Post on 16-Dec-2020
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.