One-dimensional model equations for incompressible

fluid motion

10 Jan., 2013 @ London

Hisashi Okamoto

Research Institute for Mathematical

Sciences Kyoto University

okamoto@kurims.kyoto-u.ac.jp

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A short introduction of myself • Educated in Univ. Tokyo---

Hiroshi Fujita---Tosio Kato

• Tosio Kato died in 1999.

• His wife died in 2011 and left thousands of slides taken by Tosio Kato.

T. Kato

H. Fujita

H. O. …….

S.T. Kuroda

……..

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Three sources of PDE peoples Th

ree

sou

rces

of

PD

E p

eop

les

Kôsaku Yosida Gigas, Takada, Tsutsui, Abe

Tosio Kato H.O., Kishimoto,

Masaya Yamaguti Matano, Miyaji

3

Navier-Stokes

0 div

1)(

u

uuuu pt

The parish where Stokes was born. His father was the parish minister.

4

5

3D Navier-Stokes: A bad problem

Try simpler models:

❂ Burgers (‘15 Bateman, ‘39 Burgers)

❂ Proudman--Johnson eq. (‘62)

❂ Fujita’s eq. ut = u + up

(‘66)

❂ De Gregorio (‘90)

❂ Strain-vorticity dynamics (unbounded sol.)

❂ Quasi-geostrophic eq.

❂ Many others.

Turbulence is a bad Problem!? How about the NS itself?

6

Navier-Stokes is nonlinear & nonlocal

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

ωuωωuω )()(t

dt

x

xxt ),(

|-|

-

4

1),(

SavartBiot ,curl)(

3

-1

ωu

ωu

Therefore models must be nonlinear & nonlocal.

decom. Helmoltznonlocal p

7

model ① The Proudman-Johnson equation. ‘62

• Derived from 2D Navier-Stokes

(unbounded solution of NS )

),(),,( xtyuxtu xu

)(),0( & BC periodic

)10 ,0(

xxu

xt

xx

xxxxxxxxxxtxx uuuuuu

8

Global existence or finite time blow-up?

)(),0(

,1

2

2

xx

uuudx

dxxxxt

xx

xxxxxxxxxxtxx

u

uuuuuu

ωuωωuω )()(t

SavartBiot ,curl)( -1 ωu

Order -2

Order -1 9

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

• Finite time blow up was predicted by numerical computation.

10

Global existence was proved by X. Chen

Theorem. Assume that > 0.

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

if homogeneous Dirichlet, Neumann, or the periodic

boundary condition.

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

Blow-up if non-homogeneous Dirichlet BC.????

Grundy & McLaughlin (1997). 11

Be careful for numerical solution

• Somebody may say:

12

A remark on numerical experiments

• In the case of =0, numerical experiments are sometimes misleading.

Rigorous analysis

is necessary

13

Prime suspect of the blow-up is the stretching term.

convection stretching diffusion

ωuωωuω )()(t

Conjecture: blow-up is caused by the stretching term. The convection term is the by-stander.

14

Effect of convection term

1

2

2 ,

dx

duuu xxxxt

)( ,2

1

constant2

1

2

2

tbUUUuU

uuu

uuuu

xxtx

xxxxtx

xxxxxxxtxx

The convection term is NOT important in blow-up.

Close to the Fujita eqn.

convection stretching difusion

15

A proper convection term prevents solutions from blowing-up. (O. & J. Zhu, Taiwanese J. Math., 2000)

R/: ,

BC periodic 0,),(

)11,0( ,),(2

1

222

1

1

1

1

22

LLPPUUU

dxxtU

xtdxxtUUUU

xxt

xxt

xxxtxx

xxxxtxx

u

uu

, Global existence Blow-up

Fujita+Projection

16

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

• ∃ x0 .),(

1

0

22

dxxtuuuu xxt

0),(

),(lim

)(),(lim

,),(lim

0

0

0

xtu

ytu

xyytu

xtu

Tt

Tt

Tt

　　

17

Blow-up with or without the projection

ut = uxx + u2 ut = uxx + Pu2

Everywhere blow-up is likely Proof?

18

model ② Generalized Proudman-Johnson equation

• A model:

)(),0(

,1

2

2

xx

uauudx

dxxxxtxx

a = -(m-3)/(m-1), axisymmetric exact solutions of the Navier-

Stokes eqns in Rm.

a = 1 (m=2) Proudman-Johnson eqn

a = -2, =0. Hunter-Saxton equation (’91)

a = -3 the Burgers equation (‘46)

xxxxxxxxxxtxxxxxt uuuuuuuuuudx

d 3

2

2

19

Prime suspect of the blow-up is the stretching term.

convection stretching diffusion

ωuωωuω )()(t

xxxxtxx auu

Conjecture: blow-up for large |a|

global existence for small |a| .

20

Xinfu Chen’s proof of global existence

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

• periodic boundary condition.

• THEOREM. If 0 < & -3 ≦ a ≦ 1, the solutions exist globally in time for all initial data.

Proudman-Johnson

Burgers

21

If a < -3, or 1 < a, then …

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

a = 1 is a threshold. 22

Numerical experiments

||max/

10

)2sin(0

a

x10

)2sin(300

a

x

23

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

a = -2.5

a = 1.5

24

25

• Nakagawa’s method(1976) adaptive tn

• W. Ren & X.-P. Wang’s iterative grid redistribution method(2000) adaptive xn

25

Initial data

26 26

Max norm of u & ux

27 27

Blow-up time versus a

28 28

Current Status

? ?

?

a = 1

0 <

0 =

a = -1 a = -3

O. J. Math. Fluid Mech. 2009

29

Summary for = 0

• Blow-up for -∞ < a < -1. (Remember that the solutions can exist globally in this region if > 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

O. J. Math. Fluid Mech. 2009

30

Weak sol. of the generalized PJ

Cho & Wunsch, (2010), a = -(n+2)/(n+1)

31

A necessary and

sufficient condition

is known

(Constantin, Lax,

& Majda 1985).

xx cos)(0

model ③ Constantin-Lax-Majda

Hu

u

x

xt

0

xx cos)(0

32

De Gregorio ‘90

Global existence???

Does the convection term delete the blow-up?

Hu

uu

x

xxt

0

R

aHu

uau

x

xxt

,

0

-∞ < a ≦0. Blow-up

Castro & Cordoba ‘09

O, Sakajo & Wunsch ‘08

33

Constantin-Lax-Majda & De Gregorio & Proudman-Johnson can be unified.

)(),0(

,1

2

2

xx

uauudx

dxxxxtxx

)(),0(

,2/

2

2

xx

uauudx

dxxxxtxx

= 1 & a = 1 ??? De Gregorio’s `90

= 1 & -∞ < a < 0 Blow-up Castro & Cordoba ‘09

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

34

Unified equation & b-equation

Holm & Hone 2005

Escher & Seiler 2010

35

)(),0(

,1

2

2

2

xx

mubuudx

dxxxxtxx

The generalized P-J with =0.

)(),0(

BC periodic

)10 ,0(

0

xxxxu

xt

xxxxxxtxx uauuuu

• 3D axisymmetric Euler for a = 0.

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

• Burgers for a = -3.

36

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 , xxtxx auuu

,/)1,0(),0( 2RL

,/)1,0(),0( 2RL

37

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

• The case of -∞< a < -2 is settled in Zhu & O., Taiwanese J. Math. (2000). 3

2

2

1

0

2

)()(

),()(

tbtdt

d

dxxtut x

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

38

-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()),,(,( 　　　XtXtuX t

).,(),( tXtV

　,)()( 2 VtIVVV ttt 1

0

2

)( dV

VtI t

39

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 )(

40

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

• Define

• Invariance

• Boundedness of

ass /1|| )(

1

0

1

0

1

0

.0)]()([

])[()),((

dxuuauu

dxuauuuudxxtudt

d

xxxxxxx

xxxxxxxxxx

1

0

1

0

/1),(,),( dxxtudxxtu xx

a

xx 　　

41

-1 ≦ a < 0. Continued.

•

• gives us

ctux

)(

xxxxxxxxxxtxx uuauuuu

1

0

21

0

2 )12(),( dxuuadxxtudt

dxxxxx

1

0

21

0

2 ),()12(),( dxxtuacdxxtudt

dxxxx

42

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

• Define

• Then

• is bounded.

)0(0

)0(||)(

)1/(1

s

sss

a

0)()(1

0

1

0

2 dxuuadxudt

dxxxxxxxx

1

0),( dxxtuxxx

43

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

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

.0

)(),(

FFaFFF

tT

xFxtu

44

Another

• 3D Navier-Stokes

exact sol.

• Nagayama and O., ’02 numerical experiment.

• Proof ???

),(),(

))(()(

xtfxtSf

ffSfffSfff xxxxxxxxxxxtxx

45

2D Example (with K. Ohkitani) J. Phys. Soc. Japan, vol. 74 (2005), 2737--2742

• 2D Euler

0(( ))

),(

curl

0

uχχuχ

uχ

u

u

t

xy

t

46

The convection term is now deleted.

χu

uχχ

uχχuχ

uχ

1)(

)

))

),(

0(

0((

t

t

xy

χu

uχχ

1)(

) 0(

P

t

47

L2-norm of c

48

∫0t|c(s)|∞dx

49

(-)1/2 ~ |c| t=0

t=3, Euler t=3, model

50

Conclusions

• Similarity solutions of the Navier-Stokes eqns can blow up in finite time: necessity of the energy inequality.

• A proper convection term prevent 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.

Thank you very much. 51