Hilbert-Space Integration

Transcription

1 Hilbert-Space Integration. Introduction. We often tink of a PDE, like te eat equation u t u xx =, a an evolution equation a itorically wa done for ODE. In te eat equation example two pace derivative are preent and we expect tat for eac t >, te olution ut ould be at leat in te Hilbert pace H 2. However, te time derivative require ome regularity on te temporal dependence. Moreover, te different order of derivative preent in patial and temporal derivative implie different pace are required for te time dependence tan for te patial variable. Ti lead to pace like L 2,T;H k and o on. We briefly decribe a way to deal wit tee pace, and fill in te gap in te preentation in [], [2]. Trougout H i a Hilbert pace wit inner product,. We need a few known reult about Hilbert pace to get tarted. Definition. A equence {u n } in H converge weakly to u H if u n,v u,v a n and for all v H. Teorem.2 Suppoe {u n } i a bounded equence in H. Ten {u n } a a weakly convergent ubequence. Definition.3 A linear bounded functional f on H, i a linear map from H to R. It i denoted f,v and atifie f,v f v, were f := up f, u. u = Finally, we will need Teorem.4 Riez Repreentation Teorem For every bounded linear functional on H, f, tere exit a unique element f H uc tat f,v = f,v for all v H and f = f. Proof. See P. 7 of [2]. Next we conider te pace L 2,T; H wit inner product ut,vtdt. An element in L 2,T; H i an element in H for eac T wit te property tat ut i a meaurable function on [,T] and ut 2 dt <.

2 Lemma.5 Te vector pace L 2,T; H wit te above inner product i a Hilbert pace. Proof. Ti i a omework exercie. Follow te proof owing L 2 Ω i a Hilbert pace. We may integrate ut by uing te Riez repreentation teorem. Specifically, we define f by < f,v >:= < f,v > ut,vdt. Ti define a bounded linear functional on H. Indeed, ut,v dt ut v dt CT ut 2 2 dt By te Riez repreentation teorem tere exit a unique I H acting te ame way ti particular f act. We et Hence, utdt,v = utdt := I. ut, v dt. Te pace L p,t; H are defined in a imilar way, and we ave Lemma.6 If ut L,T; H, ten ut dt ut dt. v. Proof. Set I = utdt. Ten I,v = Now take v = I/ I and te reult follow. ut,vdt v ut dt. We extend ut L 2,T; H to be zero outide of [,T] and define u t to be te unique element in H wic act like te functional t t ρ ut,vdt for all v H and eac t [,T]. Lemma.7 Suppoe ut L 2,T; H. Ten u t L 2,T; H. In fact, u t 2 dt ut 2 dt. 2

3 Proof. From te previou Lemma.6 we find u t = t t T ρ ut dt t t = ρ 2 t t ρ t t ρ ut 2 dt 2. ρ 2 ut dt ut dt Te lat inequality ue te Caucy-Scwarz inequality and te normalized property of ρ: ρ =. Integrating te quare of te previou inequality give u t 2 dt and te reult follow. ρ ut 2 dt dt ut 2 t t ρ ut 2 dt, dt dt Teorem.8 For every ut L 2,T; H, tere exit a equence {u m } uc tat i u m u in L 2,T; H a m ; ii u m C,T;L 2 Ω; iii u m t 2 C R. Proof. Let = /j for j N wit u defined a above. We compute uing Lemma.7 u t ut 2 dt = = 2 u t ut,u t utdt u t 2 + ut 2 2u t,ut dt ut 2 dt 2 u t,utdt.. Ten Lemma.7 alo ow tat {u } i a bounded equence in L 2,T; H. It terefore a a weakly convergent ubequence wic we denote {u m }. We need to compute te weak limit. Note v H i alo in L 2,T; H contant in time. We et f t := u t,v = ρ 3 ut,vdt = ut,v,

4 and ft := ut,v. Since f L,T, f converge to f in L,T Appendix C4 in []. Since te weak limit i unique, u m converge weakly in L 2,T; H to u, and te rigt ide of. goe to zero a. To prove ii we compute for a fixed u t u,v = Since ρ i uniformly continuou u t u,v t t t ρ ρ ut,vdt. ǫ ut,v dt ǫ ut L 2,T;L 2 Ω T v L 2 Ω, for and t ufficiently cloe. Te reult follow upon letting v = ut u. Finally we ow te regularity of u t. A calculation ow u t,u t = ρ ρ t u,ut d dt.2 wic i clearly infinitely differentiable. 2. Wo Care? Minimum patial regularity for a weak olution of a econd order parabolic PDE require for eac t > te olution ut H Ω. Since we are working wit Hilbert pace, we will look for a weak olution in L 2,T;H Ω. Ti owever doe not addre te time derivative. Te weak olution i motivated by taking te L 2 Ω inner product wit a tet function and te PDE. Te time derivative will be in te form u t,v, were te inner product i te one on L 2 Ω and v H Ω. So te time derivative look like, for eac t, a functional on H Ω only wit te L2 Ω inner product. Ti motivate te tale of tree Hilbert pace: Tink of H Ω L2 Ω. In fact, let ι : H Ω L2 Ω wit ιu = u. Te above inner product look like a functional on ιh Ω. Definition 2. We et H Ω to te dual et of all bounded linear functional on ιh Ω. A functional u H Ω i denoted u,v and i a real-valued function for all v H Ω. In addition, we define te norm of f H Ω to be { } f H Ω := up f, v v H Ω, v H Ω =. 4

5 Note tat f L 2 Ω i in H Ω. Indeed, Tu we ave f,v = f,v f L 2 Ω v L 2 Ω f L 2 Ω v H Ω. Te time derivative will live in H Ω. H Ω L2 Ω H Ω. An Example. To better undertand te notation, H Ω, conider H, - te cloure of C, in te H, norm. We ow function woe anti-derivative are quare integrable are in H,. Let f L,. We know te weak derivative of ux := x fdx i fx. Tat i, u xϕxdx = uxϕ xdx = fxϕx dx, for all ϕx C,. Since C, i dene in H,, ti lat expreion implie fxvxdx = uxv xdx. Since u i a continuou function, ux L 2,, and a a conequence bounded linear functional on H. Tat i, { f x } fdx = ux L 2, H,. fxvxdx i a So H conit of function woe weak derivative are quare integrable; L2 function ave zero weak derivative quare integrable; wile H include function woe anti-derivative are quare integrable - ence te in te labeling of H. Propoition 2.2 H Ω i a Hilbert pace. Proof. Since ιh i a normed pace, it dual i a Banac pace wit norm f = up v H Ω = f,v ee Problem 2 below. Tu at leat H Ω i a Banac pace. To ow it i a Hilbert pace, we need to ceck tat parallelogram law old. Tat i, a Banac pace i a Hilbert pace if and only if x + y 2 + x y 2 = 2 x y 2. In wic cae a inner product can be defined by ee Problem 3 below x,y = 2 x + y 2 x 2 y 2. 5

6 Te proof of Teorem, page 283 in [] ow tat for eac f H Ω, unique f i, i =,...,n exit in L 2 Ω uc tat n f 2 H Ω = f i 2 dω. Ω i= It follow 2 H Ω atifie te parallelogram law and H Ω i a Hilbert pace. Definition 2.3 Suppoe ut L 2,T;H Ω. We ay u t L 2,T;H Ω. Provided d dt ut,v L 2 Ω = u t,v v H Ω. Note te analogy wit te way we defined an integral on L 2,T;H Ω. Note alo tat te definition old for u L 2,T;L 2 Ω. However, to make ene of te initial data in te weak formulation of a PDE, we need ome continuity in te time variable. Te next teorem give u ti. Teorem 2.4 Suppoe u L 2,T;H Ω, wit u t L 2,T;H Ω. Ten u C[,T];L 2 Ω, ut 2 a a weak derivative wit d dt ut 2 L 2 Ω = 2 u t,ut. Moreover, we ave te etimate max ut L t T 2 Ω CT u L 2,T;H Ω + u L 2,T;H Ω We need a lemma to etabli te proof of Teorem 2.4. Below, and denote te inner product and norm on L 2 Ω. Lemma 2.5 Suppoe ut atifie te ypotei of Teorem 2.4. Ten d dt u t 2 = 2u t,u t = 2 u t,u t. Proof. Te derivative can be directly calculated from.2. Tank to te ymmetry in te expreion we find u t,u t d t t t = 2 dt ρ ρ ut,udt d t t [ ] t = 2 ρ ρ ut,ud dt t t = 2 ρ ut,u tdt = 2 d t t dt ρ ut,u tdt

7 Definition 2.3 implie ut,v ϕtdt = ut,vϕ tdt = du t,v ϕtdt, 2.4 dt for all v H Ω and ϕt C [,T]. Notice ρt t / C [,T +]. Since ut i et to zero outide of [,T], te above integral can be enlarged to [,T + ] witout canging any of teir value. Terefore, we may ue ρ for ϕ in 2.4. Alo recall tat we define du/dt to be te unique element in H Ω acting like te functional Tat i, ρ Finally returning to 2.3 ρ du dt t,u dt. du du dt t,u dt = dt t,u. u t,u t = 2 u t,u t. 2.5 Te oter equality implied in te lemma, u t = u t, i eentially proven above. See alo Problem 5. Proof of Teorem 2.4. Let u m be a in Teorem.8. Teorem.8 and Lemma 2.5 ow u m and u m converge in L 2,T;H T Ω and L2,T;H Ω to u and u repectively. Integrating 2.5 wit u replaced wit u m u n and wit m > n, we find u m t u n t 2 L 2 Ω = u m u n 2 L 2 Ω + 2 u m u n,u m u n dτ. Now pick a,t uc tat u m u in L 2 Ω. Te previou equality implie for n and m large u m t u n t 2 L 2 Ω ǫ + 2 ǫ + u m t u n t H Ω u mt u nt H Ω [ u m t u n t 2 H Ω + u mt u nt 2 H Ω ] dτ. Since te u m are continuou in L 2 Ω, Lemma.8, ti implie tat te equence {u m } i Caucy in C,T;L 2 Ω. It terefore converge to ome v C,T;L 2 Ω. Since we alo know u m converge to u in L 2,T;L 2 Ω, it follow u = v almot everywere, and we may identify u wit a continuou function. 7

8 Integrating 2.5, we find Te limit i eaily jutified and i u m t 2 L 2 Ω = u m 2 L 2 Ω + 2 u m τ,u mτ dτ. ut 2 L 2 Ω = u 2 L 2 Ω + 2 u τ,uτ dτ. 2.6 It follow tat te weak derivative of ut 2 L 2 Ω i 2 u t,ut. To obtain te etimate we integrate 2.6 wit repect to to find T ut 2 L 2 Ω u 2 L 2,T;L 2 Ω + 2T u τ,uτ dτ u 2 L 2,T;L 2 Ω + 2T u H Ω u H Ω dτ. Now ue Young inequality ab a 2 /2 + b 2 /2, te obviou inequality u 2 L 2,T;L 2 Ω u 2 L 2,T;H Ω, and ten te triangle inequality to find max ut L t T 2 Ω CT u L 2,T;H Ω + u L 2,T;H Ω. Reference [] Lawrence Evan, Partial Differential Equation, Graduate Studie in Matematic, AMS, 9, 998. [2] Robert McOwen, Partial Differential Equation, Prentice Hall 23. 8

9 Homework. Let H be a Hilbert pace. Suppoe u n H converge weakly. Prove te following a Te weak limit i unique. b Suppoe te weak limit i u, and u n converge to u a n. Sow u n converge trongly to u. c Suppoe te weak limit i u and u n u for all n. Sow u n converge trongly to u. 2. Let X be a normed pace. Prove tat it dual, X, i a Banac pace. Te pace X conit of all bounded linear functional on X wit norm f = up{ fx x X, x = }. 3. Let X be a Banac pace wit norm. Prove tat a Banac pace i a Hilbert pace if and only if te parallelogram law x + y 2 + x y 2 = 2 x y 2 old. If it doe, ten X i a Hilbert pace wit inner product x,y = 2 x + y 2 x 2 y Prove H Ω i a Hilbert pace. 5. Suppoe ut atifie te ypotei of Teorem 2.4. Ten u t = u t. Tat i, for all v H Ω d dt u t,v = d dt u t,v = u t,v. Hint Ti i baically proven in Lemma