Evoluion Equaions For 0, fixed, le U U0, where U denoes a bounded open se in R n.suppose ha U is filled wih a maerial in which a conaminan is being ranspored by various means including diffusion and convecion. Le he concenraion of conaminan a x, in U be denoed by ux, and le he vecor field F x, denoe he flux field for he conaminan. hen we can sae a conservaion principle for he conaminan as follows. For an arbirary ball B in U and an arbirary ime inerval 1, 0, we can asser ha ux, dx ux, 1 dx B F x, n dsd fx,dxd A B C D where he inegrals A,B represen he amoun of conaminan inside B a ime, 1, respecively, inegral C represens he amoun of conaminan flowing ou of B during he ime inerval 1,,(i.e., n here denoes he ouward uni normal o he boundary of B) and he inegral D represens he amoun of conaminan ha is creaed or desroyed inside B during he ime inerval 1, as a resul of a source densiy given by fx,. Using he divergence heorem o conver inegral C o an inegral over B, and using he he fundamenal heorem of calculus o wrie he difference A minus B as ux, dx ux, 1 dx ux,dxd we ge ux, divf f dxd 0 B Since his holds for all B in U and every ime inerval 1, 0,,we can conclude ha ux, divf x, fx, 0 x, U. Nex, we observe ha if he conaminan is being ranspored hrough he medium due o a combinaion of diffusion and convecion by a velociy field V x,, hen he flux field is relaed o he conaminan concenraion by, F x, ax, ux, ux,v x,, where ax, denoes a maerial dependen propery called "diffusiviy". he diffusiviy may be a ensor (n by n marix) in anisoropic maerials, a scalar in maerials which are isoropic bu no homogeneous, and for isoropic and homogeneous maerials, he diffusiviy is a consan. hen he conservaion saemen becomes, ux, div ax, u V udivv u fx, x, U. which means ha he concenraion saisfies an equaion of he form where ux, Lux, fx, in U Lux, divax,u V x, udivv x,u. he diffusiviy is only physically meaningful if i is nonnegaive, and we will assume furher ha i is sricly posiive a every poin in U.hen he operaor L is said o be ellipic and he operaor L is said o be parabolic. Evidenly, parabolic operaors arise in modelling he ime evoluion of physical processes ha behave like diffusion-convecion siuaion described here. 1
We are going o consider he problem ux, Lux, fx, ux, gx, ux,0 u 0 x in U on 0, on U 0. Such a problem is called an iniial value problem for an evoluion equaion, (IVP for shor). We shall show ha a modificaion of he approach we used for he ellipic BVP s can be employed o solve hese problems. Firs i will be necessary o describe he funcion spaces in which he variable is viewed differenly from he space variables, x. H Valued Funcions of In dealing wih evoluion equaions we will be faced wih funcions of x, where x is in R n and is in R 1 or in R 1. I will urn ou o be useful o view such funcions as funcions of, aking heir values in an appropriae space of funcions of he variable x; i.e., 0, U x ux, H u, funcion of wih values in H he funcion u(x,) will be wrien as u where his is inerpreed o mean ha he value a each is u,, a funcion which is an elemen of he funcion space H. For us, mos of he funcion spaces will be Hilber spaces. H Valued Disribuions: D [0,:H] For H a Hilber space, a linear mapping, J : C c 0, H defines a disribuion in wih values in H if and only if n 0 in C c 0, implies J n 0 in H We will use he noaion D0, o denoe C c 0,. For f : 0, H, locally inegrable, define J f 0 fd D0,. hen J f is easily seen o be a disribuion in wih values in H. he noaion for his will be J f D 0, : H, and we shall use he noaion L 1 loc 0, : H o denoe locally inegrable funcions of, wih values in H. hese are he regular H-valued disribuions. For J D 0, : H, he derivaive, J D 0, : H, is defined by J for all D0,. More generally, n 1 n n for all D0,. Now define L 0, : H o be he se of all measurable funcions of, 0, wih values in H such ha
u L 0,:H 0 uh d. heorem 1- he naural inclusion, L 0, : H D 0, : H, is a coninuous injecion; i.e., f n f in L 0, : H implies J fn f in H D0, and f 0 in H D0, implies f 0 in L 0, : H If we define linear spaces wih norms, C0, : H coninuous funcions of, 0, wih values in H u C0,:H max 0 u H H 1 0, : H u L 0, : H : u L 0, : H u H 1 0,:H 0 uh u H d hen i is shown on p 86 in he ex, ha for any u in H 1 0, : H, 1. here exiss û in C0, : H such ha u û a.e., in (0,). ii) for all s, 0,, u us 0 u zdz 3. iii) u C0,:H Cu H 1 0,:H he argumens are very close o hose ha show ha funcions in H 1 a,b are coninuous. For Hilber spaces, H 1 and H wih H 1 coninuously included in H i is easy o show ha In paricular, for he siuaion D 0, : H 1 D 0, : H V H V f L 0, : V D 0, : V implies f D 0, : V D 0, : V In he evoluion equaions we are going o sudy, he ellipic operaor L is a bounded linear map from V ono is dual, V. hen a naural choice of soluion space for he evoluion equaions is W0, u L 0, : V : u L 0, : V i.e., if u is in W0, hen u and Lu boh belong o L 0, : V. heorem - W0, is a Hilber space for he inner produc f,g W0. 0 f,gv f,g V proof- Suppose f n is a Cauchy sequence in W0,. hen he definiion of he inner produc implies 3
f n F 0 in L 0, : V : and f n G 0 in L 0, : V Now L 0, : V D 0, : V implies Bu hence However, f n F 0 in D 0, : V : and f n F 0 in D 0, : V D 0, : V D 0, : V, f n F 0 in D 0, : V. f n G 0 in L 0, : V D 0, : V, and i follows ha F 0 G 0 and F 0 W0,. Since we are sudying equaions of he form ulu for L an ellipic operaor of order, we will be paricularly ineresed in he case of W0, when V H 0 1 U (and H H 0 U, V H 1 U. I is no hard o show, using mollifiers, ha in his case C 0, : H 0 1 U is dense in W0,. hen we have he following resul, heorem 3- he naural injecion C 0, : H 0 1 U C0, : H 0 U can be exended as a coninuous injecion of Moreover, if u W0,,hen W0, C0, : H 0 U i) u, H is absoluely coninuous on 0, ii) d d u, H u,u V V and d d u,v H u,v V V v,u V V iii) here is a consan C 0 such ha u C0,:H Cu W0, proof-heurisically, his resul is nohing more han he observaion ha for u W0,, u, H u,,u, H hence d d u, H u,,u, H u,u V V where we have noed ha u L 0, : V and ha he dualiy pairing for V V is jus he Hinner produc. his can all be made rigorous as follows. Le u W0, u L 0, : H 0 1 U : u L 0, : H 1 U Exend u o, as ũ u 0 u 0 Clearly his defines a coninuous injecion of W0, ino W,. Now le a denoe a 4
smooh cu-off funcion; i.e., a C R a 1 for 0, a 0 for, and 0 a 1 for 0. Now if u C 0, : H 0 1 U, hen d/daũ, H d/daũ,,aũ, H Bu aũ,aũ V V d/daũ,s H ds aũ, H 0 from which i follows ha aũ, H aũ,s,aũ,s V V ds aũ,s aũ,s H 1 U H 1 0 U ds aũ,s H 1 U C a u,s 0 H 1 U C a u W0, aũ,s H0 1 Uds u,s H0 1 Uds where C a denoes a consan depending on he cu-off funcion, a. Now i follows ha max 0 u, H C a u W0, and, since C 0, : H 0 1 U is dense in W0,, his holds no jus for u in C 0, : H 0 1 U, bu for all u in W0,. he meaning of his resul is ha for every u in W0, here exiss a û C0, : H 0 U such ha u û a.e.,0. In paricular, for u in W0,, a soluion of an IVP, i means ha as a coninuous funcion of wih values in H 0 U, as ends o zero, u ends o u0 in H 0 U. his gives meaning o he iniial condiion, u0 u 0 H 0 U. Wih regard o boundary condiions associaed wih an IVP, any u such ha, u W0, u L 0, : H 0 1 U : u L 0, : H 1 U auomaically saisfies he homogeneous boundary condiion, 0 u u 0. An inhomogeneous boundary condiion of he form, 0 u u g, for a funcion, g, L 0, : H 1/ can be formulaed as follows. Le Gx, be such ha G L 0, : H 1 U, G L 0, : H 1 U G, G,0 in H 0 U as 0, 0 G g, a.e., 0. hen u L 0, : H 1 U saisfies he boundary condiion 0 u g, if 5
u, G, H 0 1 U,a.e., 0 ; i.e., ug u L 0, : H 0 1 U : u L 0, : H 1 U. I is worh noing ha here is no condiion of compaibiliy required beween u 0 and G,0. While he condiions u, G, H 0 1 U,a.e., 0 ; u, u 0, in H 0 U as 0 seem o require ha u 0 G, H 0 1 U, i is clear from he fac ha u 0 H 0 U ha his condiion has no meaning. In fac, if h is any funcion such ha and h u L 0, : H 0 1 U : u L 0, : H 1 U h,0 G,0, hen replacing G in he IVP wih Gh would have no effec on he boundary condiion; i.e., he value of G a 0 is no relevan o he formulaion of he IVP. For example, consider he IVP in 1-d, ux, xx ux, 0, 0 x 1, 0, ux,0 1 0 x 1, u0, u1, 0, 0. Alhough he incompaibiliy beween he iniial and boundary condiions prevens he exisence of a soluion which is coninuous on he closure of U, i does no preclude he exisence of a soluion in W0,. his soluion is hen coninuous as a funcion of wih values in H 0 U, which means ha as ends o zero, ux, ends o u 0 x 1, in he norm of H 0 U. A he same ime, for each posiive, ux, is valued in L 0, : H 0 1 U, which means ha 0 u, 0; i.e., he boundary condiion is saisfied. ha his implies no conradicion a 0,0 and 1,0 is a resul of he fac ha he funcions involved are no poinwise defined bu are defined only up o ses of measure zero. 6