C C w for every w H. Taking v (C + C ), we see tat (2.19) olds wit a = C. Given (2.19) te existence of a minimizer for F on H follows by te direct metod. Consider a minimizing sequence v j, lim F (v j) = inf F. (2.21) j H By (2.19) we ave + > inf F v j 2 ( ) u v j µ a + v v j H 2 wic immediately implies te boundedness of v j, and tus v j v for some v H. By lower-semicontinuity of F and by (2.21) we find inf H F F (v) lim inf j so tat v is indeed a minimizer of F over H. F (v j) = inf H F Step two: We prove te Lipscitz estimates for J µ and A µ, and te monotonicity of A µ. If u k H ten tere exists w k E(J µ (u k )) suc tat u k = J µ (u k ) µ w k ; ence u 1 u 2 2 = J µ (u 1 ) J µ (u 2 ) 2 + µ 2 w 1 w µ w 1 w 2 J µ (u 1 ) J µ (u 2 ) J µ (u 1 ) J µ (u 2 ) 2 + µ 2 w 1 w

17 tanks to te monotonicity of E. In particular, Lip(J µ ) 1, and as a consequence Lip(A µ ) 2/µ. Te monotonicity of A µ also follows from Lip(J µ ) 1: indeed, A µ (u 1 ) A µ (u 2 ) u 1 u 2 = 1 µ u 1 J µ (u 1 ) (u 2 J µ (u 2 )) u 1 u 2 were Lip(J µ ) 1 was used in te last inequality. = u 1 u µ µ J µ(u 2 ) J µ (u 1 ) u 1 u 2 u 1 u 2 u 1 u 2 J µ (u 2 ) J µ (u 1 ), µ Step tree: We prove (2.17) and (2.18). By definition of J µ (u), if u H ten tere exists z E(J µ (u)) suc tat u = J µ (u) µ z. In particular, for every w H E(w) E(J µ (u)) + z u J µ (u) = E(J µ (u)) + A µ (u) u J µ (u) tat is, A µ (u) E(J µ (u)). We can exploit tis last property togeter wit te monotonicity of E to find tat if u dom( E) and tus tere exists z E(u), ten u J µ (u) z A µ (u) = A µ (u) z A µ (u) = A µ (u) z A µ (u) 2 A µ (u) z A µ (u) 2 tat is te minimal slope property of A µ stated in (2.18). Finally, J µ (u) u = µ A µ (u) µ inf z : z E(u). Tis implies J µ (u) u as µ + since we are assuming u dom( E), and tus inf z : z E(u) <. Te convergence of J µ (u) u for u dom( E) follows easily by density. Proof of Teorem 2.2. Since A µ is a Lipscit map we can apply te Caucy-Picard teorem and define U µ (t) C 1 ([, ); H) in suc a way tat U µ(t) = A µ (U µ (t)) and U µ () = u. We aim to sow tat U µ (t) as a limit as µ + and tat tis limit U(t) defines te desired solution to te gradient flow defined by E. Step one: We obtain a basic estimate on U µ, namely sup U µ(t) A µ (U µ ()) = A µ (u ). (2.22) t To tis end we first compare U µ (t) wit te flow V µ(t) = A µ (V µ (t)) corresponding to anoter initial condition, V µ () = v. In tis way tat is d U µ (t) V µ (t) 2 dt 2 = U µ(t) V µ(t) U µ (t) V µ (t) = A µ (U µ (t)) A µ (V µ (t)) U µ (t) V µ (t), U µ (t) V µ (t) u v t. (2.23) We fix >, and apply tis property wit v = U µ (), tat is, we compare te flow U µ (t) wit tat lagged flow U µ (t + ) = V µ (t): we find U µ (t) U µ (t+) u v = U µ () U µ () Dividing by and letting we find (2.24). U µ(t) dt = A µ (U µ (t)) dt. 17

18 Step two: Assume now tat u dom( E), ten by (2.18) te rigt and side of (2.24) is bounded uniformly in µ, tat is z : z E(u ), (2.24) sup µ> sup U µ(t) inf t wic implies te weak sequential compactness of U µ(t) in L ((, ); H). In order to exploit tis estimate we now compare te flows U µ (t) and U ν corresponding to µ, ν >. To tis end we compute tat d U µ (t) U ν (t) 2 dt 2 = U µ(t) U ν(t) U µ (t) U ν (t) = A µ (U µ (t)) A ν (U ν (t)) U µ (t) U ν (t). Considering tat U µ = µ A µ (U µ ) + J µ (U µ ), and dropping te t-dependency for te sake of clarity, we ave d U µ U ν 2 dt 2 as A µ (U µ ) E(J µ (U µ )). Hence = A µ (U µ ) A ν (U ν ) µ A µ (U µ ) ν A ν (U ν ) A µ (U µ ) A ν (U ν ) J µ (U µ ) J ν (U ν ) A µ (U µ ) A ν (U ν ) µ A µ (U µ ) ν A ν (U ν ) A µ (U µ ) A ν (U ν ) µ A µ (U µ ) ν A ν (U ν ) = µ A µ (U µ ) 2 + ν A ν (U ν ) 2 (µ + ν) A µ (U µ ) A ν (U ν ) µ A µ (U µ ) 2 + ν A ν (U ν ) 2 µ ( A µ (U µ ) 2 + A ν(u ν ) 2 ) 4 ν ( A ν (U ν ) 2 + A µ(u µ ) 2 ), 4 were we ave used ab a 2 + b 2 /4. In conclusion, d U µ U ν 2 µ dt 2 4 A ν(u ν ) ν 4 A µ(u µ ) wic, tanks to (2.24), implies d U µ U ν 2 dt 2 µ + ν 4 inf z : z E(u ), (2.25) and ence U µ U ν 2 µ + ν inf z : z E(u ) t, 2 4 t, µ, ν >. (2.26) Now (2.26) implies tat, for eac t, U µ (t) µ> is a Caucy family in H, so tat U µ (t) U(t) in H for some U(t) H and along a subsequence µ +. By (2.26), te convergence is locally uniform on t [, ), so tat U(t) C ([, ); H). By (2.24), U admits a weak derivative U (t) L ((, ); H) wit b wenever w H and (a, b) (, ), and a b U µ (t) w dt U (t) w dt as µ + (2.27) a sup U (t) inf t z : z E(u ) Recalling tat U µ = A µ (U µ ) E(J µ (U µ )) we see tat E(w) E(J µ (U µ )) + U µ w J µ (U µ ). (2.28) 18

19 tus tat (b a)e(w) b Now, J µ (U µ (t)) U µ (t) uniformly on t as a E(J µ (U µ )(t)) + U µ (t) w J µ (U µ (t)) dt J µ (U µ (t)) U µ (t) = µ A µ (U µ (t)) µ min ence J µ (U µ )(t) U(t) as µ uniformly on (a, b). lower-semicontinuity of E we obtain (b a)e(w) b a z : z E(u ), E(U(t)) + U (t) w U(t)) dt. Dividing by b a and letting b a we conclude tat, for a.e. a >, E(w) E(U(a)) + U (a) w U(a)) Terefore by (2.27) and by w H i.e. U(a) E(U(a)) for a.e. a >, tat is (2.14). We are left to prove tat U(t) dom( E(U(t))) for every t >, and not just for a.e. t >. It suffices to pick any t j t suc tat U (t j ) E(U(t j )) and notice tat by (2.28), up to extract a subsequence, U (t j ) v weakly in H. By lower-semicontinuity of E, te inequalities E(w) E(U(t j )) + U (t j ) w U(t j ) w H, imply tat U(t) dom( E), wit v E(U(t)). Final remarks: Notice tat te uniqueness property of te convex gradient flow implies tat actually U µ (t) U(t) as µ +, and not just as µ for a subsequence µ. Also, te assumption u dom( E) is easily dropped, for, pick any u H and use te density assumption of dom( E) into H to approximate u wit u k dom( E). Denote by U k (t) te corresponding flows and notice tat by differentiating U k (t) U j (t) 2 and by monotonicity of E we get U k (t) U j (t) u k u j. Hence U k (t) k N is a Caucy sequence and it easily seen, by repeating te argument used above, tat te limit flow is indeed a gradient flow in te sense of Teorem 2.2 wit U() = u An example. We finally discuss an example of a non-linear flow, precisely, we consider a non-linear parabolic PDE of te form u t = div ( f( u)) on, t > u() = u, on (2.29) u = ψ, on, t >. wic corresponds, formally, to te gradient flow of te energy E(u) = f( u) u : R defined by an integrand f : R n R. Indeed, E(u + t φ) = E(u) + t so tat, formally, de(u)[φ] = f( u) φ + o(t) f( u) φ. 19

20 If φ = on, ten an integration by parts gives de(u)[φ] = φdiv ( f( u)) so tat, if div ( f( u)) L 2 (), ten te L 2 -gradient of E at u is given by L2 E(u) = div ( f( u)). In tis sense te nonlinear parabolic PDE (2.29) can be seen as te gradient flow of te energy E. By exploiting te H 2 -regularity teory for elliptic equations in divergence form it is not ard to sow tat (2.29) can indeed be solved in te framework described in tis section. See Section in Evans s PDE book. (Francesco Maggi) Department of Matematics, University of Texas at Austin, 2515 Speedway STOP C12, 78712, Austin, Texas, USA address: maggi@mat.utexas.edu 2