'Observe the motion of the surface of the water, whic

1
Hydrodynamic Singularities
Marco Antonio Fontelos
Universidad Rey Juan Carlos
2
1.-Navier-Stokes equations and singularities
Leonardo da Vinci, 1510
Ando Hiroshige, 1830
Angry sea at Naruto
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
3
John F. Nash, 1958
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.
The Millenium prize problems. Clay Mathematics
Institute 2000

• Birch and Swinnerton-Dyer Conjecture
• Hodge Conjecture
• Navier-Stokes Equations
• P vs NP
• Poincare Conjecture
• Riemann Hypothesis
• Yang-Mills Theory

4
1.- Navier-Stokes equations and
singularities. 2.- The quasigeostrophic
equation 3.- Break-up of fluid jets drops. 4.-
Kelvin-Helmholtz instability.
5
Mass Momentum conservation
6
Initial condition
Boundary conditions
No slip (when in contact with a solid)
Force balance (when in contact with another fluid)
Decay at infinity (no boundaries)
7
Euler equations (inviscid fluid, 1755)
Vorticity
Vorticity equation
Local existence (Kato 1972)
8
Blow-up
Non uniqueness
9
Biort-Savarts law
Then
Euler eqn
Singular integral operator acting on vorticity
10
1-D Model
Constantin, Lax, Majda, 1985
Hilbert T. in R
Hilbert T. for periodic B.C.
Properties
11
Example
12
Navier-Stokes equations (viscous fluid, 1822)
Vorticity equation
Local existence Leray 1934
13
(1)
(2)
14
(3)
Corollary Global existence in 2-D
15
Weak solutions (Leray 1934)
16
(No Transcript)
17
Hausdorff dimension of singularities
(Caffarelli, Kohn, Nirenberg 1982)
Parabolic cylinder
r2
r
t
S
18
Hausdorff dimension of singularities
(Caffarelli, Kohn, Nirenberg 1982)
t
S
19
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?
20
2.- The quasigeostrophic equation
Constantin, Majda, Tabak 1994
Temperature field.
21
Level lines of
Is there a finite time singularity in the
derivatives of
?
Formation of sharp fronts
22
Q-G equation
D. Chae, A. Córdoba D. Córdoba, MAF, 2003
A 1-D Analog
23
Hilbert T. in R
Hilbert T. for periodic B.C.
Using
One gets
Let
Then
Complex Burgers eqn.
24
Hodograph transform
()
25
Introduce
() is equivalent to the Cauchy-Riemann system
For example
26
Finite time singularity at te-1
27
3.- Break-up of fluid jets drops
28
Fluid 1
Interface
Fluid 2
Fluid 1
Interface
Fluid 2
29
The case of just one fluid (inside )
N-S
(in
)
B.C. (in )
Kinemat. (in )
30
The kinematic condition in an axisymmetric domain
n
h(z,t)
z
of the fluid
31
Mean curvature of an axisymmetric domain
R2
s
32
Rayleighs inestability of a uniform cylinder
(Rayleigh 1879)
Consider an inviscid and irrotational fluid
Bernoullis law
At the boundary.
33
Small perturbations of a uniform cylinder
R
34
Rayleighs Instab.
Savart 1833
Viscous case, Chandrasekhar 1961.
35
The one dimensional limit
n
Navier-Stokes (axisymmetric)
t
D
L
Boundary conditions
Kinematic condition
z
36
n
Navier-Stokes (axisymmetric)
t
D
L
Boundary conditions
Kinematic Condition
z
37
Taylors expansion in r divergence free vector
field
Then
N-S
p0
v2
B.C.
Kin.
38
Undo the chance of variables for h and z and
introduce
One-dimensional system (Eggers 1993)
39
Rutland Jameson, 1971
40
Numerical solution of the system (profiles)
h(z,t)
41
Numerical solution of the system (velocity)
v(z,t)
42
(No Transcript)
43
as
near
Conjecture The self-similar break-up mechanism
is universal
44
4.- Kelvin-Helmholtz Instability
Billow clouds
45
Conjecture The system presents finite-time
singularities in the curvature. Moore 1979.
Numerical and asymptotic evidence.
46
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.
47
NOTA
NOTA