Te Hille-Yosida Teorem We ave seen a wen e absrac IVP is uniquely solvable en e soluion operaor defines a semigroup of bounded operaors. We ave no ye discussed e condiions under wic e IVP is uniquely solvable. However, i is clear a S() is some sor of generalized version of epa were A is an unbounded operaor. To see is connecion beween S() and e eponenial of A, consider e scalar equaion u au, u0 u 0. Ten u Su 0 e a u 0. In is simple siuaion, we ave also i.e., ûs Lu 0 e s e a u 0 d u 0 sa ; ûs 0 e s Su 0 d sa 1 u 0 1 s u 0. We will see is resul appear again in a more general seing. An eample more general an is is provided by e sysem of linear ODE s X AX X 0 X 0, were A denoes an n by n symmeric mari of consans. Ten e soluion o is sysem is given by X n k1 X 0,V k e k V k were k,v k denoe e eigenvalues and normalized eigenvecors of A. Ten X n k1 n k1 n k1 X 0,V k X 0,V k X 0,V k k m m0 m! V n k k1 m m0 m! Am V n k k1 e A V n k e A k1 X 0,V k X 0,V k m m0 m! k m V k A m m0 m! X 0,V k V k e A X 0. We observe a e A A m and, in fac, wen A is a bounded linear operaor on a m0 m! Hilber space H (as is e case in is eample for H R 1 ) en we can epec a S lim M M m0 For any fied value of, we ave S M S N LH M mn A m m! m m! A m LH. lim M S M. 0 as M,N so e meaning of S lim M S M is o be undersood as a limi in e complee normed V k 1
linear space, LH. Clearly e limi S saisfies S0 I and S SS since ese equaliies old for every S M. Also, an easy calculaion wi e series sows a and S I LH A LH e A L S I A LH A LH S I LH. We say, in is case, a e semigroup S is uniformly coninuous on H. Of course, all is is rue under e assumpion a A is bounded. On e oer and wen A is unbounded, since DA n1 DA n, i may be a e domain in e infinie sum srinks o zero. We could make use of e fac a Su 0 solves o wrie u Au, u0 u 0, u u Au IAu from wic we find u I n A n u 0 n, n 1,2,.... Since we are again ieraing an unbounded operaor, e difficuly of e srinking domain as no disappeared. However, if we wrie en u u Au or IAu u, u IA 1 u and u I n A n u 0 so in is case we are ieraing a bounded operaor, IA 1. Te poin of is discussion is jus a i is plausible a we will be able o find a meaningful definiion for e A even wen A is unbounded bu we mus be careful ow we do i. Of course, ere mus also be some resricions on A. We will now sae and prove a eorem giving a se of necessary and sufficien condiions on A in order a A generaes a C 0 semigroup. Teorem (Hille-Yosida) Te following saemens are equivalen: 1. A : D A H generaes a C 0 semigroup of conracions on H 2. a) A is closed and densely defined b) 0 A : D A H is one o one and ono wi A 1 LH 1 Proof- We already sowed in a previous lemma a if B A, generaes a C 0 semigroup, en B A is closed and densely defined. We also found a 2
i Su 0 u 0 0 S Bu0 d u 0 D A D B ii S 0 B S d H. Now, for any 0, T e S is a C 0 semigroup of conracions a is generaed by B I, D BI D B. Apply e resuls i,ii o T o ge Tu 0 u 0 e Su 0 u 0 0 e SB Iu 0 d u 0 D A D B and T e S 0 e B I S d H. Now le and use e fac a A B is closed o conclude u 0 0 e SIBu 0 d u 0 D A D B 0 IB e S d IB 0 e S d H. Ta is, u 0 D A u 0 0 e SIAu 0 d i.e., IA is 1-1 H IA 0 e S d IA z, z D A i.e., IA is ono Finally, implies IA 1 0 e S d IA 1 H 0 e d S H 0 e d H 1 H Tis proves a 2)a),b) are necessary condiions if A is o generae a C 0 -semigroup of conracions. Noe a and IA 1 0 e S d LS, IA 1 H 1 H wic generalizes e analogous resul observed earlier for e scalar equaion. Now we suppose A saisfies 2)a),b) and we will sow A generaes a C 0 -semigroup of conracions, S(). We will accomplis is by approimaing A by a bounded linear operaor A and sowing A LH A A in H as D A e A S in LH as H. Now if A saisfies 2a) and 2b) en IA : D A H is bijecive for 0 so a IA 1 eiss. Ten we may define A : AIA 1 a bounded operaor wic will be sown o be an approimaion o A. Lemma 1 Under e condiions 2a) and 2b), e operaor A LH, 0 and 3
a. A I 2 IA 1 b. A H A H D A c. A A in H as D A Proof of lemma- For D A wrie AIA 1 I IA A 2 A 2. i.e., A I z 2 IA 1 z z H or A z z 2 IA 1 z z H. For D A, 2b) implies A H AIA 1 A H H. If we combine ese wo resuls, we ge IA 1 zz 1 H A z H 1 A z H z D A from wic i follows a IA 1 z z in H as z D A. Bu D A and IA 1 is uniformly bounded on H by 2b) ence, by coninuiy, IA 1 z z in H as z H. Ten A IA 1 A A in H as D B. is dense in H Since A LH, 0 we can define S e A n n A n0 n! for 0, 0. Lemma 2 Under e condiions 2a) and 2b), for eac 0, S : 0 is a srongly coninuous semigroup of conracions on H wi generaor equal o A. For eac D A S converges in H as. Moreover, e convergence is uniform in on all bounded inervals 0,T. Proof of lemma- Lemma 1a) implies S e A e e 2 A 1 resul of ypoesis 2b) S LH e e 2 A 1 LH e e 1. and en we ave as a Tus S is a srongly coninuous semigroup of conracions on H. Also and Ten d S d A S S S 0 d S ds ss s ds S 0 ss sa A ds. S S H A A H 0,, 0, D A. 4
i.e., S is a Caucy sequence in H, uniformly in 0,T. To complee e proof of e eorem, observe a eac S is a conracion and D A dense in H, so e limi S S in H as, 0 eends o all in H, and olds uniformly in 0,T. Since every S is a srongly coninuous conracion, clearly S LH is a conracion and in addiion, since e convergence is uniform in on bounded inervals, 0,T, i follows a S is srongly coninuous. Te semigroup ideniy also olds since S S S S ec. Ten S is a srongly coninuous semigroup of conracions on H. Now for D A and 0, and as, is becomes S 0 S A d 0, S 0 SAd and is implies a e generaor B of S() is an eension of A. Bu we ave assumed a IA is ono and since B is e generaor of S(), IB is one o one. Ten IBD A IAD A H, wic is o say, IB 1 H D A or B A. Now we can prove: Teorem- Eisence and Uniqueness for e IVP Suppose A generaes S, a srongly coninuous semigroup of conracions on H. Ten for any u 0 D A, u Su 0 saisfies i u C 0 0,; H C 1 0,; H, ii u Au 0, 0, iii u0 u 0 and u is unique. Proof- We ave already sown a for S, a srongly coninuous semigroup and u 0 D A, Su 0 saisfies i) and iii), and moreover, d Su d 0 BSu 0 ASu 0 Te Hille-Yosida eorem sows a if A en u 0 D A generaes a srongly coninuous semigroup, is or Tis implies IA 1 1 H H 0, H, 2 z 2 2 H IAz H 0, z D A. 2Az,z H A z H 2 0, z D A 5
or Az,z H 0 z D A. Ten A is accreive and u Su 0 is unique. Corollary- Suppose A generaes S, a srongly coninuous semigroup of conracions on H. Ten for any u 0 D A,and every f C 1 0,T; H, e unique soluion of u Au f, 0 T, u0 u 0, is given by u Su 0 0 S sfsds. Proof- Le g 0 S sfsds. Ten g0 0 and g g 0 S sfsds 0 S sfsds Dividing by 0 and leing 0, we find 0 Sf ds0 Sf ds 0 Sf f ds Sf ds g 0 Sf dssf0, wic sows a g is differeniable. Bu, and, as 0, g g 1 0 S I S I S sfsds 0 S sfsds 0 S sfsds 1 S sfsds g 1 S sfsds g g g and 1 S sfsds S0 f f. Since ese limis eis, i follows a e limi lim 0 S I g eiss and equals Ag. Ten we conclude g f Ag, and g saifies g Ag f, 0,T and g0 0. Evidenly, u Su 0 g solves e inomogeneous IVP. Tis soluion is unique since if ere are wo suc soluions, eir difference saisfies e IVP wi f 0, u 0 0, and en is difference is zero since A is accreive. 6
Eamples- 1. Consider e Banac space X L 1 0, wi A d d, D A u X : Au X.Tis corresponds o solving e following iniial value problem, u Au u, u, 0, u,0 u 0 D A Noe a if u and u Au bo belong o X, en u 0 as. Ten i follows a for 0, and u D A IAu u u 0 implies u 0; i.e., u Ce and u 0 as if and only if C 0. Ten IA 1 eiss. In fac, IAu v u Ce 0 e y vy dy, were u D A if C 0 e y vy dy. Ten we ave, u e y vy dy 0 e z v z dz IA 1 v. Noe a IA 1 v X e y 0 vy dy d 0 e y vy dyd y 0 e y 0 vy d dy 1 0 1e y vy dy 1 v X If we ake A o be a closed eension of e indicaed operaor en we ave an operaor a saisfies e ypoeses of e Hille Yosida eorem and i follows a -A generaes a C 0 semigroup of conracions on X. Noe a e equaion a e boom of page 3 implies IA 1 v 0 e v d 0 e Sv d, from wic i follows a for, 0, Sv v for v X. Ten e soluion of is given by speed one. u Au u, u, 0, u,0 u 0 D A u, Su 0 u 0. Tis is a wave ravelling from rig o lef wi 2. Consider e Banac space X L 1 0, wi A d d, D A u X : Au X, u0 0.Tis corresponds o solving e following iniial-boundary value problem, u Au u, u, 0, u,0 u 0 D A u0, 0, 0, Ten i follows a for 0, and u D A IAu u u 0 implies u 0 7
since u Ce and u0 0 implies C 0. Ten IA 1 eiss. In fac, IAu v, u0 0 u e y 0 vy dy IA 1 v. Ten and IA 1 v X e y 0 0 vy dy d 1 v X IA 1 v e y 0 vy dy e 0 Sv d. Bu 0 e y vy dy 0 e v d ence 0 e v d 0 e Sv d. Ten Sv v if 0 if 0, and e soluion of e iniial boundary value problem is given by u, Su 0 u 0 if 0 0 if Tis is a wave ravelling from lef o rig wi speed one. 8