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Constantin-Lax-Majda Model Equation (1-Dimension) Blow Up Problem
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Transcript of Constantin-Lax-Majda Model Equation (1-Dimension) Blow Up Problem
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Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Constantin-Lax-Majda Model Equation(1-Dimension) Blow Up Problem
Blow Up Problem Fluid motion Navier-Stokes equation Vorticity equation Euler equation Deterministic equation Stochastic equation
)()0,( 0 xwxwwHwwt
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Structures0. Historical review, fluid motion (p 4-9)1. Navier-Stokes equation in 2, 3 Dim (p 10-11)2. Euler equation of fluid motion in 2, 3 Dim (p 12)3. Vorticity equation in 2, 3-Dim (p13-14)
4. Constantin-Lax-Majda 1-D model equation (p 15-18)5. Stochastic CLM 1-D Model equation (p 19-21)6. Some model equations (p 22-24)
• Hilbert Transform• Fourier Transform• Numerical Methods
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Blow Up=Blow Up=Blow Up Fluid Mechanics
Blow Up, Turbulence, Volcano, Hurricane, Airplane, Ocean
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Archimedes of Sicily (BC 287-812)Leonardo da Vinci (1452-1519, Italy) 2300 years ago, Archimedes principle in a fluid 500 years ago, (1513) Motion of the surface of the water
Archimedes 225 B.C.
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Euler’s Equation Leonhard Euler (1707-1783, Swiss mathematician) 300 years ago, Euler equation of fluid motion
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Navier-Stokes Equation Claude-Louis Navier (1785-1836, France) George Stokes (1819-1903, Ireland) Navier 1821, modifying Euler’s equations for viscous
flow in Fluid Mechanics, 200 years ago Stokes 1842, incompressible flow
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One Million Dollar Problems Jean Leray, (1906-1998, France) 1933, Existence and smoothness of the Navier-Stokes
equation, open problem, 100 years ago Clay Mathematics Institute, Cambridge,Massachusetts 2000 (7 problems), Navier-Stokes equation, 3-Dim
Clay Mathematics Institute Dedicated to increasing and disseminating mathematical knowledge
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Nobel and Abel prize Alfred Nobel (1833-1896, Sweden) 1895, Nobel prize ($ 1 Million) for scientists Abel, Niels Henrik (1802-1829, Norway) 2002, Abel Prize ($ 1 Million) for mathematicians
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Constantin-Lax-Majda equation Peter Constantin, (1951-), University of Chicago Peter Lax, (1926- Hungary), 2005 Abel Prize, Courant
Institute Andrew J. Majda, (1949- USA), Courant Institute
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1. Navier-Stokes Equationa viscid, incompressible (like water) ideal (homogeneous) fluid
the condition of incompressibility the initial velocity field)
Divergence- Fluid density-Pressure field-
Vorticity diffusion coefficient- Gradient vector- Laplace operator-
fupuuuDtDu
t
11
0 uudiv)()0,( 0 xuxu
),( tx),( txpp
u
12
2
j jx
1j j
j
xu
udiv
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Velocity vector field
),...(),( 1 Nuutxu
From internet sources
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2. Euler Equation in 2, 3 dima nonviscid, incompressible (water) ideal (homogeneous) fluid
the condition of incompressibility the initial velocity field)
Vorticity diffusion coefficient- From Navier-Stokes equation to Euler equation
0 uudiv)()0,( 0 xuxu
puuuDtDu
t 1
0
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3. Vorticity Equation in 2, 3 dimFrom Euler equation to the Vorticity equation
the initial velocity field)
Using Biot-Savart formula
In 3 Dim Convolution operatorIn 2 Dim Conservation of vorticity, In 1 Dim There is only one Hilbert operator
)( uucurlw
uwwuwDtDw
t )()(
)()()0,( 00 xuxwxw
3
),(||4
1),( 3Rdytyw
yxyxtxu
wDwwuwt )()(
D0)( wDw
Hw
0wcurl
x
dytywtxu ),(),(
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Vorticity
From internet sources
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4. Constantin-Lax-Majda Model 1D Model Vorticity Equation 1985
1-D Model
Hilbert Transform
)()0,( 0 xwxwwHwwt
dyyxywxHw )(1)(
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Constantin-Lax-Majda model equation( 1-Dim Model Vorticity Equation, 1985)
Solution
Blow Up
T=2
)())(2()(4
),( 20
220
0
xwtxtHwxw
txw
)cos()(0 xxw
)sin()(1)( 0
0 xdyyxyw
xHw
2222 )sin(44)cos(4
)(cos))sin(2()cos(4),(
txtx
xtxtxtxw
)sin(22)cos(),(x
xtxw
2
,0)sin(22 xx
01
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Computing, Blow up Complex methods Hilbert transform Fourier transform Fast (Discrete)
Fourier transform Matlab
0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.60
200
400
600
800
1000
1200
1400y=cos(x)./(2-2.*sin(x)) Plotting example
x interval Time t=2
y(t)
01
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Blow UpBlow up
From internet sources
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5. Stochastic CLM Model Equation We attempt to extend the model equation including white noise term Brownian motion
Stochastic CLM model equation
When goes to the deterministic model equation
)(tW
)()0,()(
0 xwxwRRontWwHwwt
dttdBtW )()(
)(),(),(),( tdBdtxwHtxwtxdw )()()0,( 00 xuxwxw
0
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Stochastic calculation, BM
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Stochastic methods Hilbert transform Fourier expansion
Fast Fourier transform Stochastic CLM model equation, finite scheme
Spectral methods
k
ikxk etwtxw )(ˆ),(
2
0
),(21)(ˆ dxetxwtw ikx
k
1
)(~),(N
Nk
ikxk
NN etwtxw
)(),(),(),( tdBdtxwHtxwtxdw jjN
jjN
12,...0),()0,( 0 Njxwxw jjN
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6. Some other models Fractional Laplacian term (stochastic), not computed
Laplacian, Brownian term (stochastic), not computed
Control theory (deterministic), not computed
)()0,()(
0 xwxwwtWwHwwt
)()0,()(
0 xwxwtWwdtwHwwt
)()0,(
),()(
0
0
xwxw
dtuwLuBwHwwT
t
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Some other models Second order term (deterministic), not computed
Semigroup theory (normal cone), not computed
)()0,( 0 xwxwwwwHww xxxt
)()0,(
)(
0 xwxwUwwHwt )(wNU k
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Computed other models Generalized viscosity term added (Takashi, computed,
blows up)
Viscosity term added (Schochet, computed, blows up)
Dissipative term added (Wegert, computed, blows up)
)()0,()(
0
2
xwxwwwHwwt
)()0,( 0 xwxwwwHww xxt
)()0,( 0 xwxwHwwHww xt
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BYE BLOW UP
THANK YOU.