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 xwxwwHwwt

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

2

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)

3

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)

4

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)

5

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)

6

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)

7

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)

8

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)

9

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)

10

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

11

Velocity vector field

),...(),( 1 Nuutxu

From internet sources

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

12

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

13

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 ),(),(

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

14

Vorticity

From internet sources

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

15

4. Constantin-Lax-Majda Model 1D Model Vorticity Equation 1985

1-D Model

Hilbert Transform

)()0,( 0 xwxwwHwwt

dyyxywxHw )(1)(

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

16

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

17

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

18

Blow UpBlow up

From internet sources

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

19

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

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)

21

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

22

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

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

23

Some other models Second order term (deterministic), not computed

Semigroup theory (normal cone), not computed

)()0,( 0 xwxwwwwHww xxxt

)()0,(

)(

0 xwxwUwwHwt )(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

xwxwwwHwwt

)()0,( 0 xwxwwwHww xxt

)()0,( 0 xwxwHwwHww xt

Third Graduate Student Symposium 2005-04

UW Math Department (Batmunkh. Ts)

25

BYE BLOW UP

THANK YOU.