Well-posedness of the generalized Proudman-Johnson equation without viscosity. Hisashi Okamoto RIMS, Kyoto University

Transcription

1 Well-posedness o he generalized Prodman-Johnson eqaion wiho viscosiy Hisashi Okamoo RIMS, Kyoo Universiy okamoo@krims.kyoo-.ac.jp

2 Generalized Prodman-Johnson eqaion Proposed in by Zh and O. in order o measre he balance o he convecion and sreching erms. + a convecion sreching viscosiy < <, <. a is a parameer D Navier-Sokes + Prodman-Johnson eq. ( ) ( 6) Riabochinski ( 4) ν ( (, ), y (, )) a

3 Generalized Prodman-Johnson eqaion Why his eqaion is ineresing o me? ω, ω + ω a ω C. 3D voriciy eqaions. νω ω crl, ω + ( ) ω ( ω ) νδω 3D Navier-Sokes is ormidable o me, b, D analoge cold be solved, I hoped. However,

4 Thogh simple, i conains some known eqaions as pariclar members. a -(m-3)/(m-), aisymmeric eac solions o he Navier-Sokes eqaions in R m. (Zh & O. Taiwanese J. Mah. ) (a or 3D Eler) a (m) Prodman-Johnson eqaion ( 4, 6) a-, ν. Hner-Saon eqaion ( 9) a-3 Brgers eqaion ( 4) a ν + (, ) d.

5 The Hner-Saon eqaion is a model appearing in he nemaic liqid crysal heory. SIAM J. Appl. Mah. (99) + + ( ). (known o be inegrable) By diereniaion + +

6 The Brgers eqaion + ν Diereniae + + ( ) ν Diereniae once more ν

7 My goal: To deermine wheher blow-p occrs or no, depending on he parameer a and he iniial daa. Wha is epeced is: global eisence or small a and blowp or large a. ω, ω + ω a ω convecion sreching viscosiy Sreching is a case o blow-p, viscosiy sppresses blow-p, and convecion is neral. Are hese herisic saemens really sbsaniaed? A lile srprise: convecion erm isn a bysander. I sppresses blow-p: O & Ohkiani, J. Phys. Soc. Japan, 5. For he sake o simpliciy, we consider in << wih periodic bondary condiion. νω

8 Smmary o resls in he case o ν >. I -3 a, no blow-p occrs. Every solion ends o zero. X. Chen & O., Proc. Japan Acad., () I a < -3 or < a, nmerical eperimens srongly sgges ha: large solions blow p small solions decay o zero.

9 Nmerical eperimens (Zh & O. Taiwanese J. Mah. ) a is a hreshold.

10

11 The limi as a + a ν redeine a and le end o ininiy. a ( ) ν + γ ( ). a + a ν, ν + (, ) d. a ν a

12 Blow-p occrs in ν + (, ) d. Large solions blow-p and small solions eiss and decay o zero. Bdd e al. ( 93, SIAM J. Appl. Mah.), O.& Zh ( ) B he asympoic behavior as approach he blow-p ime is qie dieren. ν +

13 Bdd, Dold & Sar ( 93), Zh &O. ( ). ), ( + d ν ), ( ), ( lim ) ( ), ( lim, ), ( lim ) (, ), ( + y y y d d T T T

14 I ν, Theorem ( X. Chen & O., 3, J. Mah. Sci. Univ. Tokyo). Blow p i { ; (,) ma (, ) } < ½ ), ( ), ( ), ( ), ( d dy y

15 I wan o know a proo or blow-p when ν>, - < a < -3, < a <.

16 The case o ν. We have ragmenal knowledge only. Blow-ps occr i a < - (Zh & O.) No blow-p or a (Zh & O.) Blow-ps occr i a (Childress & ohers) Blow-ps occr i a -3 (Brgers, shock wave) Blow-ps occr i a- (Hner & Saon)

17 My repor oday Blow-p or - < a < -. (Remember ha he solions eis globally in his region i ν >. Viscosiy helps global eisence.) Global eisence or - a < & smooh iniial daa. Sel-similar, non-smooh blow-p solions eis or - < a <. So ar, I have no conclsion in he case o < a.

18 A remark on nmerical eperimens In he case o ν, (Eler), nmerical eperimens are someimes (b no oen) misleading. + (a, 3D Eler) Rigoros analysis is necessary

19 Saring poin: local eisence heorem Wih a help o Kao-Lai heorem (J. Fnc. Anal. 84), ω, G( ω ), ω + ω a ω νω Theorem (Zh & O. ). For all ω ω() L here eis T and a niqe solion in < T. C([, T]; L (,)) C ([, T]; H ω() (,) / R, A priori bond or is enogh or global eisence (,))

20 Analysis or global eisence/blow-p proceeds in dieren ways in dieren philosophy in - < a < -, - a <-, - a <, a < The case o - < a < - is seled in Zh & O., Taiwanese J. Mah. (). φ( ) d d φ( ) (, ) bφ( ) 3 d

21 Smmary o he resls. a<-3 a-3 a- a <a < ν ν???

22 - a < -. Follows he recipe o Hner & Saon ( 9) Use he Lagrangian coordinaes X (, X(, ξ )), X(, ξ) ξ, ( ξ ) Deine V (, ξ ) X ξ (, ξ ). VV ( V) I( ) V, I ) V ends o -. V ( dξ V Global weak solion in he case o a - (Bressan & Consanin 5).

23 Blow-p occrs boh in - < a < - and in - a < -, b Asympoic behavior is qie dieren. ( ) blow p. (- < a < -) L ( ) is bonded. blows p. L L (- a < -) ( )

24 - a <. Follows he recipe o Chen &O. Proc. Japan Acad., () Deine Φ( ) / a Invarian d d Φ( (, )) d Φ ( )[ + a ] d [ Φ( ) + a Φ ( )] d. Bondedness o / a (, ) d, (, ) d

25 - a <. Conined. ( ) c gives s d d d d + a ν (, ) ( + ) d a d + (, ) d c(a ) (, ) d

26 a <. Follows he recipe o Chen &O. Proc. Japan Acad., () Deine Then d d Φ( ) /( a ) ( ( < < ) ) Φ ( ) d a Φ ( ) d (, ) d is bonded. is bonded.

27 Non-smooh, sel-similar blow-p solions when - < a < + F ( ) (, ) T F + FF af F. Nonrivial solion eiss or all - < a < +.

28 Some proiles Periodic, b no smooh. a a.5

29 I < a, we epec blow-p occrs even or smooh iniial daa. a.5 a -.5

30 Conclsion. Inviscid generalized Prodman-Johnson eqaion is analyzed. Ecep or he case o < a <, global eisence/blow-p are deermined depending on a. Smooh iniial daa give s global solions or - < a <. B non-smooh blow-p solions co-eis. For < a, even smooh iniial daa are epeced o lead o blow-p.

31 Crren Sas a-3 a- a < ν?? ν Sel-similar blow-p Type o blow-p discree poins? everywhere?