Hydrodynamic SingularitiesMarco Antonio Fontelos

Universidad Rey Juan Carlos

“Observe the motion of the surface of the water, which resembles that of hair, which has two motions, of which one is caused by the weight of the hair, the other by the direction of the curls; thus the water has eddying motions, one part of which is due to the principal current, the other to random and reverse motion”

Leonardo da Vinci, 1510

Angry sea at Naruto

Ando Hiroshige, 1830

1.-Navier-Stokes equations and singularities

“The open problems in the area of non-linear partial differential equations are very relevant to applied mathematics and science as a whole, perhaps more so than the open problems in any other area of mathematics, and this field seems poised for rapid development. Little is known about the existence, uniqueness and smoothness of solutions of the general equations of flow for a viscous, compressible, and heat conducting fluid. Also, the relationship between this continuum description of a fluid and the more physically valid statistical mechanical description is not well understood. Probably one should first try to prove existence, smoothness, and unique continuation (in time) of flows, conditional on the non-appearance of certain gross types of singularity, such as infinities of temperature or density. A result of this kind would clarify the turbulence problem.”

John F. Nash, 1958

The Millenium prize problems. Clay Mathematics Institute 2000:1) Birch and Swinnerton-Dyer Conjecture

2) Hodge Conjecture

3) Navier-Stokes Equations

4) P vs NP

5) Poincare Conjecture

6) Riemann Hypothesis

7) Yang-Mills Theory

1.- Navier-Stokes equations and singularities.2.- The quasigeostrophic equation3.- Break-up of fluid jets: drops.4.- Kelvin-Helmholtz instability.

Mass + Momentum conservation

Boundary conditions:

Initial condition:

No slip (when in contact with a solid) :

Force balance (when in contact with another fluid)

Decay at infinity (no boundaries):

Euler equations (inviscid fluid, 1755) :

Vorticity :

Vorticity equation :

Local existence (Kato 1972)

Blow-up

Non uniqueness

Biort-Savart’s law:

Then:

Euler eqn:

Singular integral operator acting on vorticity

1-D Model

Constantin, Lax, Majda, 1985

Properties:

Hilbert T. in R:

Hilbert T. for periodic B.C.:

Example:

Navier-Stokes equations (viscous fluid, 1822) :

Vorticity equation :

Local existence: Leray 1934

(1)

(2)

(3)

Corollary: Global existence in 2-D

Weak solutions (Leray 1934)

r

r2

Parabolic cylinder

Hausdorff dimension of singularities (Caffarelli, Kohn, Nirenberg 1982)

t

Hausdorff dimension of singularities (Caffarelli, Kohn, Nirenberg 1982)

t

Euler:2-D: Global existence and uniqueness (Kato 1967)3-D: Local existence. Singularity if and only if the sup norm of vorticity is not integrable in time (Beale-Kato-Majda 1984). Nonuniqueness (Scheffer 1993).Problem: Finite time blow-up in 3-D?

Navier-Stokes:2-D: Global existence and uniqueness (Kato 1967)3-D: Local existence (Leray 1934). Singularity if and only if the square of the sup norm of velocity is not integrable in time (Serrin 1962). Global existence of weak solutions (Leray 1934).Problems: 1) Uniqueness of weak solutions? 2) Finite time blow-up in 3-D?

2.- The quasigeostrophic equationConstantin, Majda, Tabak 1994

: Temperature field.

Level lines of

Is there a finite time singularity in the derivatives of ?

Formation of sharp fronts

Q-G equation:

A 1-D AnalogD. Chae, A. CórdobaD. Córdoba, MAF, 2003

Using

One gets:

Hilbert T. in R:

Hilbert T. for periodic B.C.:

Let

Then Complex Burgers eqn.

Hodograph transform:

(*)

Introduce

(*) is equivalent to the Cauchy-Riemann system:

For example

Finite time singularity at t=e-1

3.- Break-up of fluid jets: drops

Fluid 1

Fluid 2 Interface

n

Fluid 1

Fluid 2Interface

The case of just one fluid (inside )

N-S

(in )

B.C. (in )

Kinemat.(in )

The kinematic condition in an axisymmetric domain

z

h(z,t)n

of the fluid

s

R2

Mean curvature of anaxisymmetric domain

Rayleigh’s inestability of a uniform cylinder (Rayleigh 1879)

Consider an inviscid and irrotational fluid

Bernoulli’s law:

At the boundary.

Small perturbations of a uniform cylinder:

R

x 1.210.80.60.40.20

0.2

0.1

0

-0.1

-0.2

G(x)

x

Viscous case, Chandrasekhar 1961.

Savart 1833

Rayleigh’s Instab.

D

L

The one dimensional limit

z

n

tNavier-Stokes (axisymmetric)

Boundary conditions

Kinematic condition

D

L

z

n

tNavier-Stokes (axisymmetric)

Boundary conditions

Kinematic Condition

Taylor’s expansion in r + divergence free vector field

Then:

N-S

B.C.

Kin.

p0 v2

Undo the chance of variables for h and z and introduce:

One-dimensional system (Eggers 1993):

Rutland & Jameson, 1971

-6 -4 -2 0 2 4 6-2

-1

0

1

2

0

-10 0 10 20 30 40 50

0

5

10

15

20

Numerical solution of the system (profiles)

h(z,t)

-6 -4 -2 0 2 4 6-6

-5

-4

-3

-2

-1

0

1

2

3

4

-4 -3.5 -3 -2.5 -2

-5

-4

-3

-2

-1

0

1

2

3

4

-6 -4 -2 0 2 4 6 8 10

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

Numerical solution of the system (velocity)

v(z,t)

-10 0 10 20 30 40 50

0

5

10

15

20

-6 -4 -2 0 2 4 6 8 10

-0.5

-0.4

-0.3

-0.2

-0.1

0

0.1

0.2

0.3

as

near

Conjecture: The self-similar break-up mechanism is universal

Billow clouds

4.- Kelvin-Helmholtz Instability

Conjecture: The system presents finite-time singularities in the curvature. Moore 1979. Numerical and asymptotic evidence.

Conclusions

1.- Many physical phenomena related to fluids are linked to the appearence of singularities (finite time blow up of a derivative at some point). Break-up of jets: singularity in the velocity field. Quasigeostrophic equation: singularity on the slope of the temperature field. Kelvin-Helmholtz: singularity in the curvature. Turbulence: (maybe) singularity in the vorticity.

2.- The nature of the singularities indicates the presence of regularizing effects at small length scales (possibly at molecular level).

3.-The existence of a singularity poses an important fundamental question on the consistency of the theory.

NOTA:

NOTA: