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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
1/51
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
……..
2/51
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
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
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
25
• Nakagawa’s method(1976) adaptive tn
• W. Ren & X.-P. Wang’s iterative grid redistribution method(2000) adaptive xn
25
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
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
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
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