third graduate student symposium 2005-04 uw math department (batmunkh. ts) 1 constantin-lax-majda...
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Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
1
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 xwxw
wHwwt
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Blow Up=Blow Up=Blow Up Fluid Mechanics
Blow Up, Turbulence, Volcano, Hurricane, Airplane, Ocean
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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.
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Euler’s Equation Leonhard Euler (1707-1783, Swiss mathematician) 300 years ago, Euler equation of fluid motion
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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-
fupuuuDt
Dut
11
0 uudiv)()0,( 0 xuxu
),( tx),( txpp
u
12
2
j jx
1j j
j
x
uudiv
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Velocity vector field
),...(),( 1 Nuutxu
From internet sources
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
puuuDt
Dut
1
0
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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 operator
In 2 Dim Conservation of vorticity,
In 1 Dim There is only one Hilbert operator
)( uucurlw
uwwuwDt
Dwt )()(
)()()0,( 00 xuxwxw
3
),(||4
1),(
3Rdytyw
yx
yxtxu
wDwwuwt )()(
D0)( wDw
Hw
0wcurl
x
dytywtxu ),(),(
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Vorticity
From internet sources
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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4. Constantin-Lax-Majda Model 1D Model Vorticity Equation 1985
1-D Model
Hilbert Transform
)()0,( 0 xwxw
wHwwt
dyyx
ywxHw
)(1)(
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
xwtxtHw
xwtxw
)cos()(0 xxw
)sin()(1
)( 00 xdy
yx
ywxHw
2222 )sin(44
)cos(4
)(cos))sin(2(
)cos(4),(
txt
x
xtxt
xtxw
)sin(22
)cos(),(
x
xtxw
2,0)sin(22
xx
0
1
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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)
0
1
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Blow UpBlow up
From internet sources
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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 xwxw
RRontWwHwwt
dt
tdBtW
)()(
)(),(),(),( tdBdtxwHtxwtxdw )()()0,( 00 xuxwxw
0
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
20
Stochastic calculation, BM
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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
),(2
1)(ˆ dxetxwtw ikx
k
1
)(~),(N
Nk
ikxk
NN etwtxw
)(),(),(),( tdBdtxwHtxwtxdw jjN
jjN
12,...0),()0,( 0 Njxwxw jjN
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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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 xwxw
wtWwHwwt
)()0,(
)(
0 xwxw
tWwdtwHwwt
)()0,(
),()(
0
0
xwxw
dtuwLuBwHwwT
t
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
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Some other models
Second order term (deterministic), not computed
Semigroup theory (normal cone), not computed
)()0,( 0 xwxw
wwwHww xxxt
)()0,(
)(
0 xwxw
UwwHwt )(wNU k
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
24
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
xwxw
wwHwwt
)()0,( 0 xwxw
wwHww xxt
)()0,( 0 xwxw
HwwHww xt
Third Graduate Student Symposium 2005-04
UW Math Department (Batmunkh. Ts)
25
BYE BLOW UP
THANK YOU.