ナヴィエーストークス方程式に関連した爆発問題

岡本 久

協力者：卓 建宏、濱田 信輔、大木谷 耕司、坂上貴之、朱 景輝

hisashi.okamoto@gakushuin.ac.jp

1/41

• Handbook of Mathematical Analysis in Mechanics of Viscous Fluids, Editors: Giga & Novotny to appear in Springer in 2018. Includes: O., Models and special solutions of the Navier-Stokes equations

• Bae, Chae & O., Nonlinear Analysis (2017)• H. O., T. Sakajo, and M. Wunsch, On a generalization of

the Constantin-Lax-Majda equation, Nonlinearity (2008)• H.O., Well-Posedness of the Generalized Proudman-

Johnson Equation Without Viscosity, J. Math. Fluid Mech. 0nline (2007)

• K. Ohkitani and H.O., J. Phys. Soc. Japan, 74 (2005), 2737--2742.

• H.O. & J. Zhu, Taiwanese J. Math., 4 (2000), 65—103Downloadable at http://www.kurims.kyoto-u.ac.jp/~okamoto

2/41

The Navier-Stokes equations

• Equations of motion of incompressible viscous fluid. 1827, 1845

• Many unsolved problems.

ナヴィエ・ストークスの解の正則性はなぜ面白いのか？

• 乱流の理論的理解

– J. Lerayのシナリオ

– L.D. Landauのシナリオ

– Ruelle-Takenの仮説

4

The Navier-Stokes eqs.• Incompressible viscous fluid • u : velocity, p : pressure ut + (u・∇) u = ν∆u - ∇p

div u = 0 • ν : viscosity. ν = 0 ⇒ The Euler eqs.• ωt + (u ・∇) ω – (ω ・∇) u = ν∆ω

convection stretching viscousity

ω = curl(u) --- vorticityHeated

arguments Kerr, Hou, …

5/41

It is often said that

• The stretching term amplifies the vorticity;• A nonlinear convection term may be a cause

of singularities of shock wave type but it never magnifies the function;

• As far as the indefinite amplification of the vorticity, the convection term plays no positive role: it just sits and watches.

6/41

0)()( =∇−∇+ •• uωωuωt

My goal today

• To show that the convection term plays a definite role in blow-up of solutions. The convection term may prevent vorticity from blowing-up.

• If a sol. becomes very large, stretching becomes ineffective: Depletion of nonlinearity.

• A case study for this claim

7/41

Deplete blow-up by convection

8

Naive explanation• Without convection,

the higher becomes higher still.

pxxt vvv +=

9

• With convection, the highest does not necessarily become the highest.

pxxxt vvvvfv +=+ )(

10

Stretching is weakenedby the convection

(unphysical & nonlinear) convection term may help blow-up

The direction of convection is important.

11/41

Introductory example or warming-up

• The Burgers eq. ut + uux = uxx

• Global existence; (not global if uxx is missing)

→

12/41

Burgers eqn. continued

• Burgers eq. ut + uux = uxx

• v= ux (or ; u = ∫vdx) vt + uvx + v2 = vxx (no blow-up)

convection stretching viscosity

vt + v2 = vxx (blow-up)

vt - uvx + v2 = vxx (blow-up)13/41

A thesis • Viscosity alone is not enough to

suppress the blow-up. • But blow-up can be prevented by

viscosity and/or an appropriatenonlinear convection.

14/41

15/41

Almost all solutions blow up in finite time.

Example①Constantin-Lax-Majda (‘85)

ωωωHu

u

x

xt

==− 0

xx cos)(0 =ωsdsfsHf )(2

0 2cot

21)( ∫

−

=π σ

πσ

16/41

De Gregorio’s equation

• S. De Gregorio, J. Stat. Phys. ‘90

• Constantin-Lax-Majda eq. + convection term

• is a steady-state.

ωωωω

Huuu

x

xxt

==−+ 0

xcos=ω

0)()( =∇•−∇•+ uu ωωωt

ωx is bounded.

ωxx grows rapidly.

17/41

Another exmaple

18/41

19

Local existence for De Gregorio

• Proof. Application of T. Kato’s theory of nonlinear evolution equations

)()(maxFurther,

)).(];,0([solution asuch that 0 then ),()( If

110

0,00

110

011

0

HHTtTCCt

SHTCTSHx

Theorem

ωω

ω

ω

=≤

∈

>∃∈

≤≤

20/41

A sufficient condition for global existence

• An analogue of a theorem by Beale, Kato, & Majda ’84.

Since , boundedness global existence.

]T[0,in existssolution )(0

δω +⇒∞<∞∫ dttH

L

T

2LxcH ωω ≤

∞ ⇒

A generalization

• a is an artificial parameter.

Theorem(?)Blow-up if .Global existence otherwise.

ωωωω

Huuau

x

xxt

==−+ 0

11 <≤− a

21/41

If viscosity is present,

22

Example ②• 2D Euler ; incompressible in viscid

ωt + u・∇ω = 0ω = curl u = vx – uy ; u = (u,v)

χ = ∇ωχt + (u・∇) χ – (χ・∇)u = 0

convection stretching

χ = - △u23/41

The convection term is now deleted.

χt – (χ・∇)u = 0 in R2

u = (-△)-1 χ

oru = P(-△)-1 χ

L2-norm of χ

24/41

∫ ∞t

dxs0

)(χ

25/41

(-∆)1/2ω ∼ |χ|t=0

t=3, Euler t=3, model

26

Example ③The Proudman-Johnson equation

• Derived from 2D Navier-Stokes

• ftxx + ffxxx – fxfxx = νfxxxx( 0 < t , -1 < x < 1 )

• f(t,±1) = 0, fx(t,±1) = 0• fxx(t,x) = φ(x)• u = ( f(t,x), -yfx(t,x) )

(unbounded solution of NS )27/41

Global existence or finite time blow-up?

• ftxx + ffxxx – fxfxx = νfxxxx

ω = fxx, ωt + fωx – fxω = νωxx

• Had been difficult to judge28/41

Global existence was proved by Xinfu Chen

THEOREM. Assume that ν > 0. For any initial data in L2(-1,1), a solution exists uniquely for all t and tends to zero as t →∞

Xinfu Chen and O., Proc. Japan Acad., 2000.

29/41

Effect of convection term• ftxx + ffxxx – fxfxx = νfxxxx

ω = fxx, ωt + fωx – fxω = νωxx ftxx – fxfxx = νfxxxx

• ftx – ½ fx2 = νfxxx + β ; u ≡ ½ fx

ut = νuxx + u2 + β30/41

blow-up occurs. A proper convection term prevents

solutions from blowing-up.

R/: ,

0)1,( ,0),(

)11,0(

222

1

1

2

LLPPuuu

tudxxtu

xtuuu

xxt

xxt

→+=

=±=

<<−<++=

∫−

β

31/41

Budd, Dold & Stuart (’93), Zhu &O. (’00)

• ∃ x0

.),(1

0

22 ∫−+= dxxtuuuu xxt ν

0),(),(lim

)(),(lim,),(lim

0

0

0

=

≠−∞=+∞=

→

→

→

xtuytu

xyytuxtu

Tt

Tt

Tt

　　

32/41

Blow-up with or without the projection

ut = uxx + u2 ut = uxx + Pu2

33/41

Example ④ ; Generalized Proudman-Johnson equation

• A model:

ftxx + ffxxx – afxfxx = νfxxxx( 0 < t , -1 < x < 1 )

f(t,±1) = 0, fx(t,±1) = 0fx(0,x) = φ(x)

xxxxtxx afff νωωωωω =−+−= ,

34/41

Though simple, it contains some known equations as particular members.

a= -(m-3)/(m-1), axisymmetric exact solutions of the Navier-Stokes equations in Rm. (Zhu & O. Taiwanese J. Math. 2000) (a=0 for 3D Euler)

a=1 (m=2) Proudman-Johnson equation (’24, ’62)a=-2, ν=0. Hunter-Saxton equation (’91)a=-3 Burgers equation (‘40)

xxxxxxxxxxtxx ffaffff ν=−+ 35/41

Xinfu Chen’s proof of global existence

• X. Chen and O., Proc. Japan Acad., vol. 78 (2002),

• periodic boundary condition.

• THEOREM. If -3 ≦ a ≦ 1, the solutions exist globally in time for all initial data.

36/41

If a < -3, or 1 < a, then …

• Global existence for small initial data. Blow-up for large initial data ---numerical evidence but no proof.

• Blow-up sets are [-1,1] for 1 < a, and discrete for a < -3. ( no proof )

37/41

Numerical experiments (Zhu & O. Taiwanese J. Math. 2000)

• a=1 is a threshold.

38/41

39

Numerical experiments

)2cos(2.0),10sin(

0

0

xgxf

ππ

==

)2sin(2.0),6sin(

0

0

xgxf

ππ

==

)8cos(5.0),16sin(

0

0

xgxf

ππ

==

40/41

If 1 < a, we expect blow-up occurs even for smooth initial data.

a = -2.5

a = 1.5

41/41

Current Status

? ??

a=-3 a=-1 a=1

0 < ν

0 = ν

Conclusion• Proudman-Johnson eqn’s well-

posedness is guaranteed by the convection term.

• Generalized P-J eqn may or may not blow up depending on the parameter a.

• De Gregorio’s equation does not admit blow-up, while Constantin-Lax-Majda eq. admit blow-up.

42/41

Conclusion continued• The regularity of sols. of 2D Euler eqn’s

seems to be maintained by the convection term;

• Convection term, if properly placed, prevents solutions from blowing up;

• These examples suggests: Blow-up of 3D Navier-Stokes or Euler eqs. is very subtle.

完: Thank you.43/41

Some thoughts on the role of the convection term in the fluid mechanical

PDEs.

Hisashi Okamoto

Gakushuin Universityhisashi.okamoto@gakushuin.ac.jp

Today’s goal• Handbook of Mathematical Analysis in

Mechanics of Viscous Fluids, Editors: Giga& Novotny to appear in Springer in 2018. Includes: O., Models and special solutions of the Navier-Stokes equations, in Handbook

• Bae, Chae & O., Nonlinear Analysis (2017)• Ohkitani & O., J. Phys. Soc. Japan, (2005)• H. O., T. Sakajo, and M. Wunsch, Nonlinearity (2008)• H. O., J. Math. Fluid Mech. 0nline (2007)• K. Ohkitani and H.O., J. Phys. Soc. Japan, 74 (2005),

2737--2742.• H.O. & J. Zhu, Taiwanese J. Math., 4 (2000), 65—103

A motive: 3D Navier-Stokes: A bad problem. Turbulence is a bad Problem!? How about the NS itself?

Try simpler models for blow-up:

❂ Proudman-Johnson eq. (special sol.) ❂ Constantin-Lax-Majda (model)❂ generalized CLM (model)❂ Surface QG, & many others.❂ Model equations for water waves.

Navier-Stokes is nonlinear & nonlocal

• Navier-Stokes eqns. are integro-differential eqns. rather than differential eqns.

ωuωωuω ∆=∇−∇+ •• ν)()(t

ξξξξ

πdt

xxxt ),(

|-|-

41),(

SavartBiot ,curl)(

3

-1

ωu

ωu

×−

=

−=

∫∫∫

Therefore models must be nonlinear & nonlocal.

viscosityconvection stretching

When I began my career as a professional mathematician, there was a folklore:

• The vorticity is increased by the stretching term. Convection term does not increase vorticity, although the vorticity is rearranged by that.

• As far as global well-posedness is concerned, the convection term is neutral.

Can these loose ``propositions’’ be phrased mathematically?

The Proudman-Johnson equation. ‘62

BC periodic)20 ,0(

unknown:),(

πν

<<<=−+

=

xtuuuuuu

xtuu

xxxxxxxxxxtxx

Hiemenz’s ansatz Dinglers Journal 1911Die Grenzschicht an einem in den gleichförmigen Flüssigkeitsstrom eingetauchten geraden KreiszylinderHe obtained a steady-state.

0div))(),((

=⇒−=u

u xyuxu x

xxxxxxxxxx uuuuu ν=−

Laminar boundary layer

The Proudman-Johnson equation. ‘62 • Derived from 2D Navier-Stokes

(unbounded solution of NS ) )),(),,(( xtyuxtu x−=u

IC)(),0( & BC periodic

)20 ,0(

xxu

xt

xx

xxxxxxxxxxtxx uuuuuu

φ

π

ν

−=

<<<

=−+

Equivalent re-writing

xxxxt

xx

xxxxxxxxxxtxx

uuu

uuuuuu

ωωωωω

ν

ν

=−+−=

=−+

Savart-Biot&)()( ωuωωuω ∆=∇−∇+ •• νt

Generalized Proudman-Johnson equation

A model: ( ))(),0(

,

0

1

22

xx

uauudxd

xxxxt

ωω

ωωωωω ν

=

−==−+−

1. a = -(m-3)/(m-1), axisymmetric exact solutions of the Navier-Stokes eqns in Rm.

2. a = 1 (m=2) Proudman-Johnson eqn3. a = -2, ν = 0. Hunter-Saxton equation (’91)4. a = -3 the Burgers equation (‘46)

xxxxxxxxxxtxxxxxt uuuuuuuuuudxd

νν =++⇒=+ 3 2

2

Zhu & O. Taiwanese J. Math., vol. 4 (2000),

Order -2

Order -1

Global existence or finite time blow-up?

( ))(),0(

,

0

1

22

xx

uuudxd

xxxxt

ωω

ωνωωωω

=

−==−+−

xx

xxxxxxxxxxtxx

uuuuuuu

−==−+

ων

ωuωωuω ∆=∇−∇+ •• ν)()(t

SavartBiot ,curl)( -1 −= ωu

In 1989, a paper appeared in J. Fluid Mech.

• Finite time blow-up was predicted by numerical computation.

• For ten years, I was wondering if that is true.

( ))(

)(2

2

twwuwwtuuuuu

uuuuuu

xxxt

xxxxxxtx

xxxxxxxxxxtxx

γ

γ

ν

ν

ν

++=+

+=−+

=−+

xuw =

Nonlinear heat equation with nonlocal nonlinear convection

Max norm of uxx

ν = 0.01

∞•Lxx tu ),(

Theorem. Assume that viscosity ν > 0. For any initial data in L2(-1,1), a solution exists

uniquely for all t and tends to zero as t → ∞ .

Global existence was proved for PJ:

if homogeneous Dirichlet, Neumann, or the periodic boundary condition.

Xinfu Chen and O., Proc. Japan Acad., 2000.

Blow-up if non-homogeneous Dirichlet BC.????u(t,1)= a, uxx(t,1)= b, u(t,-1)= c, uxx(t,-1)= d

Grundy & McLaughlin (1997).

proof: a priori estimatexxxxxxxxxxtxx uuuuuu ν=−+

( )

xxxt

Vxxxxtxxx

Vxxxxxxtxxx

uuuuu

uuuuu

ζνζζν

ν

≥+≥+

=−+)(

)(2

Maximum principle

∫ ≡π2

00),( dxxtuxxx

13時56分 60

Intuitive explanation • Without convection, the higher becomes higher still.

13時56分 61

• With convection, the highest does not necessarily become the highest.

13時56分 62

Surgery on PDE

• If appendix is removed, we can live.⇒ Appendix is unnecessary as far

as life/death is concerned.

• Is the convection term an appendix, or not?

Surgery on convection term

• Remove

( )xxxxt

xxxxxxtx

xxxt

uwwwuwwuuuuu

uuuu

==++

=++

=+

ν

ν

ν

2

2

xxt www ν=+ 2

xxxt wwuww ν=+− 2

Surgery on convection term

( ) ωνωωωω1

22 ,

−−==−+

dxduuu xxxxt

)( U,2121

2

2

tbUUuU

uuu

uuuu

xxtx

xxxxtx

xxxxxxxtxx

−+==

=−

=−

ν

ν

ν

A proper convection term prevents solutions from blowing-up. (O. & J. Zhu 1999, Taiwanese J. Math., 2000Cf. Budd, Dold & Stuart (’93), )

R/: ,

BC periodic 0,),(

)11,0( ,),(21

222

1

1

1

1

22

LLPPUUU

dxxtU

xtdxxtUUUU

xxt

xxt

→+=

=

<<−<−+=

∫

∫

−

−

xxxtxx

xxxxtxx

uuu

νωωωνωωωω

=−=−+

,

Blow-up occurs

Global existenceBlow-up

Blow-up with or without the projection

ut = uxx + u2 ut = uxx + Pu2

Generalized Proudman-Johnson equation

A model:

( ))(),0(

,1

22

xx

uauudxd

xxxxtxx

φω

ωνωωωω

=

−==−+−

a = -(m-3)/(m-1), axisymmetric exact solutions of the Navier-Stokes eqns in Rm.

a = 1 (m=2) Proudman-Johnson eqn a = -2, ν=0. Hunter-Saxton equation (’91) a = -3 the Burgers equation (‘46)

xxxxxxxxxxtxxxxxt uuuuuuuuuudxd νν =++⇒=+ 3 2

2

No singularity if the convection term is dominant.

• If a is large ⇒ blow-up• If a is small ⇒ no blow-up

( ) ωνωωωω1

22 ,

−−==−+

dxduauu xxxxtxx

But how large it must be?

Converges to Hiemenz’s steady-

state

1),( −=∞tu

Conjecture

• If u0 ≦ 1 everywhere, global.

• If u0 > 1 somewhere, blow-up?

• u = Fx is bounded, but F is unbounded. 12 −=−+ xxxt uuFuu ν

0)0,(1),(

==∞

tutu

12 −+=+ uuFuu xxxt ν

∫==x

x uFFu0

,

No steady-state

Unimodal conjecture on the PJ

Kim & Okamoto, IMA J. Appl. Math. (2013)

( )

kxkFx

kxFFFFFF xxxxxxxxxxtxx

sinBCPeriodic,

sin

4−=

<<−−=−+

ππν

10sinsol,...3,2,1

<<<=∃=∀

νforxkFk

R = 5000

a(1) -1.999270373875

a(2) 0.000044444742

a(3) 0.000006247917

a(4) 0.000001776474

a(5) 0.000000693595

a(6) 0.000000325941

a(7) 0.000000173180

a(8) 0.000000100453

∑=

=N

nnxnax

1sin)()(ψ

75

--------------------------------------------------------------------------

Systeminput output

systeminput output

Linear & weakly nonlinear eqns.

76

2D stationary Navier-Stokes eqns.

kxk

u sin1

12 +

=

kxuu sin'' =+−

kxsin

unimodal

TheoremThere exists a sol. of the fol. form.

77

)()(sin)( 21 −− ++±= ROxhRxkxψ

...008888.02252...,2222.0

92

...4sin22523sin02sin

92sin)( 1

==

++×++= xxxxcxh

)( ∞→R

k = 4, R = 10000

a1 3.99887570831577a2 0.00002222222222a3 0.00000000054426a4 0.00000088889289

R92

( )kxR xxxxxxxxxxtxx sin1

+=−+ ψψψψψψ

3D flows

• C.C. Lin, Arch. Rat. Mech. Anal. (1957)• Grundy & McLaughlin, IMA J. Appl. Math. (‘99)• J. Zhu, Japan J. Indust. Appl. Math. (2000)• Ansatz

( )),(),,(),,(),( xtzgxtyfxtgxtf xx−−=u

( ) ( )( ) ( ) xxxxxxxxxxxtxx

xxxxxxxxxxxtxx

gggfggfgffgffgff

νν

=++−+=+−−+

Blow-up for 3D

(No blow-up for PJ)

19/5/12 13時56分 80

A necessary and sufficient condition is known (Constantin, Lax, & Majda 1985).

xx cos)(0 =ω

model ③Constantin-Lax-Majda

ωωωHu

u

x

xt

==− 0

xx cos)(0 =ω

Constantin-Lax-Majda & De Gregorio & Proudman-Johnson

( ))(),0(

,1

22

xx

uauudxd

xxxxtxx

φω

ωνωωωω

=

−==−+−

( ))(),0(

,2/

22

xx

uauudxd

xxxxtxx

φω

ωνωωωωβ

=

−==−+−

β= 1 & a = 1 ??? De Gregorio’s `90β= 1 & -∞ < a < 0 Blow-up Castro & Cordoba ‘09

β= 1 & a = ∞ Blow-up Constantin-Lax-Majda `85

The generalized P-J with ν=0.

)(),0(

BC periodic

)10 ,0(

0

xxxxu

xt

xxxxxxtxx uauuuu

φ−=

<<<

=−+

03 0 2

2

=++⇒=+ xxxxxxtxxxt uuuuuuuudxd

• 3D axisymmetric Euler for a = 0.

• Hunter-Saxton model for nematic liquid crystal for a = -2.

• Burgers for a = -3.

Summaryfor ν = 0

• Blow-up for -∞ < a < -1. (Remember that the solutions exist globally in this region if ν > 0. Viscosity helps global existence.)

• Global existence if -1 ≦ a < 1 & if smooth initial data.

• Self-similar, non-smooth blow-up solutions exist for -1 < a < ∞.

• So far, I have no conclusion in the case of 1 < a.

?a = -1 a = 1

Starting point: local existence theorem

• With a help of Kato & Lai’s theorem (J. Func. Anal. ’84),

• Locally well-posed if

• Global existence if

0 , =−+−= ωωωω xxtxx auuu

,/)1,0(),0( 2 RL∈•ω

,/)1,0(),0( 2 RL∈•ω

Different methods were needed for global existence/blow-up in

-∞< a < -2, -2 ≦ a <-1, -1 ≦ a < 0, 0 ≦ a < 1

• The case of -∞< a < -2 is settled in Zhu & O., Taiwanese J. Math. (2000).

32

2

1

0

2

)()(

),()(

tbtdtd

dxxtut x

φφ

φ

≥

≡ ∫

-2 ≦ a < -1. Follows the recipe of Hunter & Saxton ( ’91)

• Use the Lagrangian coordinates

• Define

• V tends to -∞.• Global weak solution in the case of a= -2 (Bressan & Constantin ‘05).

)10(,),0()),,(,( ≤≤== ξξξξ 　　　XtXtuX t

).,(),( ξξ ξ tXtV =

　,)()( 2 VtIVVV ttt −= ∫=1

0

2

)( ξdVVtI t

Blow-up occurs both in -∞< a < -2 and in -2 ≦ a< -1, but

• Asymptotic behavior is quite different.

• blow up. (-∞ < a < -2)

• is bounded. blows up.

(-2 ≦ a < -1)

2)(Lx tu

2)(Lx tu ∞Lx tu )(

-1 ≦ a < 0. Follows the recipe of Chen &O. Proc. Japan Acad., (2002)

• Define

• Invariance

• Boundedness of

ass /1|| )( −=Φ

∫

∫ ∫=Φ′+Φ=

+−Φ′=Φ

1

0

1

0

1

0

.0)]()([

])[()),((

dxuuauu

dxuauuuudxxtudtd

xxxxxxx

xxxxxxxxxx

∫∫− 1

0

1

0

/1 ),(,),( dxxtudxxtu xxa

xx 　　

-1 ≦ a < 0. Continued.

•

• gives us

ctux ≤∞

)(

xxxxxxxxxxtxx uuauuuu ν=−+

∫∫ +=1

0

21

0

2 )12(),( dxuuadxxtudtd

xxxxx

∫∫ +≤1

0

21

0

2 ),()12(),( dxxtuacdxxtudtd

xxxx

0 ≦ a < 1. Follows the recipe of Chen &O. Proc. Japan Acad., (2002)

• Define

• Then

• is bounded.

<<

=Φ−

)0(0)0(||

)()1/(1

sss

sa

0)()(1

0

1

0

2 ≤Φ′=Φ∫ ∫ dxuuadxudtd

xxxxxxxx

∫1

0),( dxxtuxxx

Non-smooth, self-similar blow-up solutions when -1 < a < +∞

Nontrivial solution exists for all -1 < a < +∞.

.0

)(),(

=′′′−′′′+′′−

=

FFaFFFtT

xFxtu

Another • 3D Navier-Stokes

exact sol.

• Nagayama and O., ’02 numerical experiment.

• Proof ???

),(),())(()(

xtfxtSfffSfffSfff xxxxxxxxxxxtxx

−==−−−+ ν

Conclusions• A proper convection term prevents the

solution from blowing-up. Or, at least, rapid growth is slowed down by a convection term

• There are some cases where proof is needed.• Blow-up behavior is very different from a

nonlinear heat eqn: the yoke of non-locality.

• More problems in Bae, Chae & O. Nonlinear Analysis 2017. O. in Handbook

Thank you very much.