## Abstract

In this paper I investigate the dynamics of vortex patches in the Yudovitch phase space. I derive an approximation for the evolution of the vorticity in the case of nested vortex patches with distant boundaries, and study its long-time behaviour.

## 1. Introduction

The long-time behaviour of solutions of two-dimensional incompressible Euler equations is an interesting and highly non-trivial subject. It is well known that smooth and localized initial data lead to a global-in-time well-posed evolution in spaces of smooth functions. Beyond this, little is known about the long-time dynamics. In this paper, I consider the evolution of non-smooth solutions, in a well-known phase space of functions with a limited degree of non-smoothness. The equations of ideal incompressible fluids in two dimensions can be described in terms of a single scalar field, the vorticity *ω*, which is a function of space and time, *ω*=*ω*(*x*,*t*), with and . The vorticity is transported by a flow it creates: it is an active scalar. The transport
1.1
is done by an incompressible velocity field *u*(*x*,*t*) whose curl is the vorticity, *ω*=∂_{1}*u*_{2}−∂_{2}*u*_{1}. This linear relation can be inverted by writing *u*=∇^{⊥}*ψ* and seeking *ψ* whose gradient decays at infinity and solves Δ*ψ*=*ω*. The global existence and uniqueness of solutions of (1.1) for vorticity in the class is a classical result of Yudovitch [1]. The evolution (1.1) results in a rearrangement of the vorticity distribution by a volume-preserving transformation with quasi-Lipschitz classical trajectories. If the initial datum *ω*(*x*,0) is a step function, then it remains a step function, with only the plane domains of constant value evolving in time. An equation of evolution for the boundary of such a domain, termed ‘contour dynamics’, was derived and studied numerically by Zabusky *et al.* [2]. If the initial vorticity equals a constant *Ω* in a simply connected bounded domain with smooth boundary (a vortex patch), then evolution of vorticity is reduced to a non-local evolution equation for a complex-valued function *z*(*α*,*t*) representing the boundary of the vortex patch at time *t*, parametrized by a parameter *α*∈[0,2*π*],
1.2
The derivative *z*′(*α*,*t*)=∂*z*(*α*,*t*)/∂*α* obeys
1.3
This equation resembles very much the simple equation ∂_{t}*ω*=*ωHω* [3], where *H* is the Hilbert transform, an equation that served as a one-dimensional scalar model for the three-dimensional vectorial vortex stretching equation. The simple equation blows up in finite time. Motivated by this, it was conjectured [4] that the vortex patch equation develops singularities in *z*′. It turned out [5,6] that the boundaries of vortex patches remain smooth, if they were initially so. If the initial patch is an ellipse, then it remains an ellipse for all time, and the evolution consists of a rigid rotation with constant angular velocity, around a fixed centre, the symmetry centre of vorticity. The stability of these Kirchhoff ellipses under strain or local perturbations was investigated [7,8]. In this paper, we study the effect of far-field perturbations. We derive equations for a couple of contours which approximate the Eulerian evolution when one contour is far from the other. The system becomes almost uncoupled: the outer curve has a self-determined evolution influenced by the inner curve only via a constant coefficient computed from the area of the region surrounded by the inner curve. That area is conserved under the evolution. The effect that the evolving inner curve has on the outer curve is one of pure rotation around the conserved vorticity field centre. The rotation, however, is not rigid: its angular velocity depends on radius, is constant at fixed radius, but decreases with increased radius. The evolution of the inner curve is influenced by the outer curve via a time-dependent complex coefficient *ζ*(*t*). Remarkably, if the inner curve is an ellipse, it stays an ellipse. The nonlinear stability of this ellipse is determined by the long-time correlation of *ζ*(*t*) with a geometric quantity representing the inner ellipse. If the outer curve is initially an ellipse, it does not stay one, except in the case it was a circle. If the outer curve is initially an ellipse of small eccentricity, then its evolution can be approximated for long time by that of an ellipse, and in that case *ζ* can be computed explicitly. The resulting system can be investigated in detail and instability can be proved. The instability is strong, in the sense that the perturbed ellipse's aspect ratio degenerates, while keeping constant area. The proof of this instability is done by studying the dynamics of a complex quantity that represents the aspect ratio of the inner ellipse and the angle it makes with a coordinate system. Degenerate ellipses are represented by the boundary of the unit disc, and the dynamics is such that there can be a stable fixed point on the boundary of the unit disc which attracts trajectories from inside the circle. This means that non-degenerate ellipses degenerate in infinite time.

## 2. Vortex patches

We consider the evolution of a two-dimensional incompressible inviscid fluid. We describe first the vorticity distribution. We take *N* smooth, disjoint, oriented, closed plane curves, , *j*=1,…,*N*. The complement of their union is an open set . We denote by *D*_{j} the connected components of *D*, . We denote by *D*_{N+1} the unbounded connected component. Each curve *Γ*_{j} divides into two connected open sets. We denote the bounded one *U*_{j}. We orient *Γ*_{j} such that the vectors (*n*_{j},*τ*_{j}), where *n*_{j} is the outer normal to *U*_{j} and *τ*_{j} is tangent to *Γ*_{j}, define the same orientation in as the standard basis (*e*_{1},*e*_{2}). This is the same as saying that an observer travelling on *Γ*_{j} in the sense of the parametrization has *U*_{j} on his left side, or that e^{iπ/2}*n*_{j}=*τ*_{j}. (We identify with .) We consider the vorticity
2.1
with , *k*=1,…,*N*, *Ω*_{N+1}=0 and *χ*_{Dk} the characteristic (indicator) function of *D*_{k}. As , it is well known [1] that the incompressible Euler equations with initial data like in (2.1) possess global unique weak solutions in *Y* . Moreover, the solution is given implicitly by
2.2
with *D*_{k}(*t*) obtained from *D*_{k}(0) by the Lagrangian transformation
2.3
where
2.4
and *u* is the velocity vector *u*=∇^{⊥}*ψ*, with ∇^{⊥}=e^{iπ/2}∇,
i.e.
2.5
with
2.6
and boundary condition ∇*ψ*→0, as , that is,
2.7
As e^{iπ/2}*n*_{k}=*τ*_{k}, we obtain by the divergence theorem
2.8
where *ω*_{k} are the numbers
2.9
i.e. *ω*_{k} is the jump in *ω*(⋅,*t*) as we cross from *U*_{k} to . Because each *Γ*_{k} intersects exactly two sets , there is no ambiguity in the definition. If *Γ*_{k} is parametrized by *z*_{k}(*s*), with *s*∈[0,2*π*] and *z*_{k}(0)=*z*_{k}(2*π*), , then the integrals in (2.8) are
The velocity field defined by (2.8) is Hölder continuous, and, in particular, (2.8) is well defined for *x*∈*Γ*_{j}. The vortex patch equations are the equations of evolution of the curves *Γ*_{j}. If
2.10
and
2.11
then the vortex patch equations are
2.12
i.e.
2.13
The centre of the vorticity field is defined by
We check that *x* is conserved during the motion:
by incompressibility (). Then
Now
and thus
The expression
is antisymmetric in *k* and *l*, and thus d*x*/d*t*=0. We note that, if *ψ*∈*C*^{2}, then
2.14
and the integral because *u* decays like |*y*|^{−1}. This argument requires though the compactly supported *ω* to be smoother than a vortex patch (*C*^{α} suffices). If the configuration of the *Γ*_{k} is a collection of concentric ellipses (in the geometric sense), then (0,0), the centre of the vorticity field, coincides with the geometric centre.

Let us observe that, for any vorticity in the Yudovitch class , we have that
2.15
Indeed, this is easily verified by writing first
then splitting the integral in two pieces, one for |*x*−*y*|≤*R* and one for |*x*−*y*|≥*R*, and then optimizing in *R*. Note that if *ω* solves the Euler equations, then the right-hand side of (2.15) is time independent. On the other hand, it is easy to see that a velocity given by (2.10) is bounded by
2.16
where |*Γ*| is the length of the curve *Γ*. Indeed, parametrizing
with *s*∈[0,|*Γ*|] and *r*(0)=*r*(|*Γ*|), *θ*(0)=*θ*(|*Γ*|), we have, denoting *d*/d*s* by ′ and integrating by parts twice,
The inequality (2.16) follows because |*r*′|≤|*ζ*′|.

## 3. Elliptical vortex patches

An ellipse centred at the origin of Cartesian coordinates in the plane can be represented as
3.1
with , *α*∈[0,2*π*]. If we write *z*_{j}=*r*_{j} e^{iθj}, then the ellipse is
3.2
with
3.3
Thus, (*θ*_{1}−*θ*_{2})/2 is a phase shift, which of course is a redundant parameter, (*θ*_{1}+*θ*_{2})/2 represents the angle the ellipse makes with the coordinate system, and *a* and *b* are major and minor semi-axes. The convention |*z*_{1}|≥|*z*_{2}| corresponds to a choice of positive trigonometric orientation (anticlockwise). A Kirchhoff ellipse is a solution of the two-dimensional incompressible Euler equations whose vorticity is a non-zero constant *Ω* in a region bounded by an ellipse, and zero outside that region. The parametric representation of a Kirchhoff ellipse is [9]
3.4
with
3.5
where *A* is the area of the ellipse
3.6
The Kirchhoff ellipse has time-independent |*z*_{1}| and |*z*_{2}|, and therefore constant length of its semi-axes, and constant area. It rotates rigidly with angular velocity
3.7

## 4. Far-field perturbations of vortex patches

Let us consider a base vorticity
4.1
and a perturbed vorticity
4.2
Because, by definition *D*_{2}∩*D*_{1}=∅, we have
where |*D*_{2}| is the area of *D*_{2}. The boundaries *Γ*_{1} and *Γ*_{2} are described by functions *z*_{1}(*α*,*t*) and *z*_{2}(*α*,*t*) satisfying the vortex patch equations. We assume that *z*_{2} is situated far away,
4.3
with *ϵ*>0 very small. The fact that *Γ*_{2} is far from *Γ*_{1} does not stop *η* from being a small perturbation in *Y* of *ω*. The vortex patch system is
4.4
and
4.5
with *ω*_{1} of order one and *ω*_{2} very small. Let us write, in (4.4),
and, in (4.5),
The system (4.4) and (4.5) is thus
4.6
where
4.7
and
4.8
Now we use the assumption that any |*z*_{2}| is much larger than any |*z*_{1}| and approximate the system by
4.9
where
4.10
and *U*_{1} is given in (4.7). Let us make a few observations regarding quantities in (4.9). First,
where ind(0,*Γ*_{2}) is the index (winding number) of *Γ*_{2} at zero. Second,
4.11
where *A*_{j}(*t*) is the normalized area of the region *U*_{j} bounded by the curve *Γ*_{j}:
4.12
Collecting these observations, the system (4.9) becomes
4.13
Now we claim that solutions of (4.13) have constant normalized areas *A*_{j}. Indeed,
The terms *I*_{jj}(*α*,*t*) in the integrals cancel because they lead to integrals
which are zero because of the antisymmetry of the integrand in (*α*,*β*). The rest of the terms cancel because they are integrals of derivatives of periodic functions:
and
Note the effect of the separation of *Γ*_{2} from *Γ*_{1}: the equation for *Γ*_{2} decouples,
4.14
where *A*_{1} is a constant, determined once and for all from the area enclosed by the initial curve *Γ*_{1}. On the other hand, *z*_{2} influences the evolution of *z*_{1} only through constant (in *α*) terms, *U*_{1}(*t*), given in (4.7), the winding number around zero of *Γ*_{2}, ind(0,*Γ*_{2}) and
4.15
and
4.16
The same decoupling occurs if we have a system of *N* widely separated curves where the vorticity jumps from one constant value to another. Now we are going to restrict our attention to the case in which the curve *z*_{2} has antipodal reflection symmetry
4.17
It is easy to see that, if the initial curve *z*_{2}(⋅,0) has antipodal reflection symmetry, then the solution of (4.14) has antipodal reflection symmetry for all time. This follows because the derivative also has antipodal reflection symmetry. If a curve *z* has antipodal reflection symmetry then
In particular, if *z*_{2} has antipodal reflection symmetry then
4.18
If the winding number of the outer curve around the origin is 1, then the equation for *z*_{1} becomes
4.19
and this equation respects antipodal symmetry: if initially present, the symmetry persists as long as the solution is smooth. We note that, if the winding number of the outer curve is non-zero, then, under our assumption of separation of contours, the outer curve surrounds the inner curve. Let us denote
4.20
the velocity of the origin
4.21
If the initial curves have antipodal reflection symmetry then *U*(0,*t*)=0, because both integrals vanish. This means that 0 is a stagnation point, i.e. a fixed point of the Lagrangian path, for all time. An example of such a configuration is formed with two concentric ellipses (not necessarily aligned). The centre of vorticity coincides with the origin in these cases. The system in which *z*_{2} has antipodal reflection symmetry is therefore
4.22
with *ζ* given by (4.15).

## 5. Inner ellipse

The system (4.22) has the remarkable property that, if the initial curve *Γ*_{1} is an ellipse
then it remains an ellipse
5.1
where *w*_{j}(*t*) solve the ordinary differential equation (ODE) system
5.2
The proof of this fact is based on the following lemma.

### Lemma 5.1

*Let* *with |ζ*_{1}*|>|ζ*_{−1}*|. Then*
5.3

### Proof.

The proof of the lemma is based on a calculation done already in [9], but for the sake of completeness we present it below. The first observation is that
where
is the circular Hilbert transform. This follows from the properties of the logarithm and
which is obtained by integration by parts. Now we write
with
and expand
Raising *δ*(*α*,*β*) to a power *k*, we obtain
and the only non-zero contribution to the integral
comes from the second term, so
Therefore,
and (5.3) follows from
Now, this proof seems to work only if |*ζ*_{1}−1|+|*ζ*_{−2}|<1, but the linear scaling property
valid for any , reduces the problem to this case. Indeed, if *ζ*_{1}=*r* e^{iϕ} and we choose *c*= (*r*+*ϵ*)^{−1} e^{−iϕ} then
This concludes the proof of the lemma. ▪

Noting that
5.4
we can write (5.2) as
5.5
The conservation in time of *A*_{1} (given in (5.4)) can be checked independently in (5.2). The system (4.22) now is reduced to the ODE system (5.5), where *ζ* is obtained from (4.15), coupling the ODE to equation (4.14). Kirchhoff ellipses are obtained by turning off the coupling, i.e. setting *ω*_{2}=0. The ODE system reduces further by considering the variable
5.6
This is a geometric quantity (see (3.2) and (3.3)):
The system (5.5) implies
In view of (5.4)
5.7
and therefore the equation for *w* is self-contained:
5.8
The variables *w*_{1} and *w*_{2} are easily obtained once *w* is known. In view of the geometric interpretation, we expect |*w*|=1 to be an invariant circle for the ODE. Indeed,
5.9
This shows that |*w*|=1 is an invariant circle for the equation. Moreover, in view of (5.4), if this set attracts a trajectory from inside (|*w*|<1), this means that the inner ellipse evolves in time and degenerates into a line . Indeed,
and |*w*|=1 implies *b*_{1}=0 and (because *A*_{1}=*a*_{1}*b*_{1} is finite). This can happen only if
Let us write now
5.10
with and consider the evolution of *w* in a co-moving frame, i.e. we introduce the variable
5.11
Equation (5.8) becomes
5.12
and equation (5.9) becomes
5.13
We rescale time in order to have non-dimensional quantities. Setting *τ*=(*ω*_{1}/2)*t* we have
5.14
Writing *u*=*x*+i*y*, we arrive at
5.15
where we denoted
5.16
Recall that *δ* and Δ are computed from *ζ*, which is computed from the outer curve *z*_{2}. Our choice of variable *u* is motivated in the next section, where we compute an approximation of *ζ* explicitly, and obtain *δ* and Δ explicitly and, in addition, constant in time.

## 6. Two ellipses

We saw that, if the initial curve *z*_{1}(⋅,0) in (4.22) is an ellipse, then it remains an ellipse for all time, all be it with changing length of semi-axis. This is no longer the case for the evolution of *z*_{2}, unless the initial curve is a circle, in which case it stays a circle. If the initial data is an ellipse with small eccentricity, the evolution away from the ellipse will take a long time; the farther the curve and the smaller the eccentricity, the longer the time. More precisely, if we start with
6.1
the right-hand side of equation (4.14) (which is the same as the second equation of (4.22)) introduces higher harmonics. These are introduced not by the nonlinear term, but by the term , which is small. We therefore approximate the evolution of *z*_{2} by projecting it on the elliptical modes. This is done in order to compute *ζ*(*t*) explicitly. Thus, if *z*_{2}=*ζ*_{1} e^{iα}+*ζ*_{2} e^{−iα} then we approximate
(Recall from (3.3) that circles correspond to *ζ*_{2}=0 with our orientation convention that |*ζ*_{1}|≥|*ζ*_{2}|.) With this, equation (4.14) with initial data (6.1) has solutions approximated by
6.2
with
6.3
Indeed, the approximate system is
and
so, from the second equation |*ζ*_{2}|^{2}=|*ζ*_{2}(0)|^{2}, and substituting in the first equation we arrive at (6.3). Now, it is elementary to check that, if *z*_{2}=*ζ*_{1} e^{−iα}+*ζ*_{2} e^{−iα}, then *ζ* given by (4.15) is computed by
because the series converges if |*ζ*_{2}|<|*ζ*_{1}|. We have that
6.4
where
6.5
and, without loss of generality, we assumed that *γ* is real. Indeed, in view of (3.2) and (3.3),
where *a*_{2} and *b*_{2} are the major and minor semi-axes of *Γ*_{2}(0) and is the angle *Γ*_{2}(0) makes with the coordinate system. So, assuming that *γ* is real amounts to choosing the axis so that *Ox* is in the direction of the major semi-axis of *Γ*_{2}(0). If *γ* is real, then
6.6
and (6.4) follows from (6.3). Thus *γ*=(*a*_{2}−*b*_{2})/(*a*_{2}+*b*_{2}) is time independent. The variable *u* defined in (5.11) describes the parameters of the inner ellipse *Γ*_{1}(*t*) in a frame which rotates with angular velocity −(1−*γ*^{2})(*ω*_{1}*A*_{1}+*ω*_{2}*A*_{2})/2*A*_{2}. In particular,
6.7
gives the ratio *a*_{1}/*b*_{1} of the major to minor semi-axes of *Γ*_{1}(*t*). The quantities *δ* and Δ defined in (5.16) and giving the coefficients in the system (5.15), which describes the evolution of *u*=*x*+i*y*, are
6.8
and
i.e.
6.9
They are constant because the lengths of the semi-axes of *Γ*_{2}(*t*), *a*_{2} and *b*_{2}, are constant in time.

## 7. The ODE system

We investigate the system (5.15),
7.1
Recall that *u*=*x*+i*y*, where *u* parametrizes the inner ellipse via (5.1), (5.6) and (5.11). In view of (6.7), we are interested in *x*^{2}+*y*^{2}≤1. The quantities *δ* and Δ are constants, fixed by the outer ellipse via (6.8) and (6.9). The fixed points of (7.1) are given by
7.2
and
7.3

By ‘attracts trajectories’, we mean that there exist (*x*_{0},*y*_{0}) with such that the solution (*x*(*t*),*y*(*t*)) of (7.1) with initial data (*x*_{0},*y*_{0}), satisfies

### Proof.

The quantity
7.5
is a conserved quantity for (7.1), as is easily verified. If we assume
then, from the hypothesis , , we derive a contradiction. Indeed, on the trajectory (*x*(*t*),*y*(*t*)), *K* takes the finite value *K*_{0} computed at (*x*_{0},*y*_{0}). Multiplying by (1−*x*^{2}−*y*^{2}) we obtain
and, on a sequence on which *x*(*t*_{j})^{2}+*y*(*t*_{j})^{2}→1 we deduce that ∈[−1,1] which is absurd. On the other hand, if (7.4) is satisfied, then the fixed points
7.6
lie on *x*^{2}+*y*^{2}=1. Analysing the linear stability we find that the linearized system is
7.7
and the fixed point (7.6) is stable if *δy*<0, and unstable if *δy*>0. By ODE theory, the stable fixed point has a non-empty open basin of attraction which therefore intersects *x*^{2}+*y*^{2}<1. This finishes the proof of the theorem except for the borderline case of |Δ/2*δ*|=1. This case necessitates a further study of the phase portrait of (7.1), which we also perform for other reasons. Before we do so, let us note that (7.4) holds if and only if
7.8
Note that (7.8) does not involve the aspect ratio of the inner ellipse. We investigate the system further. We take *δ*>0: in view of (6.8) and (7.8), this is the only possible case for instability if *ω*_{2} has the same sign as *ω*_{1}. We need to find out how many solutions of the cubic equation in (7.2) lie in *x*^{2}≤1. We look therefore for intersections of the curves *f*(*x*)=*x*−*x*^{3} and *g*(*x*)=*δx*^{2}−Δ*x*+*δ* in −1≤*x*≤1. The minimum of *g* is attained at *x*=Δ/2*δ* and is positive if *g*_{min}=*δ*−Δ^{2}/4*δ*>0. The maximum of *f* is obtained at and equals . All intersections will be in 0≤*x*≤1. There will be two intersections if and only if the point is situated above the graph of the parabola *y*=*g*(*x*). The reason for this is that *f*(0)=0<*g*(0)=*δ* and *f*(1)=0<*g*(1)=2*δ*−Δ. In this case there will be two roots, . If the point is situated below the graph of the parabola, there will be no intersections, and if it is on the parabola, there will be one intersection point. The conditions are thus
7.9
When there are two solutions, then the smaller one *x*_{1} is stable, and the larger one *x*_{2} is unstable. Numerically, it is easy then to see that there is a homoclinic orbit connecting *x*_{2} to itself and surrounding *x*_{1}. The circle *x*^{2}+*y*^{2}=1 is composed of two heteroclinic orbits going from the unstable fixed point on the circle to the stable one. There are also heteroclinic orbits connecting the unstable fixed point on the circle to *x*_{2} and *x*_{2} to the stable fixed point on the circle. If there is only one fixed solution then the previous picture simplifies, and, in addition to the two heteroclinic orbits on the unit circle, there are only heteroclinic orbits connecting the unstable fixed point on the circle to *x*_{1}=*x*_{2}, and connecting the latter to the stable fixed point on the circle. If there is no solution inside, then all orbits connect the unstable fixed point on the circle to the stable one. If |Δ/2*δ*|=1 then there are no orbits connecting the circle with the interior of the disc.

In view of the fact that |*u*|=|*w*| (see (5.11)), the upshot is that in all the cases obeying (7.4), when the unit circle attracts trajectories, it follows that where *w* is related to (5.1) by (5.6). Consequently, there is unbounded growth of the inner ellipse. Indeed, from the conservation of *A*_{1} and from (5.7) it follows that , and that means, in view of (3.3), that the sum of lengths of semi-axes of the inner ellipse diverges. ▪

## Competing interests

I declare I have no competing interests.

## Funding

Research partially supported by NSF-DMS grants nos 1209394 and 1265132.

## Footnotes

One contribution of 15 to a theme issue ‘Free boundary problems and related topics’.

- Accepted February 2, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.