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Constantin-Lax-Majda Model Equation (1-Dimension) Blow Up Problem

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. Structures. 0. Historical review, fluid motion (p 4-9)

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Constantin-Lax-Majda Model Equation (1-Dimension) Blow Up Problem

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  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 UW Math Department (Batmunkh. Ts)

  2. Structures • 0. 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 UW Math Department (Batmunkh. Ts)

  3. Blow Up=Blow Up=Blow Up Fluid Mechanics • Blow Up, Turbulence, Volcano, Hurricane, Airplane, Ocean 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 Eureka UW Math Department (Batmunkh. Ts)

  5. Euler’s Equation • Leonhard Euler (1707-1783, Swiss mathematician) • 300 years ago, Euler equation of fluid motion UW Math Department (Batmunkh. Ts)

  6. Navier-Stokes Equation • Claude-Louis Navier (1785-1836, France) • GeorgeStokes (1819-1903, Ireland) • Navier 1821, modifying Euler’s equations for viscous flow in Fluid Mechanics, 200 years ago • Stokes 1842, incompressible flow 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 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 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 UW Math Department (Batmunkh. Ts)

  10. 1. Navier-Stokes Equation • a 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- UW Math Department (Batmunkh. Ts)

  11. Velocity vector field • From internet sources UW Math Department (Batmunkh. Ts)

  12. 2. Euler Equation in 2, 3 dim • a nonviscid, incompressible (water) ideal (homogeneous) fluid • the condition of incompressibility • the initial velocity field) • Vorticity diffusion coefficient- • From Navier-Stokes equation to Euler equation UW Math Department (Batmunkh. Ts)

  13. 3. Vorticity Equation in 2, 3 dim • From 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 UW Math Department (Batmunkh. Ts)

  14. Vorticity • From internet sources UW Math Department (Batmunkh. Ts)

  15. 4. Constantin-Lax-Majda Model 1D Model Vorticity Equation 1985 • 1-D Model • Hilbert Transform UW Math Department (Batmunkh. Ts)

  16. Constantin-Lax-Majda model equation( 1-Dim Model Vorticity Equation, 1985) • Solution • Blow Up • T=2 UW Math Department (Batmunkh. Ts)

  17. Computing, Blow up • Complex methods • Hilbert transform • Fourier transform • Fast (Discrete) Fourier transform • Matlab UW Math Department (Batmunkh. Ts)

  18. Blow Up Blow up • From internet sources 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 UW Math Department (Batmunkh. Ts)

  20. Stochastic calculation, BM UW Math Department (Batmunkh. Ts)

  21. Stochastic methods • Hilbert transform • Fourier expansion • Fast Fourier transform • Stochastic CLM model equation, finite scheme • Spectral methods 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 UW Math Department (Batmunkh. Ts)

  23. Some other models • Second order term (deterministic), not computed • Semigroup theory (normal cone), not computed 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) UW Math Department (Batmunkh. Ts)

  25. BYE BLOW UP THANK YOU. GooD LucK UW Math Department (Batmunkh. Ts)

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