The Navier-Stokes equations with spatially nondecaying data II

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1 The Naier-Sokes equaions wih spaially nondecaying daa II Yoshikazu Giga Uniersiy of Tokyo Winer

2 Table of lecures I. The Naier-Sokes equaions wih -daa II. The Sokes seigroup in space of bounded funcions III. Geoeric regulariy crieria and he Liouille ype heores 2

3 II. The Sokes seigroup in space of bounded funcions Join work wih Ken Abe (U. Tokyo 3

4 Conens 1. Inroducion Probles and Resuls 2. A priori esiaes and blow-up arguens 3. A priori esiaes for haronic pressure 4

5 1. Inroducion Sokes syse: B.C. I.C. q 0 di 0 in (0 T on in Here is a uniforly -doain in : unknown elociy field q : unknown pressure field : a gien iniial elociy 0 C 3 R n ( n 2 5

6 Proble. Is he soluion operaor (called he Sokes seigroup S( : ( 0 is an analyic seigroup in -ype spaces? L In oher words is here C 0 s.. d d S( f X C f (01 X L where X is an -ype Banach space. Analyiciy is a noion of regularizing effec appeared in parabolic probles in an absrac leel. f X 6

7 Definiion of analyiciy Definiion 1 (seigroup. Le beafaily of bounded linear operaors in a Banach space.in oher words. We say ha is a seigroup in if (i (seigroup propery for (ii (srong coninuiy in as for all (iii (non degeneracy for all iplies. (i (boundedness for 7

8 Definiion 2 (non analyic seigroup. Le be a seigroup in. We say ha is analyic if such ha d d C S( (01. op See a book [ABHN] W. Arend Ch. Bay M. Hieber F. Neubrander Vecor-alued Laplace ransfors and Cauchy probles Birkhäuser (2011 8

9 Definiion 3 ( -seigroup. A seigroup is called -seigroup if as for all. Reark. The nae of analyiciy ses fro he fac ha can be exended as a holoorphic funcion o a secorial region of i.e. wih soe. 9

10 A siple exaple Hea seigroup (Gauss-Weiersrass seigroup 10

11 Proposiion 1. The faily is a non -analyic seigroup in (and also in bu a -analyic seigroup in (and also in Here : uniforly coninuous -closure of 11

12 Spaces for diergence free ecor fields he space of all sooh solenoidal ecor fields wih copac suppor -closure of in on If is bounded. (Mareoni 09 -closure of 12

13 More spaces Helholz decoposiion ( bounded -doain. for all e.g. Fujiwara-Morioo 79 Galdi s book 11 Here for all. 13

14 Typical ain resuls Theore 1. (K. Abe Y. G. Aca Mah o appear Le be a bounded -doain in. Then he Sokes seigroup is a -analyic seigroup in. I can be regarded as a non seigroup in. C C0 ( ( BUC ( C 0 Reark. Whole space case is reduced o he hea seigroup. This ype of analyiciy resul had been only known for half space where he soluion is wrien explicily (Desch-Hieber- Prüss 01 Solonniko 03 3 L C 0 ( 14

15 Analyiciy of seigroup > regularizing effec Known resul for ellipic operaors (i 2 nd order operaor on (one di: K. Yosida 66 (ii 2 nd order ellipic operaor K. Masuda book in 75 heory cuoff procedure for resolen (iii higher order H. B. Sewar Masuda-Sewar ehod (i degenerae + ixed B. C. K. Taira 04 See also: P. Acquisapace B. Terrani (1987 A. Lunardi (1995 Book. More recen. nonsooh coefficien / nonsooh doain Heck-Hieber-Saarakidis (2010 VMO coeff. higher order Arend-Schaezle ( nd order Lipschiz doain 15

16 Sokes proble in (= -closure of 2 L (i : easy since he Sokes operaor is nonnegaie self-adljoin. p (ii L : V. A. Solonniko '77 Y. G. 81 (bdd doain (ax regulariy / resolen esiae H. Abels Y. Terasawa 09 (ariable coefficien bdd exerior ben half space. L r (iii space L p p L 2 ~ L r L r 2 2 L r 2 L r C c W. Farwig H. Kozono and H. Sohr General uniforiy C -doain / All excep Solonniko appeals o he resolen esiae 16

17 Theore 2. (K. Abe Y. G. 12 C 3 Le be an -exerior doain in. Then he Sokes seigroup is a -analyic seigroup in and exends o a non - analyic seigroup in C 0 C0 ( L ( C BUC (. I can be regarded as a -analyic seigroup in. 0 C 0 C0 ( BUC ( f C0 ( x Noe ha for an unbounded doain is sricly saller han because iplies f ( x 0 as. 17

18 18 2. A priori esiaes and blow-up arguens Theore 3 (A priori esiae. Le be a bounded doain wih -boundary. There exiss and such ha for soluion we hae (This esiae iplies Theore 1 3 C 0 0 T C r L ( q ( ( ( ( ( ( ( 2 2 1/ x q x x x x x q N. ( ( ( sup c T C C q N A key a priori esiae

19 Idea of he proof a blow-up arguen a key obseraion (Haronic pressure gradien esiae by elociy gradien sup d ( x q( x C L ( x d ( x dis( x. ( 19

20 A blow-up arguen (Arguen by conradicion Suppose ha he priori esiae were false for any choice of T and C. Then here would exis a soluion ( 0 q wih 0 and a sequence 0 such ha N( There is q ( N( q (. 0 ( 0 M such ha / 2 M sup 0 N( q 20 (.

21 21 We noralize by diiding o obsere wih We rescale around a poin saisfying ( q M. M q q M ~ ~ 4. 1 ( ~ ~ ( / x q N ~ ~ ( q x 0 ~ 2 1/ ( ~ ( ~ 1 ( ~ ( ~ sup 0 0 q N q N

22 22 soles he equaion in a rescaled space-ie doain ( is expanding (01] Blow-up sequence ( ( ~ ( ( ~ ( 2 1/ 2 1/ 2 1/ p u x x q x p x x x u

23 Basic sraegy A. Copacness: Proe ha ( u p (subsequenly conerges o ( u p srong enough so ha N( u p(01 1/ 4. B. Uniqueness: The blow-up lii ( u p soles he Sokes proble wih zero iniial daa so if he soluion is unique i us be u 0 p 0 which conradics. N( u p(01 1/ M.-H. Giga Y. Giga J. Saal Nonlinear PDEs

24 Blow-up arguen E. De Giorgi (1961 regulariy of a inial surface; popular in nonlinear probles B.Gidas J. Spruck 81 a priori bound for seilinear ellipic probles Y. Giga 86 Firs applicaion o a priori bound for parabolic proble (Giga Kohn 87 P. Quiner Ph. Souple 07 Superlinear parabolic probles 24

25 Less for Naier-Sokes equaions Koch Nadirashili Seregin Šerák 09 (nonexisence of ype I axisyeric singulariy Miura Y. G. 11 (nonexisence of ype I singulariy haing coninuous oriciy direcion In our case he proble is linear so we rescale he physical space and elociy in an unrelaed way. 25

26 26 Wha esiae is aailable for? The pressure gradiens esiae iplies Se ( u p Copacness 1. ( ( 2 1/ x p x d / (01 ( 1 ( ( sup u p u N p u N. ( / 2 1/ x d d d c

27 Case 1. li c ( R n Case 2. li c ( half space Case 2 is ore inoled. Een case 1 i is nonriial because he proble canno be localized copleely. 27

28 Ex. Inerior regulariy of he hea eq u u u M If is bdd by hen u u C k M k 0 in B ( x ( R 0. in B ( x ( R / 40. R / 2 This is no longer rue for he Sokes equaions een if we assue R 2 2 sup 0T N ( q ( (. 28

29 Ex. ( x g( p( x g'( x g C 1 [0 Eidenly is fulfilled. Howeer is no Hölder if is no Hölder. ( g' 29

30 Lea 1 (Conrol of pressure gradien. If he haronic pressure gradien esiae d holds hen such ha [ d q ( x q] L ( L ( M M Q ( T M inarian under dilaion and ranslaion. wih. The consan is C M 2 sup N( q ( Q (1/ 0T 30

31 Uniqueness Lea 2. (Solonniko 03 C syse in If wih sup N ( q 0 0T soles he Sokes (weak in L 1/ 2 and if sup x q hen 21 q C n n R (0 T C R (0 T R n ( 0 T n R 0T 0 q 0. n 0 on x 0. 0 n 31

32 Exaple of nonriial soluions Wihou decay esiae for q his is no rue. i i i i g 0 i ( on in R R n n (0 T (0 T ( 1 i n 1 i i 0 ( x (independen of '. 0 n x 1 n1 Then ( 0 and p ( x g( x' soles he Sokes syse in a half space wih zero iniial daa and zero boundary daa. Here. x' ( x1 xn 10 32

33 3. A priori esiae for haronic pressure gradien Consider Take diergence o ge since wih Equaions for he pressure in in. Take inner produc : (uni exerior noral and use o ge on 33

34 Lea 3. If hen wih In hree diensional case where. In any case is a angen ecor field. 34

35 Neuann proble The pressure soles (NP in Enough o proe ha for all angenial ecor field. 35

36 Sricly adissible doain Definiion 4 (Weak soluion of (NP. (Ken Abe Y. G. 12 Le be a doain in wih boundary. We call a weak soluion of (NP for wih if wih fulfills for all on. saisfying 36

37 Definiion 5 (Sricly adissible doain. Le be a uniforly doain. We say ha is sricly adissible if here is a consan such ha holds for all weak soluion of (NP for angenial ecor fields. Noe ha sricly adissibiliy iplies adissibiliy defined below. 37

38 38 Le be he Helholz projecion and. Applying o he Sokes equaion o ge Here for. Adissible doain ( ~ ( ~ : r r L L P Q Q[ ]. q P I Q 2 2 ~ ~ L L L L L L r r r r 2 r

39 Adissible doain (coninued Definiion 6. (Ken Abe Y. G. Aca Mah o appear Le be a uniforly C 1 -doain. We say ha is adissible if here exiss r n and a consan such ha C C hold for all arix alue saisfy sup d ( x Q[ and for all i j l 1 n. f ]( x C ~ r f ( f L ( r f 0 j j ij f l f ( ij f ij f L C j f ( 1 ( il 39

40 Reark. (i This is a propery of he soluion of he Neuann proble for he Laplace operaor. In fac q Q[ f ] is forally equialen o q di ( f in q/ n n ( f on. Under he aboe condiion for we see ha is haronic in since q f di ( f f f 0. i j i j ij j j ii 40

41 (ii The consan depends on C bu independen of dilaion ranslaion and roaion. (iii If is adissible we easily obain he pressure gradien esiae by aking. f ij j i (i I urns ou ha q / n di n ( f f. 41

42 Reark. Sricly adissibiliy iplies adissibiliy. Exaple of sricly adissible doains (a half space (b bounded doain (c exerior doain Noe ha layer doain is no sricly adissible. Consider. Conjecure: Is sricly adissible if i is NOT quasi-cylindrical? 42

43 A siple exaple half space Proposiion 2. A half space sricly adissible for. Skech of he proof: The soluion (NP is of he for is of in is he Poisson seigroup. where 43

44 Poisson seigroup Thus Clearly Moreoer... 44

45 Basic esiae and copleion of he proof This is explicily proed by esiaing he Poisson kernel. Thus This is wha we wan o proe. 45

46 Nonriial exaples Proposiion 3. A bounded doain is sricly adissible for. Skech of he proof: We shall proe his esiae by arguen by conradicion and blow-up arguen. Suppose ha he esiae holds here is a sequence of funcion such ha By noralizaion we ay assue ha 46 and.

47 Blow-up arguen We race axiu poin of Le be a axiu poin. By aking a subsequence we ay assue ha as. Case 1 This conradics uniqueness of (NP since he lii of is a nonriial soluion of (NP. Case 2 We blow up so ha disance beween and he boundary equals 1. Then we yield a nonriial soluion (NP as a lii of conradicing he 47 uniqueness of (NP in a half space.

48 Suary The Sokes seigroup is analyic in when uniforly doain is adissible. I can be exended o a analyic seigroup in when is exerior and bounded. Blow-up arguen is useful o proe esablish necessary esiae. Noe: Proof by resolen esiae is now aailable. (Ken Abe Y. G. M. Hieber 12 (Abe s presenaion I is applicable o oher boundary condiions like Naier boundary condiion. 48

49 Open probles We hae discussed regularizing effec by proing analyiciy of he Sokes seigroup. We do no know well abou large ie behaior. Proble. (1 Is bounded in ie? i.e. for all. (2 Is a bounded analyic seigroup? i.e. for all. [(1 (2 yes for a bounded doain: Abe-Giga Aca Mah] [(1 yes for an exerior doain: Mareoni 12] 49

50 Open probles (Solabiliy of he Naier-Sokes equaions Proble. (3 Do he Naier-Sokes equaions adi a local sooh soluion een if iniial daa is in or for a doain haing a boundary? [(3 yes for a half space: Solonniko 03 Bae- Jin 12] [(3 yes for a hree diensional exerior doain proided ha is Hölder and bounded: Galdi- Mareoni-Zhou 12] 50 [Ken Abe work in progress]

On the Stokes semigroup in some non-helmholtz domains

On the Stokes semigroup in some non-helmholtz domains Yoshikazu Giga University of Tokyo June 2014 Joint work with Ken Abe (Nagoya U.), Katharina Schade (TU Darmstadt) and Takuya Suzuki (U. Tokyo) Contents