The Two-dimensional Navier-Stokes Equations

and the Oseen Vortex

C. E. Wayne

March 18, 2014

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 Abstract

This lecture will focus on the Oseen vortex, an explicit solution of the
two-dimensional Navier-Stokes equation. Using methods from dynamical
systems theory I will explain how one can prove that any solution of the
Navier-Stokes equation whose initial vorticity distribution is integrable
will asymptotically approach an Oseen vortex. These are results originally
obtained in collaboration with Th. Gallay

Work supported in part by the US National Science Foundation.

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 Introduction

1 Understanding the long-time evolution of fluid motion is often

facilitated by studying the coherent structures of the flow.
2 In physical flows, these structures are often vortices.
3 From a mathematical point of view these structures may be
invariant manifolds in the phase space of the system.

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 Two-dimensional fluids

1 Although we live in a three dimensional world, many fluid flows

behave in an essentially two-dimensional way.
(a) In many physical circumstances (e.g. the ocean or the
atmosphere), one dimensional of the domain is much smaller
than either the other two dimensions, or the dimensions of
typical features of interest.
(b) This effect is compounded by the effects of stratification and
rotation.
2 There are fascinating physical and mathematical differences between
two and three dimensional flows.

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 Two-dimensional fluids

What are some typical phenomena in two dimensional fluids?

Figure: A variety of atmospheric vortices. (All images from NASA)

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 Two-dimensional vortices

One of the characteristic features of two-dimensional flows is the

tendency of large vortices to form regardless of the initial state of the
fluid.
• This is in marked contrast to three-dimensional fluids where energy
flows from large scales to small scales.
• This is an example of the “inverse cascade” of energy in
two-dimensional fluids.

How do we characterize vortices?

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 The Navier-Stokes Equations

A system of nonlinear partial differential equations which describe the

motion of a viscous, incompressible fluid.
If u(x, t) describes the velocity of the fluid at the point x and time t then
the evolution of u is described by:

∂u
+ (u · ∇)u = ν∆u − ∇p , ∇·u=0 ,
∂t

The first of these equations is basically Newton’s Law; F = ma while the

second just enforces the fact that the fluid is incompressible.

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 Vorticity

The velocity of the fluid is not the best way to visualize or characterize
vortices, however, for that it is better to use the vorticity!
Roughly speaking, the vorticity describes how much “swirl” there is in
the fluid.

ω(x, t) = ∇ × u(x, t)
Note that for two-dimensional fluid flows, u(x, t) = (u1 (x, y ), u2 (x, y ), 0),
so

ω = ∇ × u = (0, 0, ∂x u2 − ∂y u1 ) .

Note that in two dimensions we can treat the vorticity as a scalar!

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 The Vorticity Equation

To find out how the vorticity evolves in time we can take the curl of the
Navier-Stokes equation. We find quite different equations, depending on
whether we are in two or three dimensions. In three dimensions one has
the systems of equations

∂t ω(x, t) − ω · ∇v(x, t) + v · ∇ω(x, t) = ν∆ω(x, t)

while in two dimensions one has only the single, scalar equation

∂t ω(x, t) + v · ∇ω(x, t) = ν∆ω(x, t) .

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 Vortex Stretching

The presence of the “vortex stretching” term

−ω · ∇v

in the three-dimensional equation is a crucial physical as well as

mathematical difference - it is literally the million dollar term. Because of
its presence it is not known whether or not solutions of the
three-dimensional Navier-Stokes even exist for all time.

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 The two dimensional vorticity equation

For the remainder of the lecture I’ll focus on the two-dimesional vorticity
equation
∂t ω(x, t) + v · ∇ω(x, t) = ν∆ω(x, t) .
For this equation, proving the existence and uniqueness of solutions is
possible even for initial vorticity distributions that have little regularity.
A complicating factor is the presence of the velocity field in the equation
for the vorticity:
1 One can recover the velocity field from the vorticity via the
Biot-Savart operator - a linear, but nonlocal, operator.
2 As a consequence, we can think of the two-dimensional vorticity
equation as the heat equation, perturbed by a quadratic nonlinear
term.

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 Vortex formation in two-dimensional fluids
Let’s begin by looking at typical phenomena present in solutions of the
two-dimensional vorticity equation - or at least in the numerical
approximation of solutions of this equation.

Figure: A numerical simulation of a two-dimensional turbulent flow. The

figures display the vorticity field (with blue and red representing fluid
“swirling” in opposite directions) at successively later and later times and
clearly indicate the tendency of regions of vorticity of like sign to coalesce
into a smaller and smaller number of larger vortices. From the Technical
University of Eindhoven; Fluid mechanics lab

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 Emergence of Vortices

Our goal will be to try and understand the emergence and stability of
these large vortices from very general initial conditions for
two-dimensional flows – – or more poetically,
When little whirls meet little whirls,
they show a strong affection;
elope, or form a bigger whirl,
and so on by advection.

This is quoted without attribution on

http://www.fluid.tue.nl/WDY/vort/2Dturb/2Dturb.html

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 The 2D Vorticity Equation
Let’s see what insight we can obtain into the behavior of the 2D vorticity
equation by considering two different limiting cases:

∂t ω(x, t) + v · ∇ω(x, t) = ν∆ω(x, t) .

1 First limiting case - ignore the dissipative term:

∂t ω(x, t) + v · ∇ω(x, t) = 0 .

This is known as Euler’s equation - but not that if we “forget” the

velocity is in fact determined by the vorticity, it is just the transport
equation which says that the vorticity is carried along by the
background velocity field.
2 Second limiting case - ignore the nonlinear term:

∂t ω(x, t) = ν∆ω(x, t) .

In this case we just have the heat equation.

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 The point vortex model
Helmholtz and Kirchhoff studied the equation without dissipation and
assumed that the vorticity could be written as a sum of finitely many
point vortices (... not always a good assumption, but let’s see where it
leads ...)
In this case, the vortices are just swept along by the velocity field -
however, the velocity field itself must respond to the alteration in the
vorticity field caused by the motion of the vortices.
It turns out that one can compute this response and one finds a simple
and explicit system of equations for the motion of the centers
zj = (xj , yj ) of the vortices:

1 X yj − yk 1 X xj − xk
ẋj (t) = − Γk , ẏ j (t) = Γk
2π |zj − zk |2 2π |zj − zk |2
k6=j k6=j

The use of methods of celestial mechanics to investigate solutions of this

system of (Hamiltonian) ordinary differential equations will be the subject
of my third lecture.
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 Onsager’s idea

Given the Hamiltonian nature of the equations of motion and the chaotic
nature of their solutions for large numbers of vortices, it is natural (at
least in retrospect) to attempt to understand the behavior of large
collections of vortices with the aid of statistical mechanics.
Lars Onsager seems to have been the first person to adopt this point of
view and it lead him to a remarkable conclusion.

• Onsager found that the statistical mechanical description of a

collection of point vortices moving according to the H-K equations
could support states of negative absolute temperature.
• He then realized that a consequence of these negative temperature
states was that vortices of like sign would tend to attract each other
and that this could explain the tendency of large vortices to form,
regardless of the initial conditions.

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 Drawbacks

The limitation of Onsager’s idea is that even now, sixty years after
Onsager first proposed this method of explaining the formation of large
vortices, we have no idea of whether or not the hypotheses that underly
the theory of statistical mechanics are actually satisfied by the dynamical
system defined by the H-K equations.

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 The heat equation

If we ignore the nonlinear term and focus just on the linear terms in the
vorticity equation - this yields the heat equation:

∂t ω(x, t) = ν∆ω(x, t) .

If we assume again that the initial vorticity is concentrated in a

delta-function, it will not remain a point vortex - the viscosity will cause
it to spread with time. In fact, if we assume that the initial vorticity is
given by
ω(z, 0) = αδ(z)
the solution at later times is found to be
α −|z|2 /(4νt)
ω(z, t) = e .
4πνt

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 Oseen vortices

Remarkably, this explicit Gaussian turns out to be an exact solution of

the full, 2D vorticity equation, not just the linear approximation.
1 Note that the Gaussian solution corresponds to a vorticity
distribution that depends only on the radial variable.
2 Inserting this into the Biot-Savart law yields a purely tangential
velocity field.
3 This combination insures that the nonlinear term in the vorticity
equation
v · ∇ω = 0

Thus, the Gaussian vorticity profile is an exact solution of the 2D

vorticity equation known as the Oseen-Lamb vortex.

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 Oseen vortices (cont)

Recall that in the numerical simulations we considered earlier, the

system seemed to tend to a small number of large vortices which
increase in size with time:

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 Scaling variables

Note that the formula for the Oseen

√ vortices shows that the size of the
vortex increases with time (like t ). This is consistent with the
simulations we looked at above and suggests that the analysis of these
vortices may be more natural in rescaled coordinates. With this in mind
we introduce “scaling variables” or “similarity variables”:
x
ξ=√ , τ = log(1 + t) .
1+t

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 Scaling variables (cont.)

Also rescale the dependent variables. If ω(x, t) is a solution of the

vorticity equation and if v(t) is the corresponding velocity field, we
introduce new functions w (ξ, τ ), u(ξ, τ ) by

1 x
ω(x, t) = w(√ , log(1 + t)) ,
1+t 1+t
and analogously for u.

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 Scaling variables (cont.)

In terms of these new variables the vorticity equation becomes

∂τ w = Lw − (u · ∇ξ )w ,
where
1
Lw = ∆ξ w + ξ · ∇ξ w + w
2
Note that the Oseen vortices take the form
A − ξ2
W A (ξ, τ ) = AG (ξ) = e 4 ,
4π
in these new variables. Thus, they are fixed points of the vorticity
equation in this formulation.

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 Function Spaces

We will discuss some further advantages of the scaling variables in

Lecture 2, but let me note for the time being that we can compute the
spectrum of the linearized operator
1
Lw = ∆ξ w + ξ · ∇ξ w + w
2
explicitly in the weighted L2 spaces,

L2 (m) = f ∈ L2 (R2 ) | kf km < ∞ ,



where Z 1/2
kf km = (1 + |ξ|2 )m |f (ξ)|2 dξ ,
R2

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 Spectrum of L

In these spaces, the spectrum of L consists of two pieces:

• Eigenvalues σd = {− k2 | k = 0, 1, 2, . . . }, and

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 Spectrum of L

In these spaces, the spectrum of L consists of two pieces:

• Eigenvalues σd = {− k2 | k = 0, 1, 2, . . . }, and
• Continuous spectrum σc = {λ ∈ C | <λ ≤ −( m−1
2 )}.

(m−1)/2
           
           

      

  
  


 
  

      

  
  


 
  

     

−2 −1

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 Dynamical Systems

It is natural to inquire whether or not these fixed points are stable. It

turns out (somewhat remarkably) that they are actually globally stable.
Any solution of the two-dimensional vorticity equation whose initial
velocity is integrable will approach one of these Oseen vortices.
There are (at least) two approaches that we could use to study the
stability of these vortex solutions:
• A local approach, based linearization about the fixed point.
• A global approach based on Lyapunov functionals.

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 Global Stability

Recall that a Lyapunov function is a function that decreases along

solutions of our dynamical system. In the present case it will be a
functional of the vorticity field w (ξ, τ ) which is monotonic non-increasing
as a function of time.
We’ll look for the ω-limit set of solutions of the 2D vorticity equation.
1 Describes the long-time behavior of solutions.
2 Can be a fixed point, periodic orbit, or even a chaotic attractor.
3 Always exists provided the system satisfies certain compactness
properties.

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 ω-limit set

Definition: Given a semi flow Φt on a Banach space X , consider the

forward orbit of a point w0 - i.e. the set {Φt (w0 )}{t>0} . Then w is in the
ω-limit set of w0 if for every ε > 0 and every T > 0 there exists t > T
such that
kΦt (w0 ) − w k < ε .

Note that it is possible that the ω-limit set is empty. However, if the
forward orbit is compact in X , then the ω-limit set will be non-empty.

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 LaSalle Invariance Principle

A key tool in determining the ω-limit set is the LaSalle Invariance

Principle. - i.e. the ω-limit set of a trajectory must lie in the set on which
the Lyapunov function is constant (when evaluated along an orbit). More
precisely, if the points in the phase space of the dynamical system are
denoted by w , if the flow, or semi-flow defined by the dyanamical system
is denoted by Φt and if the our Lyapunov functional is denoted by H(w )
(and it is differentiable), then the ω-limit set must lie in the set of points

d
E = {w | H(Φt (w ))|t=0 = 0} (1)
dt

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 The Lyapunov functionals

We choose two Lyapunov functions, each motivated by one of the two

different points of view:
1 The H-K model, and Onsager’s idea of treating it with statistical
mechanics ideas, suggests a Lyapunov function based on the
entropy.
2 The linearization which yields the heat equation suggests a
Lyapunov function based on the maximum principle.

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 The (relative) entropy

R
The classical entropy function is S[w ](τ ) = R2 w (ξ, τ ) ln w (ξ, τ )dξ.
However, this would typically be unbounded for the sorts of solutions we
wish to consider. Thus, we study the relative entropy
Z  
w (ξ, τ )
H[w ](τ ) = w (ξ, τ ) ln dξ
R2 G (ξ)
where G is the Gaussian that describes the Oseen vortex.

Note that H[w ] is only defined for vorticity distributions which are
everywhere positive. This is not a problem in statistical mechanics
(where w would typically be a probability distribution) but it is a very
unnatural restriction in fluid mechanics.

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 The relative entropy (cont)

We need to show first that H[w ] is defined and continuous:

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 The relative entropy (cont)

We need to show first that H[w ] is defined and continuous:

• Define Σ+ = {w ∈ L2 (R2 ) | w (x) ≥ 0 almost everywhere.}. Then
H[w ] is well-defined, continuous and bounded from below on
Σ+ ∩ L2 (m), if m > 3.

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 The relative entropy (cont)

We need to show first that H[w ] is defined and continuous:

• Define Σ+ = {w ∈ L2 (R2 ) | w (x) ≥ 0 almost everywhere.}. Then
H[w ] is well-defined, continuous and bounded from below on
Σ+ ∩ L2 (m), if m > 3.
• Also, the forward orbit of initial conditions in Σ+ ∩ L2 (m) are
relatively compact (so that the ω-limit set is well defined.) This
depends on two facts:

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 The relative entropy (cont)

We need to show first that H[w ] is defined and continuous:

• Define Σ+ = {w ∈ L2 (R2 ) | w (x) ≥ 0 almost everywhere.}. Then
H[w ] is well-defined, continuous and bounded from below on
Σ+ ∩ L2 (m), if m > 3.
• Also, the forward orbit of initial conditions in Σ+ ∩ L2 (m) are
relatively compact (so that the ω-limit set is well defined.) This
depends on two facts:
• Smoothing properties of the semi-group.

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 The relative entropy (cont)

We need to show first that H[w ] is defined and continuous:

• Define Σ+ = {w ∈ L2 (R2 ) | w (x) ≥ 0 almost everywhere.}. Then
H[w ] is well-defined, continuous and bounded from below on
Σ+ ∩ L2 (m), if m > 3.
• Also, the forward orbit of initial conditions in Σ+ ∩ L2 (m) are
relatively compact (so that the ω-limit set is well defined.) This
depends on two facts:
• Smoothing properties of the semi-group.
• Localization of the solution - one can show by estimating the
integral equation satisfied by solutions that if w0 ∈ L2 (m),
then w (t) consists of a piece that is in L2 (m + 1) and a piece
that goes to zero in H 1 (m).

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 The relative entropy (cont)

To show that H[w ] is a Lyapunov function compute:

Z   
d w (ξ, τ ) ∂w
H[w ](τ ) = 1 + ln dξ
dτ R2 G (ξ) ∂τ

Z   
w (ξ, τ )
= 1 + ln (Lw − v · ∇w )dξ
R2 G (ξ)
Z   
w (ξ, τ )
= 1 + ln (Lw )dξ
R2 G (ξ)
Z   
w (ξ, τ )
− 1 + ln (v · ∇w )dξ
R2 G (ξ)

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 The relative entropy (cont)

If we integrate by parts (several!) times we find:

Z    Z   w 2
w (ξ, τ )
1 + ln (Lw )dξ = − w (ξ) ∇  dξ
 
R2 G (ξ) R2 G

On the other hand, we can rewrite the integral involving the nonlinear
term as:

Z   
w (ξ, τ )
1 + ln (v · ∇w )dξ
R2 G (ξ)
Z   
w (ξ, τ )
= 1 + ln ∇ · (vw )dξ
R2 G (ξ)

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 The relative entropy (cont)

Integrating by parts this last expression becomes:

Z Z
1
− (v · ∇w )dξ − (ξ · v)wdξ
R2 2 R2

Each of these terms vanishes!

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 The relative entropy (cont)

Integrating by parts this last expression becomes:

Z Z
1
− (v · ∇w )dξ − (ξ · v)wdξ
R2 2 R2

Each of these terms vanishes!

Thus we have:
Z   w 2
d
H[w ](τ ) = − w (ξ) ∇  dξ
 
dτ R2 G

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 The ω-limit set of positive solutions

Z   w 2
d
H[w ](τ ) = − w (ξ) ∇  dξ
 
dτ R2 G
Let’s now consider the implications of this calculation for non-negative
solutions.
If we assume that w (ξ, τ ) ≥ 0 we see:
d
1 dτ H[w ](τ ) ≤ 0 (so H is a Lyapunov function.)
d
2 dτ H[w ](τ ) = 0 only if w is a constant multiple of G .
Recalling the LaSalle invariance principle, we see that the only possibility
for the ω-limit set of positive solutions of the vorticity equation is some
multiple of the Gaussian - i.e. one of the Oseen vortices.
The same result also holds for solutions that are everywhere negative, but
what about solutions that change sign?

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 The maximum principle for the vorticity
equation

One of the most powerful qualitative properties of solutions of the heat

equation is the maximum principle. Closer inspection shows that just like
the heat equation, solutions of the 2D vorticity equation also satisfy a
maximum principle. In particular:
• A solution that is positive for some time t0 will remain positive for
any later time t > t0 , and
• If the initial condition for the vorticity equation satisfies ω(z, 0) ≥ 0
then the solution will be strictly positive for all times t > 0.
Note that these remarks also hold for solutions of the rescaled vorticity
equation. As a consequence of these two observations, we find a second,
surprisingly simple, Lyapunov functional, namely the L1 (R2 )-norm of the
solution!

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 The L1 norm as a Lyapunov function

To show that the L1 norm is a Lyapunov function one splits a solution

that changes sign into two pieces t

w (τ ) = w1 (τ ) − w2 (τ )

where w1,2 solve:

∂τ w1 + v · ∇w1 = Lw1
∂τ w2 + v · ∇w2 = Lw2

where the initial conditions for are given by

w1,0 = max(w0 (ξ), 0) , w2,0 = − min(w0 (ξ), 0) .

Note that both initial conditions are non-negative.

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 The L1 norm as a Lyapunov function
(cont)
Applying the maximum principle to each piece, one can conclude:
1 The L1 norm of the solution cannot increase with time.
2 In fact, the L1 norm is strictly decreasing unless the solution is
either everywhere positive, or everywhere negative.
This last remark follows from the fact that
Z Z
|w (ξ, τ )|dξ ≤ (w1 (ξ, τ ) + w2 (ξ, τ ))dξ
R2 2
ZR Z
= (w1,0 (ξ) + w2,0 (ξ))dξ = |w0 (ξ)|dξ
R2 R2

Once again, we appeal to the LaSalle Principle and conclude that the
ω-limit set of a solution whose initial condition changes sign, must lie in
the set of functions that is either everywhere positive or everywhere
negative.
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 Putting the pieces together

Putting together our two Lyapunov functionals we have the following

conclusion,
1 For general solutions the ω-limit set must lie in the set of solutions
that are everywhere positive or everywhere negative.
2 However, for such solutions, the relative entropy function implies
that the ω-limit set must be a multiple of the Oseen vortex.

However, so far, we can only conclude that solutions for which the initial
conditions are decaying relatively rapidly at infinity (i.e. solutions for
which w0 ∈ L2 (m), with m > 3) converge to an Oseen vortex.

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 Extending this solution to all of L1

• We begin by proving that the forward orbit of an initial condition in

L1 is relatively compact:

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 Extending this solution to all of L1

• We begin by proving that the forward orbit of an initial condition in

L1 is relatively compact:
• Using decay estimates that go back to Carlen and Loss, one shows
that any point in the ω-limit set is actually in L2 (m).

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 Extending this solution to all of L1

• We begin by proving that the forward orbit of an initial condition in

L1 is relatively compact:
• Using decay estimates that go back to Carlen and Loss, one shows
that any point in the ω-limit set is actually in L2 (m).
• We then apply the argument above to conclude that this point must
be a multiple of the Oseen vortex

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 Extending this solution to all of L1

• We begin by proving that the forward orbit of an initial condition in

L1 is relatively compact:
• Using decay estimates that go back to Carlen and Loss, one shows
that any point in the ω-limit set is actually in L2 (m).
• We then apply the argument above to conclude that this point must
be a multiple of the Oseen vortex

Thus, we conclude:
Theorem (Th. Gallay and CEW) Any solution of the two-dimensional
vorticity Requation whose initial vorticity is in L1 (R2 ) and whose total
vorticity R2 ω(z, 0)dz 6= 0 will tend,
R as time tends to infinity, to the
Oseen vortex with parameter α = R2 ω(z, 0)dz.

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 Extensions and Conclusions
This theorem implies that with even the slightest amount of viscosity
present, two-dimensional fluid flows will eventually approach a single,
large vortex.
1 However, if the viscosity is small, this convergence may take a very
long time. (Much longer than observed in the numerical
experiments, for example.)
2 Furthermore, Onsager’s original calculations of vortex coalescence
were for an inviscid fluid model which suggests that some sort of
coalescence should occur independent of the viscosity
Thus, while Gallay’s and my theorem says that eventually, all
two-dimensional viscous flows will approach an Oseen vortex, there
should be a variety of interesting and important behaviors that manifest
themselves in the fluid prior to reaching the asymptotic state described in
the theorem.

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 Vortex Merger

One of the most important physical effects, and one of the hardest to
understand from a mathematical point of view concerns the merger of
two or more vortices. Clearly such mergers must take place in order for
the multitude of small vortices present in an initially turbulent flow to
coalesce into the small number of large vortices observed in numerics and
experiments.

While the Oseen vortex which characterizes the long-time asymptotics of

the flow has the property that the effects of the nonlinear terms in the
vorticity equation vanish, both numerical and experimental studies show
that the merger process is highly nonlinear and involves the filamentation
and inter-penetration of the two vortices into one another.

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 Vortex Merger (cont)

Figure: An experimental dye visualization of the merger of

two-dimensional vortices, from Meunier, et al, Comptes Rendues Phys.
vol. 6, p. 431 (2005).

While physically based criteria exist to predict when merger will occur a
rigorous mathematical understanding of this phenomenon is so far almost
completely absent.

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 Metastability

A second interesting phenomenon that is particularly noticeable in the

numerical simulations of two-dimensional flows on bounded domains is
the creation and persistence of metastable structures.
The origin and properties of these states in the two-dimensional
Navier-Stokes equation is still not understood but statistical mechanical
ideas have again been used to propose an explanation associated with the
different time scales on which energy and entropy are dissipated.

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 Metastability in Burgers Equation

Similar metastable phenomena also occur in the weakly viscous Burgers

equation which is often used as a simplified testing ground for
understanding the Navier-Stokes equations. Because of the simpler
nature of Burgers equation, one can show that the metastable states
form a one-dimensional attractive invariant manifold in the phase space
of the equation and one can speculate that a similar dynamical systems
explanation might account for the metastable behavior observed in the
two-dimensional Navier-Stokes equation, as it has for the long-time
asymptotics of solutions. (Joint work with Margaret Beck at BU.)

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 Summary

A distinctive feature of two-dimensional flows is the “inverse cascade” of

energy from small scales to large ones. Lars Onsager first sought to
explain this phenomenon by studying the statistical mechanics of large
collections of inviscid point vortices. While Onsager’s observation about
inviscid flows remains unexplained, dynamical systems ideas - in this case
Lyapunov functionals inspired by kinetic theory - have been used to show
that in the presence of an arbitrarily small amount of viscosity, essentially
any two-dimensional flow whose initial vorticity field is absolutely
integrable will evolve as time goes to infinity toward a single, explicitly
computable vortex.

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