VORTEX DYNAMICS FOR THE GINZBURG-LANDAU-SCHRÖDINGER EQUATION

J. E. COLLIANDER AND R. L. JERRARD

Abstract. The initial value problem for the Ginzburg-Landau-Schrödinger equation is examined in the ϵ → 0 limit under two main assumptions on the initial data ϕ^{ϵ}. The first assumption is that ϕ^{ϵ} exhibits m distinct vortices of degree ±1; these are described as points of concentration of the Jacobian [Jϕ^{ϵ}] of ϕ^{ϵ}. Second, we assume energy bounds consistent with vortices at the points of concentration. Under these assumptions, we identify “vortex structures” in the ϵ → 0 limit of ϕ^{ϵ} and show that these structures persist in the solution u^{ϵ}(t) of GLS_{ϵ}. We derive ordinary differential equations which govern the motion of the vortices in the ϵ → 0 limit. The limiting system of ordinary differential equations is a Hamitonian flow governed by the renormalized energy of Bethuel, Brezis and Hélein. Our arguments rely on results about the structural stability of vortices which are proved in a separate paper.

1. Introduction

We investigate the ϵ → 0 behavior of the initial value problem

\begin{cases}iu_{t}^{ϵ} − Δu^{ϵ} + \frac{1}{ϵ^{2}} (|u^{ϵ}|^{2} − 1)u^{ϵ} = 0, & u^{ϵ} : 𝕋^{2} × [0, T) ⟼ ℝ^{2} \\ u^{ϵ}(x, 0) = ϕ^{ϵ}(x), & x ∈ 𝕋^{2} \end{cases} GLS_{ϵ}

exhibiting a finite number of vortices, under appropriate assumptions on the associated Hamiltonian

I^{ϵ}[u] = ∫_{𝕋^{2}} E^{ϵ}(u)dx; E^{ϵ}(u) = \frac{1}{2}|Du^{ϵ}|^{2} + \frac{1}{4ϵ^{2}} (|u^{ϵ}|^{2} − 1)^{2}. (1.1)

The ϵ → 0 asymptotics of minimizers of I^{ϵ}[u] subject to Dirichlet boundary constraints were studied by Bethuel, Brezis and Hélein in [1]. The corresponding parabolic evolution problem was studied by Lin [13] and Jerrard and Soner [11]. These works identify “vortex structures” emerging in the ϵ → 0 limit whose location is governed by a renormalized energy. The renormalized energy is obtained by removing the divergent part of I^{ϵ}[u]. This note and the forthcoming paper [4] establish similar results for the conservative evolution GLS_{ϵ}.

The Landau theory of second order phase transitions [12] consists of expanding the energy in terms of a parameter which encodes the “order” in the phase and then exploiting energy properties to determine the evolution of the “order parameter”. This theory was applied by Ginzburg and Pitaevskii [8] and Pitaevskii [16] to argue that the order parameter describing superfluid helium II evolves according to GLS_{ϵ}. In this context, I^{ϵ} is the free energy and u^{ϵ} is the order parameter “which plays the role of ‘the effective wave function’ of the superfluid part of the liquid”

[8]. The motion of u^{ϵ} under the GLS_{ϵ} evolution conserves I^{ϵ}[u^{ϵ}]. If we express u^{ϵ}(x_{1}, x_{2}) = ρ(x_{1}, x_{2})e^{iθ(x_{1},x_{2})}, with ρ, θ ℝ-valued, then ρ^{2} represents the density of the superfluid and Dθ is the velocity of the superfluid. Gross [9] also derived GLS_{ϵ} as the Schrödinger equation for a wave function describing a system of interacting bosons. The equation GLS_{ϵ} is often called the Gross-Pitaevskii equation in the physics literature.

Because we are interested in studying the motion of vortices in the ϵ → 0 limit, we choose initial data ϕ^{ϵ} : 𝕋^{2} ⟼ ℂ possessing vortices. This heuristic notion is implemented by imposing two conditions. First, we assume the measure [Jϕ^{ϵ}] = det (Dϕ^{ϵ}) dx is close to a weighted sum of Dirac masses. More precisely, we assume

[Jϕ^{ϵ}] ⇀ π∑_{i=1}^{m}d_{i}δ_{α_{i}} (1.2)

where d_{i} = ±1 with ∑_{i=1}^{m}d_{i} = 0, and α_{i} are distinct points in 𝕋^{2}.

Second, we assume energy bounds. For example, certain of our results require that ϕ^{ϵ} satisfy

I^{ϵ}[ϕ^{ϵ}] ≤ πm log \frac{1}{ϵ} + γ_{1}. (1.3)

for some γ_{1} ∈ ℝ. It follows from results that are cited and discussed in Section 3 that assuption (1.2) implies the lower bound

I^{ϵ}[ϕ^{ϵ}] ≥ πm log \frac{1}{ϵ} − C, (1.4)

where C is independent of ϵ. Thus (1.3) asserts that the energy of ϕ^{ϵ} is O(1), given the prescribed configuration of vortices. Our strongest results need more stringent hypotheses, which amount to assuming that the energy is o(1), relative to the prescribed configuration of vortices.

We briefly outline the contents of this paper. Since we have not yet defined many of the terms we are using, this description is necessarily a little impressionistic.

Section 2 introduces notation and records some useful identities satisfied by solutions of GLS_{ϵ}.

Section 3 states and briefly discusses some results providing conditions under which the Jacobian of a function u (which in our context may be interpreted as the vorticity) and the Ginzburg-Landau energy density concentrate around a collection of points. These results provide a key element in our analysis. The proofs are long and technical and will appear in [4]. Section 3 also states some refined estimates related to the renormalized energy introduced by Bethuel, Brezis, and Hélein [1]. These too are proven in [4].

In Section 4 we prove that solutions of GLS_{ϵ} exhibit Lipschitz paths along which the energy and the vorticity concentrate. The main point is to control vortex mobility. We also describe weak limits of the “supercurrent” associated with solutions.

Finally, in Section 5 we give a complete description of limiting vortex dynamics, under somewhat stronger assumptions on the initial data. We also sharpen our characterization of the limiting behavior of solutions away from the vortices.

We ultimately show that, under appropriate hypotheses, the limiting vortices solve a system of ODEs, which is exactly that satisfied by classical point vortices in an ideal two dimensional fluid. This result was predicted as early as 1965 by Fetter [7], based on physical arguments, and more recently by Neu [15] and E [5], who arrived at the same conclusion through matched asymptotics.

2. Notation and identities

We introduce some vector notation and the quantity j(ϕ^{ϵ}). We then define measures [Jϕ^{ϵ}] and μ_{ϕ^{ϵ}}^{ϵ} which will be used to locate the singularities of ϕ^{ϵ}. Then a norm is introduced that permits us to say the singularities of two functions ψ^{ϵ} and ϕ^{ϵ} are close if the associated measur es [Jψ^{ϵ}], μ_{ψ^{ϵ}}^{ϵ} and [Jϕ^{ϵ}], μ_{ϕ^{ϵ}}^{ϵ} are close. Next, we derive some identities for [Ju^{ϵ}] and μ_{u^{ϵ}}^{ϵ} assuming u^{ϵ} evolves according to GLS_{ϵ}. Finally, we introduce the renormalized energy W and a canonical harmonic map H with singularities of degrees {d_{1}, …, d_{m}} located at the points {a_{1}, …, a_{m}} mapping 𝕋^{2} \ {a_{1}, …, a_{m}} ⟼ S^{1}.

We introduce some notation for vector fields u : 𝕋^{2} ⟼ ℝ^{2} mapping (x_{1}, x_{2}) ∈ 𝕋^{2} to (u^{1}(x_{1}, x_{2}), u^{2}(x_{1}, x_{2})) ∈ ℝ^{2}. The scalar product in ℝ^{2} is denoted by “·”,

u · v = u^{1}v^{1} + u^{2}v^{2}.

Sometimes we will view u ∈ ℝ^{2} as u ∈ ℂ when we write iu or e^{iα}u. These expressions are interpreted in the obvious way, e.g.

iu = −𝕁u; 𝕁 = (\begin{matrix}0 & 1 \\ −1 & 0 \end{matrix}) .

We define for u, v ∈ ℝ^{2},

u × v = u^{1}v^{2} − u^{2}v^{1} = 𝕁_{ij}u^{i}v^{j},
∇ × u = ∂_{x_{1}}u^{2} − ∂_{x_{2}}u^{1}.

Repeated indices are implicitly summed throughout this paper. Note that (iu) · v = u × v and (iu) · u = 0. For a scalar function ϕ we define

∇ × ϕ = (ϕ_{x_{2}}, −ϕ_{x_{1}}).

For a sufficiently differentiable u : 𝕋^{2} ⟼ ℝ^{2}, we define

j(u) = (u × u_{x_{1}}, u × u_{x_{2}}).

We sometimes write j(u) = (iu) · Du. In the superfluid model, j(u^{ϵ}) is interpreted as a current. If we write u = ρe^{iθ}, for ρ, θ ∈ ℝ, then j(u) = ρ^{2}Dθ. In particular, if |u| = 1, j(u) is the phase gr adient.

We define the signed Jacobian of u,

Ju = det Du. (2.1)

The quantities j(u) and Ju are related by the identity

∇ × j(u) = 2u_{x_{1}} × u_{x_{2}} = 2Ju, (2.2)

so the signed Jacobian can also be interpreted as the vorticity. We write [Ju] to denote the distributional signed Jacobian, defined if j(u) ∈ L^{1} via

∫ ϕ[Ju] = \frac{1}{2} ∫ (∇ × ϕ) · j(u), ϕ ∈ C_{c}^{1}(𝕋^{2}). (2.3)

In particular, [Ju] is well defined for u ∈ H^{1}(ℝ^{2}).

For a given u ∈ H^{1}(𝕋^{2}; ℝ^{2}), define the measure

μ_{u}^{ϵ}(A) = \frac{1}{| log ϵ|}∫_{A} E^{ϵ}(u) dx; E^{ϵ}(u) = \frac{1}{2}|Du|^{2} + \frac{1}{4ϵ^{2}} (|u|^{2} − 1)^{2},

for subsets A ⊂ 𝕋^{2}. The renormalization factor \frac{1}{| log ϵ|} appears naturally upon considering u = \frac{x}{|x|} smoothly cutoff in a ball of radius ϵ centered at x = 0.

We let M(𝕋^{2}) denote the dual of C(𝕋^{2}), i.e. the space of signed Radon measures on 𝕋^{2}. Similarly, we let M^{1}(𝕋^{2}) denote the dual of C^{1}(𝕋^{2}), equipped with the dual norm. We also define a seminorm

∥μ∥_{M̂^{1}(𝕋^{2})} = sup {∫ ϕ dμ : ∥Dϕ∥_{L^{∞}} ≤ 1, ∫ ϕ = 0} .

If μ(𝕋^{2}) = 0, we can compute ∥μ∥_{M^{1}(𝕋^{2})} by testing μ against functions ϕ such that ∫ ϕ dx = 0. In this case it follows that

c^{−1}∥μ∥_{M̂^{1}(𝕋^{2})} ≤ ∥μ∥_{M^{1}(𝕋^{2})} ≤ ∥μ∥_{M̂^{1}(𝕋^{2})}.

This is a consequence of the fact that ∥ϕ∥_{C^{1}(𝕋^{2})} ≤ C∥Dϕ∥_{L^{∞}(𝕋^{2})} whenever ∫ ϕ = 0. In particular, if μ has the form

μ = ∑_{i=1}^{n}δ_{ξ_{i}} − ∑_{i=1}^{n}δ_{η_{i}} (2.4)

for some points ξ_{1}, …, ξ_{n}, η_{1}, …, η_{n} ∈ 𝕋^{2}, not necessarily distinct, then Brezis, Coron and Lieb [3] show that

∥μ∥_{M̂^{1}(𝕋^{2})} = min_{π∈S_{n}}∑_{i=1}^{n}|ξ_{i} − η_{π(i)}| (2.5)

where S_{n} is the symmetric group of permutations of n objects. For measures of the form (2.4) we thus have

c^{−1}min_{π∈S_{n}}∑_{i=1}^{n}|ξ_{i} − η_{π(i)}| ≤ ∥μ∥_{M^{1}(𝕋^{2})} ≤ min_{π∈S_{n}}∑_{i=1}^{n}|ξ_{i} − η_{π(i)}|.

This estimate permits us to prove a fact illustrating the usefulness of the M^{1} norm.

Proposition 1. Suppose that for every t ∈ [0, T), μ_{t} is a measure of the form

∑_{i=1}^{n}δ_{a_{i}(t)}, for certain points a_{1}(t), …, a_{n}(t). The μ_{(·)} is a countiuous (resp. Lipschitz)

function from [0, T) into M^{1} if and only if the points a_{i}(t) can be labelled in such a way th at a_{i}(·) is continuous (resp. Lipschitz) for each i.

Proof. For any s, t, the measure μ_{t} − μ_{s} has integral zero and so satisfies

∥μ_{t} − μ_{s}∥_{M^{1}(𝕋^{2})} ∼ min_{π∈S_{n}}∑_{i}|a_{i}(t) − a_{π(i)}(t)|.

The lemma follows immediately.

Remark 1. The M̂^{1} seminorm can be interpreted as the minimum cost in a Monge-Kantorovich mass transfer problem. Indeed, when ∫ dμ = 0 as above, μ can be written in the form μ = ν_{1} − ν_{2}, where ν_{1} and ν_{2} are positive, mu tually singular measures and ν_{1}(𝕋^{2}) = ν_{2}(𝕋^{2}). Then ∥μ∥M̂^{1} is precisely the minimum cost of “transporting” ν_{1} to ν_{2}, subject to an appropriate cost functional. See for example [6] f or a more precise statement and more details.

Remark 2. Any reasonable weak norm on measures would be equally suitable for our purposes. The M^{1} norm is a convenient choice, but is certainly not the only possible choice.

We employ O and o notation in some of the analysis below. We write, for example, O_{a,b,c}(1) to indicate a quantity is O(1) with respect to the interesting limit ϵ → 0, with the implicit constant depending only upon the parameters a, b, c.

We assume in this section, for fixed ϵ > 0, that u solves

iu_{t} − Δu + \frac{1}{ϵ^{2}} (|u|^{2} − 1) u = 0

and derive evolution equations for certain functions of u and its derivatives. These are mostly well known. We formally calculate the evolution equation for E^{ϵ}(u) = \frac{1}{2}|Du|^{2} + \frac{1}{4ϵ^{2}} (|u|^{2} − 1)^{2}.

\frac{d}{dt}E^{ϵ}(u) = u_{x_{i}} · u_{x_{i}t} + \frac{1}{ϵ^{2}} (|u|^{2} − 1)u · u_{t}
= (u_{x_{i}} · u_{t})_{x_{i}} − (u_{x_{i}x_{i}} − \frac{1}{ϵ^{2}} (|u|^{2} − 1)) · u_{t}

so that

\frac{d}{dt}E^{ϵ}(u) = (u_{x_{i}} · u_{t})_{x_{i}} (2.6)

using the equation and (iu_{t}) · u_{t} = 0.

Similarly, we calculate

\frac{d}{dt} \frac{1}{2}|u|^{2} = u · u_{t} = −(iu) · Δu

to obtain

\frac{d}{dt} \frac{1}{2}|u|^{2} = −((iu) · u_{x_{i}})_{x_{i}} = −div j(u). (2.7)

Next, we consider the evolution of a component of j(u).

\frac{d}{dt}j^{k}(u) = \frac{d}{dt}(iu) · u_{x_{k}} = (iu_{t}) · u_{x_{k}} + (iu) · u_{tx_{k}}
= (iu_{t}) · u_{x_{k}} − u · (iu_{tx_{k}})
= u_{x_{j}x_{j}} · u_{x_{k}} − u · u_{x_{j}x_{j}x_{k}} + \frac{1}{ϵ^{2}} |u|^{2}|u|_{x_{k}}^{2}
= (u_{x_{k}} · u_{x_{j}} − u · u_{x_{k}x_{j}})_{x_{j}} + \frac{1}{ϵ^{2}} |u|_{x_{k}}^{4}.

We rewrite as

\frac{d}{dt}j^{k}(u) = 2(u_{x_{k}} · u_{x_{j}})_{x_{j}} − (|u_{x_{j}}|^{2} − u · u_{x_{j}x_{j}})_{x_{k}} + \frac{1}{2ϵ^{2}} |u|_{x_{k}}^{4}. (2.8)

Note that this shows _{∫𝕋^{2}}j(u)dx is time-independent.

We calculate the evolution equation of Ju using (2.2)

\frac{d}{dt}Ju = ∇ × (Du · u_{x_{j}})_{x_{j}},

which we write out explicitly,

\frac{d}{dt}Ju = 𝕁_{jk}(u_{x_{k}} · u_{x_{l}})_{x_{l}x_{j}}. (2.9)

Finally, we express (2.9) in a weak form. Multiply (2.9) by η ∈ C_{c}^{∞}(𝕋^{2}) and integrate to find

\frac{d}{dt}_{∫𝕋^{2}}η Ju dx = _{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk}u_{x_{k}} · u_{x_{l}} dx. (2.10)

This is the key identity in our analysis of vortex dynamics of GLS_{ϵ}.

Remark 3. The preceding calculations apply equally well to u^{ϵ}(·) ∈ H^{1}(𝕋^{2}). For ϵ > 0 fixed, GLS_{ϵ} is a defocussing nonlinear Schrödinger equation. Bourgain has established [2] global wellposedness for GLS_{ϵ} below H^{1}. We validate the preceding calculations for u^{ϵ} ∈ H^{1}(𝕋^{2}) as follows using various aspects of Bourgain’s result. By continuous dependence on the data, a different solution ũ^{ϵ}(t) is close to u^{ϵ}(t) in H^{1}(𝕋^{2}) provided the corresponding initial data ϕ̃^{ϵ} and ϕ^{ϵ} are close in H^{1}(𝕋^{2}). Let ϕ̃^{ϵ} be a smooth approximator to ϕ^{ϵ}. The preceding calculations apply to ũ^{ϵ} since it remains smooth for all time. The various identities above, in particular (2.10), are then validated for u^{ϵ} ∈ H^{1}(𝕋^{2}) by considering a sequence of smooth approximators.

We reformulate for the torus 𝕋^{2} some of the notions introduced by Bethuel, Brezis and Hélein [1] for the study of Dirichlet minimizers of I^{ϵ}[u]. Let F solve

ΔF = 2π(δ_{0} − 1) on 𝕋^{2}. (2.11) with the normalization

lim_{x→0}(F (x) − log |x|) = 0 (2.12)

For points a_{1}, ..., a_{m} ∈ 𝕋^{2} (sometimes denoted by (a)) and nonzero integers d_{1}, ..., d_{m} satisfying ∑ d_{i} = 0, define the renormalized energy

W (a, d) := −π ∑_{i≠j} d_{i}d_{j}F (a_{i} − a_{j}). (2.13)

Define

Φ = ∑_{i=1}^{m} d_{i}F (x − a_{i}) (2.14)

and notice that

ΔΦ = 2π∑_{i=1}^{m}d_{i}δ_{a_{i}}, ∫ Φ = 0.

The following proposition is proved in [1] (Theorem 1.1., Step 2 of proof).

Proposition 2. There is a map H ∈ Cloc^{∞}(𝕋^{2} \ (a); S^{1}) ∩ W^{1,1}(𝕋^{2}; S^{1}) satisfying

j(H) = −∇ × Φ. (2.15)

Moreover, if H̃ is any other function satisfying (2.15), then H̃ = e^{iα}H for some α ∈ ℝ.

It follows from (2.15) that H satisfies

div j(H) = 0, (2.16)

2[JH] = ∇ × j(H) = 2π ∑_{i=1}^{m} d_{i}δ_{a_{i}}. (2.17)

in the sense of distributions and pointwise away from the singluarities (a). The map H also satisfies

∫_{𝕋^{2}} j(H)dx = 0. (2.18)

The map H has singularities of degree (d) located at the points (a) as seen in (2.17). The identity (2.16) reveals that H is a harmonic map into S^{1} since j(H) = Dθ when we express H = e^{iθ}. Therefore, we refer to H as the canonical harmonic map^{1} with singularities of degree (d) located at the points (a). We will sometimes highlight the dependence of H upon (d) and (a) by writing H(x; a, d).

For a collection of points a_{1}, ..., a_{m} ∈ 𝕋^{2} define

𝕋_{ρ}^{2} := 𝕋^{2} \ ∪_{i}B_{ρ}(a_{i}).

1We have taken the name “canonical harmonic map” from the work of Bethuel, Brezis, and Hélein on the Dirichlet problem, where it is entirely appropriate. In the periodic context it is clearly something of a misnomer, since H is not unique. Nonetheless, it seems easiest to use the familiar terminology.

We will be interested in the case

0 < ρ < \frac{1}{4} min_{i≠j} |a_{i} − a_{j}|. (2.19)

A configuration of vortices (a), (d) determines (up to a constant phase factor e^{iα}) a canonical harmonic map H(·, a, d). The map H has logarithmically divergent Dirichlet energy. The renormalized energy W (a, d) describes to leading orde r the finite part of the energy associated with the configuration (a), (d). The following proposition, essentially proven in [1] (Theorem I.7), makes these observations precise.

Proposition 3. Let H = H(·, a, d). For ρ satisfying (2.19),

∫_{𝕋_{ρ}^{2}} \frac{1}{2}|DH|^{2}dx = mπ log \frac{1}{ρ} + W (a, d) + O(ρ).

We define, following Bethuel, Brezis, and Hélein [1], the quantity

I(ϵ, ρ) = min \{∫_{B_{ρ}} E^{ϵ}(u)dx : u ∈ H^{1}(B_{ρ}), u(x) = \frac{x}{|x|} for x ∈ ∂B_{ρ}\} . (2.20)

It is shown in [1] that

I(ϵ, ρ) = π log(\frac{ρ}{ϵ}) + O(1) (2.21)

as ϵ → 0, for ρ fixed.

An easy adaptation of the construction given in ([1], Lemma VIII.1) establishes

Proposition 4. Given any distinct a_{1}, ..., a_{m} ∈ 𝕋^{2} and d_{1}, ..., d_{m} ∈ {±1}, there exists a family of functions v^{ϵ} ∈ H^{1}(𝕋^{2}; ℝ^{2}) such that

[Jv^{ϵ}] ⇀ ∑_{i=1}^{m}πd_{i}δ_{a_{i}} weakly in M

and for every ρ > 0

∫_{𝕋^{2}} E^{ϵ}[v^{ϵ}]dx ≤ m(π log \frac{1}{ρ} + I(ϵ, ρ)) + W (a, d) + Cρ + o(1)

as ϵ → 0.

Remark 4. The condition [Jv^{ϵ}] ⇀ ∑ πd_{i}δ_{a_{i}} weakly in M implies the energy lower bound

∫𝕋^{2}E^{ϵ}(u^{ϵ})dx ≥ m (π log \frac{1}{ρ} + I(ϵ, ρ)) + W (a, d) + C̃ρ + o(1)

for some constant C̃ and every ρ > 0 satisfying (2.19). This follows from Theorem 2 below. Therefore, the functions v_{ϵ} of Proposition 4 are asymptotically energy minimizing, for the given configuration of vortices.

3. Concentration and structure

This section identifies two hypotheses on a function u^{ϵ} which imply detailed structural properties of u^{ϵ} as ϵ → 0. The first hypothesis is that [Ju^{ϵ}] is somewhat close to a weighted sum of Dirac masses. The second is an energy upper bound consistent with the number of Dirac masses. Theorem 1 says that these hypotheses imply the measures [Ju^{ϵ}] and μ_{u^{ϵ}}^{ϵ} concentrate to Dirac masses and that u^{ϵ} is uniformly regular away from the points of concentration. A more stringent energy upper bound on u^{ϵ} controls the current j(u^{ϵ}) and D|u^{ϵ}| as stated in Theorem 2. The proofs, which appear in [4], rely heavily on techniques developed in [10]. Similar techniques have been used by Sandier [17].

Theorem 1. (Global structure) Suppose that u ∈ H^{1}(𝕋^{2}; ℝ^{2}), and that there exist points x_{1}, ..., x_{m} ∈ 𝕋^{n}, integers d_{1}, ..., d_{m} ∈ \{±1\}, and ϵ ≤ r := \frac{1}{4} min_{i≠j} |x_{i} −x_{j}| such that

|| [Ju] − π ∑_{i=1}^{m} d_{i}δ_{x_{i}} ||_{M^{1}(𝕋^{2})} ≤ \frac{π}{200}r, (3.1)

and

∫_{𝕋^{2}} E^{ϵ}(u) dx ≤ πm log (\frac{r}{ϵ}) + γ_{1} (3.2)

for some constant γ_{1}. Then there exists points a_{i} ∈ B_{r/2}(x_{i}), i = 1, ..., m such that

||μ_{u}^{ϵ} − π∑_{i=1}^{m}δ_{a_{i}}||_{M^{1}(𝕋^{2})} ≤ O_{γ_{1}} (\frac{1}{| log ϵ|}) , (3.3)

||[Ju] − π∑_{i=1}^{m}d_{i}δ_{a_{i}}||_{M^{1}(𝕋^{2})} ≤ o_{γ_{1}}(1), (3.4)

Moreover, for ρ fixed and 0 < ρ < \frac{r}{2},

∫_{𝕋_{ρ}^{2}} E^{ϵ}(u)dx ≤ O_{ρ,γ_{1}}(1). (3.5)

∥Du∥_{L^{2}(𝕋_{ρ}^{2})} ≤ O_{ρ,γ_{1}}(1), (3.6)

Finally for any p ∈ [1, 2),

||Du||_{L^{p}(𝕋^{2})} ≤ O_{p,γ_{1}}(1) (3.7)

and

||j(u)||_{L^{p}(𝕋^{2})} ≤ O_{p,γ_{1}}(1) (3.8)

We have some remarks before commenting on the proof.

Remark 5. The estimate (3.1) says [Ju] is O(1)-close to a weighted sum of Dirac masses located at the points x_{i}. The energy upper bound (3.2) allows us to locate points a_{i} near x_{i} and conclude [Ju] is o(1)-close to a weig hted sum of Dirac masses located at the points a_{i}. This observation is exploited in the proof of Theorem 3 below.

Remark 6. Theorem 1 implies several other useful estimates. For u satisfying the hypotheses of the theorem, we have by (3.2) that

∥|u|^{2} − 1∥_{L^{2}(𝕋^{2})} ≤ Cϵ(log \frac{r}{ϵ} + 1)^{\frac{1}{2}} . (3.9)

Interpolating with (3.7) gives for any p < ∞

∥u∥_{L^{p}(𝕋^{2})} ≤ C_{p}. (3.10)

For any fixed ρ with 0 < ρ < \frac{r}{2}, the estimate (3.6) implies

∥Ju∥_{L^{1}(𝕋_{ρ}^{2})} ≤ C_{ρ}. (3.11)

Remark 7. Related results for vector fields from ℝ^{n} ⟼ ℝ^{n} generalizing the two dimensional results above appear in [4].

We briefly describe the idea of the proof of Theorem 1 appearing in [4]. The theorem follows from a localized version with the hypotheses

∥[Ju] − πdδ_{x_{i}}∥_{M^{1}(B_{r}(x_{i}))} ≤ \frac{π}{200}r

∫B_{r}(x_{i})E^{ϵ}(u)dx ≤ π log \frac{r}{e} + γ_{1}.

The Jacobian condition implies u must exhibit nonzero degree in many subsets of B_{r}. One way this condition can be satisfied, which is consistent with the assumed upper bound, is if u has a single isolated vortex of degree d near the center x_{i}. Careful lower bounds show that, if u deviates significantly from this description — for example, if u has more than one vortex, or if the energy or vorticity of u is spread out too much — then the total energy of u would be too large, contradicting the energy bound. The main technical point is thus lower bounds. These are proved by identifying some set which contains all concentrations of vorticity, and then covering it by balls in some optimal fashion. One can use this covering to relate the energy of the function to the distribution of vorticity. As noted above, this sort of argument has appeared before in [10] and in the work of Sandier [17].

The next result provides additional information about u^{ϵ} under a stronger energy upper bound hypothesis.

Theorem 2. Suppose that u^{ϵ} ∈ H^{1} is a sequence such that

(3.12) [Ju^{ϵ}] ⇀ π ∑_{i=1}^{m} d_{i}δ_{a_{i}} weakly as measures, and that there exists some γ_{2} ≥ 0 such that

∫ E^{ϵ}(u^{ϵ})dx ≤ m (π log \frac{1}{ρ} + I(ϵ, ρ)) + W (a, d) + Cρ + γ_{2} + o(1) (3.13)

as ϵ → 0, for every ρ > 0. Then

lim sup_{ϵ→0} ||\frac{1}{|u^{ϵ}|}j(u^{ϵ}) − j(H)||_{L^{2}(𝕋_{ρ}^{2})}^{2} ≤ Cγ_{2} (3.14)

and

lim sup_{ϵ→0} || D|u^{ϵ}| ||_{L^{2}(𝕋_{ρ}^{2})}^{2} ≤ Cγ_{2} (3.15)

for every ρ > 0. Here H = H(·; a, d) is the canonical harmonic map and C is a universal constant.

Remark 8. Observe that the conclusion of the theorem is ridiculous if γ_{2} is negative which implies the claim in Remark 4 .

We conclude this section with a brief discussion of the m = 0 case. Consider GLS_{ϵ} with initial data ϕ^{ϵ} = ρ^{ϵ}e^{iθ} with ρ^{ϵ}, θ ∈ ℝ. The solution is

u^{ϵ}(x, t) = ρ^{ϵ}e^{i[θ+ \frac{1}{ϵ^{2}} ((ρ^{ϵ})^{2}−1)t]}.

If |(ρ^{ϵ})^{2} − 1| > O(ϵ^{α}) for α < 2 then u^{ϵ} oscillates rapidly in the t variable and u^{ϵ} ⇀ 0 weakly in L^{2}(dxdt) as ϵ → 0. The canonical harmonic map in this case is H(x) = e^{iθ_{1}} for some constant θ_{1} ∈ [0, 2π). The rapid temporal oscillations require |(ρ^{ϵ})^{2} − 1| = O(ϵ^{2}) to have u^{ϵ} → H in any strong sense.

The assumptions (3.2) and m = 0 imply |(ρ^{ϵ})^{2} − 1| = O(ϵ). Contrast this with assumption (3.13) under the conditions γ_{2} = 0 and m = 0 which yields |(ρ^{ϵ})^{2} − 1| = o(ϵ).

The quantity j(u) is insensitive to the temporal oscillations in u since j(u) = j(e^{iα}u) for α ∈ ℝ. It is therefore natural to expect j(u) to retain more information in the ϵ → 0 limit. This expectation is verified later in Theorem 3(iv).

4. Vortex paths and convergence

Theorem 1 describes a function u^{ϵ} (through the measures [Ju^{ϵ}] and μ_{u}^{ϵ} ) provided that u^{ϵ} satisfies the hypotheses (3.1) and (3.2). The theorem identifies Dirac masses as the ϵ → 0 limits of [Ju^{ϵ}], μ_{u}^{ϵ}. The hypotheses (3.1) and (3.2) are chosen to allow for a similar description of u^{ϵ}(t), the solution of GLS_{ϵ} with initial data u^{ϵ}(0) = ϕ^{ϵ}, at each t in a time interval [0, T). In this section, we exploit the identity (2.10) and the results stated in Section 3 to show the Dirac masses in Theorem 1 move along Lipschitz paths provided u^{ϵ} solves GLS_{ϵ}. The paths are characterized as solutions of certain ordinary differential equations in the next section under stronger assumptions on the initial data.

We assume the initial data ϕ^{ϵ} ∈ H^{1}(𝕋^{2}; ℝ^{2}) satisfies the hypotheses (1.2) and (1.3). So, there exist distinct points {α_{1}, …, α_{m}} ⊂ 𝕋^{2}, integers {d_{1}, …, d_{m}} with d_{i} = ±1 and ∑ d_{i} = 0, fixed independently of ϵ such that

[Jϕ^{ϵ}] ⇀ π∑_{i=1}^{m}d_{i}δ_{α_{i}} (4.1)

∫_{𝕋^{2}} E^{ϵ}(ϕ^{ϵ}) dx ≤ πm log \frac{1}{ϵ} + γ_{1}, (4.2)

We will also assume

∫𝕋^{2}j(ϕ^{ϵ})dx = 0. (4.3)

In light of (2.8), this implies ∫ j(u^{ϵ}(t))dx = 0 for all time. This assumption is not essential but simplifies the exposition. There are modifications to the statements of Theorems 3 and 4 if ∫ j(u^{ϵ}(t))dx ≠ 0 which appear in [4].

For ϵ > 0, Bourgain’s recent work [2] shows GLS_{ϵ} is globally (in time) wellposed below H^{1}. More precisely, there exists s_{0} < 1 such that for all s > s_{0} there is a uniquely defined continuous map

H^{s} ⟼ X ⊂ C([0, ∞); H^{s})

taking

ϕ^{ϵ} ⟼ u^{ϵ},

where u^{ϵ} is the solution of GLS_{ϵ}.

We can now state our first main theorem.

Theorem 3. Let u^{ϵ} be the solution of GLS_{ϵ} with initial data ϕ^{ϵ} satisfying (4.1) and (4.2) and (4.3). Then, after passing to a subsequence as ϵ → 0, there exists a T > 0 (independent of ϵ) and Lipschitz pat hs a_{i} : [0, T) ⟼ 𝕋^{2}, a_{i}(0) = α_{i} such that

(i) [Ju^{ϵ}(t)] ⇀ π∑_{i=1}^{m}d_{i}δ_{a_{i}(t)} weakly as measures for all t ∈ [0, T).

(ii) μ_{u^{ϵ}(t)}^{ϵ} ⇀ π∑_{i=1}^{m}δ_{a_{i}(t)} weakly as measures for all t ∈ [0, T).

(iii) |u^{ϵ}(t)|^{2} → 1 in L^{2}(dx) for all t ∈ [0, T). (iv) j(u^{ϵ}) = (iu^{ϵ}) · Du^{ϵ} ⇀ j(H)) weakly in L^{p}(dxdt) for all 1 ≤ p < 2 where H(·, t) = H(·, a(t), d) is the canonical harmonic map with singularities of degree (d) located at (a(t)).

Proof. 1. Define for r = min_{i≠j} \frac{1}{4}|α_{i} − α_{j}| with 0 < ϵ ≪ r ≤ 1 the quantity

T^{ϵ} = sup \{t ≥ 0 : ∥[Ju^{ϵ}(s)] − [Jϕ^{ϵ}]∥_{M^{1}(𝕋^{2})} ≤ \frac{π}{400}r for all 0 ≤ s ≤ t\} .

Recall that Ju = det Du = u_{x_{1}}^{1}u_{x_{2}}^{2} − u_{x_{2}}^{1}u_{x_{1}}^{2}. The estimates

∥[Ju^{ϵ}] − [Jϕ^{ϵ}]∥_{M^{1}} ≤ ∥Ju^{ϵ} − Jϕ^{ϵ}∥_{L^{1}} ≤ C∥u^{ϵ} − ϕ^{ϵ}∥_{H^{1}}∥u^{ϵ} + ϕ^{ϵ}∥_{H^{1}}

and continuity of the flow of ϕ^{ϵ} ⟼ u^{ϵ}(t) through H^{1} guarantee T^{ϵ} > 0.

2. Claim: For s, t satisfying 0 ≤ s, t ≤ T^{ϵ} we have

∥[Ju^{ϵ}(s)] − [Ju^{ϵ}(t)]∥_{M^{1}(𝕋^{2})} ≤ c|s − t| + o_{γ_{1}}(1).

Proof of Claim: The definition of T^{ϵ} guarantees for all t ∈ [0, T^{ϵ}) that the hypothesis (3.1) of Theorem 1 holds if ϵ is sufficiently small. Therefore, for each t ∈ [0, T^{ϵ}), we can find points a_{i}(t) ∈ B\frac{r}{2} (α_{i}), i = 1, …, m, for which

∥[Ju^{ϵ}(t)] − π∑_{i=1}^{m}d_{i}δ_{a_{i}(t)}∥_{M^{1}(𝕋^{2})} ≤ o_{γ_{1}}(1), (4.4)

by (3.4). Of course, the a_{i}(t) may depend upon ϵ. So we can estimate

∥[Ju^{ϵ}(s)] − [Ju^{ϵ}(t)]∥_{M^{1}(𝕋^{2})} ≤ ∥π∑_{i=1}^{d}d_{i}(δ_{a_{i}(s)} − δ_{a_{i}(t)})∥_{M^{1}(𝕋^{2})} + o_{γ_{1}}(1).

By (2.5), we can estimate by

≤ π∑_{i=1}^{m}|a_{i}(s) − a_{i}(t)| + o_{γ_{1}}(1).

The claim will be established once we show for i = 1, …, m,

|a_{i}(s) − a_{i}(t)| ≤ c|s − t| + o_{γ_{1}}(1). (4.5)

3. We prove (4.5) by using the identity (2.10). Fix i and observe that a_{i}(s), a_{i}(t) ∈ B\frac{r}{2} (α_{i}) for all s, t ∈ [0, T^{ϵ}). There exists an η ∈ C_{c}^{∞}(B_{r}(α_{i})) satisfying

η(x) = ν · x for x ∈ B_{\frac{3r}{4}} (a_{i}(0)), ν ∈ S^{1}

and

π|a_{i}(s) − a_{i}(t)| = πd_{i} ∫ η(δ_{a_{i}(s)} − δ_{a_{i}(t)}).

The conditions on η guarantee that supp(D^{2}η) ⊂ B_{r}(α_{i}) \ B_{\frac{3r}{4}} (α_{i}). Notice that η depends upon the index i.

Insert the function η described above into (2.10) and again use (4.4) to observe

π|a_{i}(s) − a_{i}(t)| = ∫_{s}^{t} ∫_{B_{r}(α_{i})} η_{x_{j}x_{l}}𝕁_{jk} u_{x_{k}} · u_{x_{l}} dx dτ + o_{γ_{1}}(1).

The support properties of η_{x_{j}x_{l}} permit us to replace B_{r}(α_{i}) by B_{r}(α_{i}) \ B_{\frac{3r}{4}} (α_{i}). Finally, we estimate by

≤ |s − t| ∥D^{2}η∥_{L^{∞}} sup_{τ∈[s,t]}∥Du(τ)∥_{L^{2}(B_{r}\B_{\frac{3r}{4}} )}.

The size of ∥D^{2}η∥_{L^{∞}} depends upon r but is independent of ϵ and (3.6) permits us to control the Du term by a constant independent of ϵ, so (4.5) follows and the claim is proven. We also note that the claim implies T^{ϵ} may be taken independently of ϵ, so we denote this quantity by T from now on.

4. The remaining convergence claims follow from the bounds stated in Theorem 1 and passing to subsequences, except for (iii) which follows directly from (4.2). We prove (iv). Fix any p ∈ [1, 2). Since the conditions of Theorem 1 hold for every t ∈ [0, T), we deduce from (3.8) that

∥j(u^{ϵ})∥_{L^{p}(𝕋^{2}×[0,T))} ≤ O_{p,γ_{1},T} (1).

It follows, upon passing to a subsequence as ϵ → 0, that

j(u^{ϵ}) ⇀ j̄ weakly in L^{p}(dxdt)

for some j̄. We wish to identify j̄.

Let ϕ ∈ C_{0}^{∞}(𝕋^{2} × [0, T)). The identity (2.7) implies

∫ j(u^{ϵ}) · Dϕdxdt = ∫ ϕ_{t} \frac{|u^{ϵ}|^{2}}{2} dxdt
= ∫ ϕ_{t} \frac{1}{2}(|u^{ϵ}|^{2} − 1)dxdt
→ 0

as ϵ → 0 for every t, by (iii). Therefore div j̄ = 0. Moreover, from (i) we have ∇ × j̄ = 2[J̄] ⊗ dt = 2π ∑ d_{i}δ_{a_{i}(t)} ⊗ dt weakly.

Let H(x, t) = H(x, a(t), d). If we define V = j̄ − j(H), we have

div V = ∇ × V = 0

weakly. Let η^{δ} be a standard mollifier and set V^{δ} = V ∗ η^{δ}. The convolution here is in space and time. The above considerations imply

div V^{δ} = ∇ × V^{δ} = 0

in 𝕋^{2} for every t < T. Since V^{δ} is smooth, this implies V^{δ}(x, t) = g^{δ}(t). Letting δ → 0, we find that V is also constant in x for each fixed t. For any fixed t we have

∫ V (x, t)dx = ∫ (j̄ − j(H)) (x, t)dx
= ∫ j̄(x, t) dx
= lim_{ϵ→0} ∫ j(u^{ϵ})(x, t)dx
= 0

using (4.3).

Remark 9. The proof given above may be iterated until

T = inf{t > 0 : |a_{i}(t) − a_{j}(t)| → 0 for some i ≠ j}.

5. Vortex dynamics

Under the more restrictive upper bound on the energy (3.13), we characterize the vortex paths a_{i}(·) as solutions of a system of ordinary differential equations that conserves the renormalized energy. More precisely, we have the follo wing result.

Theorem 4. Suppose ϕ^{ϵ} satisfies

(5.1) [Jϕ^{ϵ}] ⇀ π∑_{i=1}^{m}d_{i}δ_{α_{i}} weakly as measures,

and for every ρ > 0 as ϵ → 0,

∫𝕋^{2}E^{ϵ}(ϕ^{ϵ})dx ≤ m (π log \frac{1}{ρ} + I(ϵ, ρ)) + W (α, d) + Cρ + o(1). (5.2)

Let {a_{1}(·), …, a_{m}(·)} and H be as in Theorem 3. Then the following statements hold:

(i) For each i and for t ∈ [0, T_{1}),

\begin{cases}\frac{d}{dt}a_{i} = 2 ∑_{j:j≠i}d_{j}∇ × F (a_{i} − a_{j}) = −\frac{1}{π}d_{i}𝕁D_{a_{i}}W (a, d) \\ a_{i}(0) = α_{i}. \end{cases}

Here T_{1} > 0 can be taken to be the largest time such that the above ODE has a solution on [0, T_{1}). (ii) For every t ∈ [0, T_{1}),

\frac{1}{|u^{ϵ}|}j(u^{ϵ}) → j(H) strongly in Lloc^{2}(𝕋^{2} \ (a(t))).

(iii) For every ρ > 0 and t ∈ [0, T_{1}),

lim_{ϵ→0} min_{α∈[0,2π)}∥u^{ϵ}(·, t) − e^{iα}H(·, a(t), d)∥_{H^{1}(𝕋_{ρ}^{2})} = 0

Remark 10. For an example of a point vortex system which develops a singularity in finite time, see Marchioro and Pulvirenti [14]. We expect, however, that for generic initial data the ODE does not develop singularities, in which case the above result is valid globally in time.

Remark 11. The example given at the end of Section 3 shows that the sort of convergence in (iii) is more or less optimal, unless (5.2) is strengthened still further.

Proof. 1. Let a_{i}(t), i = 1, …, m denote the paths of Theorem 2. Let b_{i}(t) denote the solution of the system

\begin{cases}\frac{d}{dt}b_{i} = 2 ∑_{j:j≠i}d_{j}∇ × F (b_{i} − b_{j}) \\ b_{i}(0) = α_{i}. \end{cases} (5.3)

A calculation shows that

D_{a_{i}}W = −2π ∑_{j:j≠i}d_{i}d_{j}DF (a_{i} − a_{j}).

Therefore, the ODE in (5.3) may be reexpressed, as in (i), in Hamiltonian form showing that the renormalized energy W is conserved. This ODE system has a unique solution on a nontrivial time interval [0, T′). Let

T_{1} = min(T, T′)

where T is as in Remark 9. Note that T_{1} is independent of ϵ. We wish to show for all i that b_{i}(t) coincides with a_{i}(t) on the time interval [0, T_{1}). Observe that this will imply T_{1} = T′.

For t ∈ [0, T_{1}), let

ζ(t) = ∑_{i}|b_{i}(t) − a_{i}(t)|.

It suffices to prove that, given any T̃ < T_{1}, we can find some small δ(T̃) > 0 and a constant C = C(T̃) such that

\frac{d}{dt}ζ(t) ≤ Cζ(t) (5.4)

for a.e. t ∈ [0, T̃] whenever ζ(t) ≤ δ. We will show that (5.4) holds at each point where a_{i}(·) is differentiable for all i; by Rademacher’s theorem, this condition is satisfied on a set of full measure.

Fix T̃ < T_{1}. By Remark 9, there is some r = r(T̃) > 0 for which

min_{i≠j, t≤T̃}|a_{i}(t) − a_{j}(t)| ≥ 4r. (5.5)

2. We use the fact that b_{i} solves (5.3) and the triangle inequality to estimate

\frac{dζ}{dt} ≤ ∑_{i}|b_{i,t} − a_{i,t}|

≤ 2∑_{i} |∑_{j:j≠i}d_{j}∇ × F (b_{i} − b_{j}) − ∑_{j:j≠i}d_{j}∇ × F (a_{i} − a_{j})|

+ ∑_{i} |a_{i,t} − 2∑_{j:j≠i}d_{j}∇ × F (a_{i} − a_{j})|
= T erm 1 + T erm 2.

We immediately dispose of T erm 1. Fix s < T̃ and a pair of indices i ≠ j. Let h = |(b_{i}(s) − b_{j}(s)) − (a_{i}(s) − a_{j}(s))|. Note that by assumption h ≤ ζ(s) ≤ δ. By Taylor’s theorem, at the fixed time s, we have

|∇ × F (b_{i} − b_{j}) − ∇ × F (a_{i} − a_{j})| ≤ h  max_{{x:|x−(a_{i}−a_{j})|≤h}}|D^{2}F |
≤ Cζ(s).

The last inequality follows from (5.5) provided δ < r. Therefore, T erm 1 satisfies the desired estimate (5.4).

3. We turn our attention to T erm 2, which is a sum of terms (T erm 2)_{i}, with i = 1, …, m. Suppose that each function a_{i}(·) is differentiable at s ∈ [0, T̃). Fix η ∈ C_{0}^{∞} such that supp(η) ⊂ B_{r(T̃)}(a_{i}(s)) and η(x) = ν · x in a neighborhood of a_{i}(s). Here we take ν ∈ S^{1} to satisfy

(T erm 2)_{i} = d_{i}ν · (a_{i,t}(s) − 2∑_{j:j≠i}d_{j}∇ × F (a_{i}(s) − a_{j}(s))) . (5.6)

We will write ϵ_{n} for an appropriate subsequence as ϵ → 0. Since

[Ju^{ϵ_{n}}(t)] ⇀ π ∑ d_{i}δ_{a_{i}(t)} weakly as measures

we can rewrite

d_{i}ν · a_{i,t}(s) = lim_{h→0}d_{i}ν · \frac{1}{h}(a_{i}(s + h) − a_{i}(s))
= lim_{h→0} lim_{n→∞} \frac{1}{πh}_{∫𝕋^{2}} (η[Ju^{ϵ_{n}}(s + h)] − η[Ju^{ϵ_{n}}(s)]) dx

= lim_{h→0} lim_{n→∞} \frac{1}{πh} ∫_{s}^{s+h}_{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk}u_{x_{k}}^{ϵ_{n}} · u_{x_{l}}^{ϵ_{n}}dxdt.

We used (2.10) in the last step.

Let H(x, t) = H(x, a(t), d). We reexpress the remaining term in (5.6) using Lemma 1 which is stated and proven below,

d_{i}ν · (2∑_{j:j≠i}d_{j}∇ × F (a_{i}(s) − a_{j}(s)))

= lim_{h→0} \frac{2}{h} ∫_{s}^{s+h}d_{i}ν · (∑_{j:j≠i}d_{j}∇ × F (a_{i}(t) − a_{j}(t))) dt

= lim_{h→0} \frac{1}{πh} ∫_{s}^{s+h}_{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk}j^{k}(H)j^{l}(H)dxdt.

Therefore,

(T erm 2)_{i} = lim_{h→0} lim_{n→∞} \frac{1}{πh} ∫_{s}^{s+h}_{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk} (u_{x_{k}}^{ϵ_{n}} · u_{x_{l}}^{ϵ_{n}} − j^{k}(H)j^{l}(H)) dxdt. (5.7)

Inside the integral, H = H(·, a(t), d) and Du^{ϵ} = Du^{ϵ}(t).

On any set where |u| > 0, we can decompose u_{x_{k}}

u_{x_{k}} = [\frac{iu}{|u|} · u_{x_{k}}] \frac{iu}{|u|} + [\frac{u}{|u|} · u_{x_{k}}] \frac{u}{|u|}

= \frac{j^{k}(u)}{|u|} \frac{iu}{|u|} + |u|_{x_{k}} \frac{u}{|u|}. (5.8)

Since |Du| = 0 a.e. on the set |u| = 0, the representation above holds a.e. on supp(Du) and

u_{x_{k}}^{ϵ} · u_{x_{l}}^{ϵ} = \frac{1}{|u^{ϵ}|^{2}} j^{k}(u^{ϵ})j^{l}(u^{ϵ}) + |u^{ϵ}|_{x_{k}}|u^{ϵ}|_{x_{l}}. (5.9)

We will have proved (5.4) when we show

lim_{h→0} lim_{n→∞} \frac{1}{h} ∫_{s}^{s+h}_{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk} (|u^{ϵ_{n}}|_{x_{k}}|u^{ϵ_{n}}|_{x_{l}}) dxdt ≤ Cζ(s), (5.10)

and

lim_{h→0} lim_{n→∞} \frac{1}{h} ∫_{s}^{s+h}_{∫𝕋^{2}}η_{x_{j}x_{l}}𝕁_{jk} (\frac{1}{|u^{ϵ_{n}}|^{2}} j^{k}(u^{ϵ_{n}})j^{l}(u^{ϵ_{n}}) − j^{k}(H)j^{l}(H)) dxdt ≤ Cζ(s). (5.11)

These estimates will follow from the tight upper bound (5.2) (see also (3.13)) on the energy and energy conservation.

4. The renormalized energy W is conserved for solutions b(·) of (5.3). This is apparent from the fact that (5.3) can be written in Hamiltonian form. Also_{∫𝕋^{2}}E^{ϵ}(u^{ϵ}(·, t))(x)dx is conserved for solutions u^{ϵ} o f GLS_{ϵ}. Therefore, for every t ≤ T̃ and every ρ > 0, the upper bound (5.2) gives

∫𝕋^{2}E^{ϵ}(u^{ϵ}(·, t))(x)dx = _{∫𝕋^{2}}E^{ϵ}(ϕ^{ϵ}(·))(x)dx
≤ m(π log \frac{1}{ρ} + I(ϵ, ρ)) + W (a(0), d) + Cρ + o(1)
= m(π log \frac{1}{ρ} + I(ϵ, ρ)) + W (b(t), d) + Cρ + o(1).

Arguing as in the estimate of T erm 1, we see that

W (b(t)) − W (a(t)) ≤ C ∑ |b_{i}(t) − a_{i}(t)| = Cζ(t)

provided δ is small enough. Therefore

∫𝕋^{2}E^{ϵ}(u^{ϵ}(·, t))(x)dx ≤ m (π log \frac{1}{ρ} + I(ϵ, ρ)) + W (a(t), d) + Cρ + Cζ(t) + o(1), (5.12)

as ϵ → 0 for every ρ > 0. We have from Theorem 3 that

(5.13) [Ju^{ϵ_{n}}(t)] ⇀ π∑_{i=1}^{m}d_{i}δ_{a_{i}(t)} weakly as measures.

The conditions (5.12), (5.13) are precisely the hypotheses of Theorem 3 with γ_{2} = Cζ(t). So, for every t ∈ [s, s + h],

lim sup_{n→∞}∥D|u^{ϵ_{n}}|(·, t)∥_{L^{2}(𝕋_{ρ}^{2})}^{2} ≤ Cζ(t) (5.14)

and

lim sup_{n→∞}∥\frac{1}{|u^{ϵ_{n}}|}j(u^{ϵ_{n}}) − j(H)∥_{L^{2}(𝕋_{ρ}^{2})}^{2} ≤ Cζ(t) (5.15)

These estimates allow us to prove (5.10), (5.11).

We quickly estimate (5.10) by observing

(5.10) ≤ lim_{h→0} lim_{n→∞}∥D^{2}η∥_{L^{∞}(𝕋_{ρ}^{2})} \frac{C}{h} ∫_{s}^{s+h}∥D|u^{ϵ_{n}}|(·, t)∥_{L^{2}(𝕋_{ρ}^{2})}^{2}dt (5.16)

and using (5.14).

5. We now establish (5.11). First, we show that

(5.17) (\frac{1}{|u^{ϵ_{n}}|}j(u^{ϵ_{n}}) − j(H)) ⇀ 0 weakly in L^{2}(𝕋_{ρ}^{2} × [s, s + h]).

To see this, note that \frac{1}{|u^{ϵn}|}j(u^{ϵ_{n}}) is uniformly bounded in L^{2}(𝕋_{ρ}^{2} × [s, s + h]) and hence converges weakly to some limit j̄. We know from Theorem 3(iv) that j(u^{ϵ_{n}}) ⇀ j(H) weakly in L^{p}(dxdt) for all 1 ≤ p < 2. We also know from Theorem 3(iii) that |u^{ϵ_{n}}|^{2} rightarrow1 strongly in L^{2}(dxdt). Thus

j(H) = weak L^{1} lim_{n→∞}j(u^{ϵ_{n}})
= weak L^{1} lim_{n→∞} (\frac{j(u^{ϵ_{n}})}{|u^{ϵ_{n}}|}|u^{ϵ_{n}}|)
= ( strong L^{2} lim_{n→∞}|u^{ϵ_{n}}|) ( weak L^{2} lim_{n→∞} \frac{j(u^{ϵ_{n}})}{|u^{ϵ_{n}}|})
= j̄

which proves (5.17).

For fixed k, l, observe that the quadratic term in (5.11) can be written

(\frac{1}{|u^{ϵ_{n}}|}j^{k}(u^{ϵ_{n}}) − j^{k}(H)) (\frac{1}{|u^{ϵ_{n}}|}j^{l}(u^{ϵ_{n}}) + j^{l}(H))

and reexpressed as

(\frac{1}{|u^{ϵ_{n}}|}j^{k}(u^{ϵ_{n}}) − j^{k}(H)) (\frac{1}{|u^{ϵ_{n}}|}j^{l}(u^{ϵ_{n}}) − j^{l}(H)) + 2j^{l}(H) (\frac{1}{|u^{ϵ_{n}}|}j^{k}(u^{ϵ_{n}}) − j^{k}(H)) .

Since (\frac{1}{|u^{ϵn}|}j^{k}(u^{ϵ_{n}}) − j^{k}(H)) ⇀ 0 weakly in L^{2}(𝕋_{ρ}^{2} × [s, s + h]) and j^{l}(H) does not depend upon n, the second expression contributes nothing as n → ∞. The first expressi on is controlled using (5.15).

6. Since we have appropriately bounded T erm 1 and T erm 2, we have proven (5.4). Gronwall’s inequality implies ζ = 0 which gives (i) of the Theorem. Since ζ = 0, (5.15) implies (ii). We conclude by proving (iii). Fix t and ρ > 0. Let u^{ϵ} be a

20 J. E. COLLIANDER AND R. L. JERRARD

subsequence which converges in L^{2}(𝕋_{ρ}^{2}) to some limit ū. We may assume, by (3.6), that Du^{ϵ} ⇀ Dū weakly in L^{2}(𝕋_{ρ}^{2}). Also, (ii) gives

lim inf_{ϵ→0} ∥Du^{ϵ}∥_{L^{2}(𝕋_{ρ}^{2})} ≥ lim_{ϵ→0}∥\frac{j(u^{ϵ})}{|u^{ϵ}|}∥_{L^{2}(𝕋_{ρ}^{2})} = ∥Dū∥_{L^{2}(𝕋_{ρ}^{2})}.

Therefore Du^{ϵ} → Dū strongly in L^{2}(𝕋_{ρ}^{2}). Finally, since j(ū) = j(H), Proposition 2 implies ū = e^{iα}H for some α ∈ ℝ.

We state and prove the lemmas employed above. Lemma 1. Suppose that η ∈ C^{2} and that

supp(η) ∩ {a_{1}, ..., a_{m}} = {a_{i}}; D^{2}η ≡ 0 in a neighborhood of a_{i}. Let H := H(·; a, d) be the canonical harmonic map. Then

∫_{𝕋^{2}} η_{x_{j}x_{l}}𝕁_{jk}j^{k}(H)j^{l}(H) = d_{i}Dη(a_{i}) · (2π ∑_{j:j≠i} d_{j} ∇ × F (a_{i} − a_{j}))

Remark 12. This computation remains valid if d_{1}, ..., d_{m} assume arbitrary integer values, that is, if we lift the assumption that d_{i} = ±1 for all i. Proof. 1. We reexpress the integral in the lemma. Recall that j(H) = −∇ × Φ where Φ satisfies

ΔΦ = ∑_{i=1}^{m}2πd_{i}δ_{a_{i}}, (5.18)

and, using (2.14), we write Φ(x) = d_{i}F (x − a_{i}) + G(x) (5.19) where

G(x) = ∑_{j:j≠i}d_{j}F (x − a_{j}).

Since j^{k}(H) = 𝕁_{km}Φ_{x_{m}}, we have 𝕁_{jk}j^{k}(H)j^{l}(H) = −𝕁_{ln}Φ_{x_{n}}Φ_{x_{j}}. (5.20) Fix any number ρ so small that D^{2}η = 0 on B_{ρ}(a_{i}). We have

∫_{𝕋^{2}} η_{x_{j}x_{l}}𝕁_{jk}j^{k}(H)j^{l}(H)dx = − ∫_{𝕋^{2}\B_{ρ}(a_{i})} η_{x_{j}x_{l}}𝕁_{ln}Φ_{x_{n}}Φ_{x_{j}}

= ∫_{𝕋^{2}\B_{ρ}} η_{x_{l}}𝕁_{ln}Φ_{x_{n}x_{j}}Φ_{x_{j}}dx + ∫_{∂B_{ρ}} η_{x_{l}}𝕁_{ln}Φ_{x_{n}}Φ_{x_{j}}ν^{j}dH^{1}

where ν = (ν^{1}, ν^{2}) is the outward unit normal to ∂B_{ρ}. We recognize Φ_{x_{n}x_{j}}Φ_{x_{j}} = \frac{1}{2}(Φ_{x_{j}}Φ_{x_{j}})_{x_{n}} and integrate by parts again to find

= − ∫_{𝕋^{2}\B_{ρ}} η_{x_{l}x_{n}}𝕁_{ln} \frac{1}{2}Φ_{x_{j}}Φ_{x_{j}}dx − ∫_{∂B_{ρ}} η_{x_{l}}𝕁_{ln} \frac{1}{2}Φ_{x_{j}}Φ_{x_{j}}ν^{n}dH^{1}

+ ∫_{∂B_{ρ}} η_{x_{l}}𝕁_{ln}Φ_{x_{n}}Φ_{x_{j}}ν^{j}dH^{1}.

Since η_{x_{l}x_{n}}𝕁_{ln} = 0, the integral over 𝕋^{2} \ B_{ρ} vanishes and we are left with two boundary integrals I_{ρ}, II_{ρ}.

2. We calculate the boundary integrals. We begin with

I_{ρ} = − ∫_{∂B_{ρ}} η_{x_{l}}𝕁_{ln} \frac{1}{2}Φ_{x_{j}}Φ_{x_{j}}ν^{n}dH^{1}.

By using (5.19), we observe

Φ_{x_{j}}Φ_{x_{j}} = F_{x_{j}}F_{x_{j}} + 2d_{i}F_{x_{j}}G_{x_{j}} + G_{x_{j}}G_{x_{j}}. (5.21)

Since F is even, the first term integrates to zero. We exploit the fact that G_{x_{j}} is nearly constant on B_{ρ}(a_{i}) to calculate the contribution to I_{ρ} arising from the remaining two terms in (5.21). The cross term contributes

−d_{i}η_{x_{l}}(a_{i})G_{x_{j}}(a_{i})𝕁_{ln} ∫_{∂B_{ρ}} F_{x_{j}}ν^{n}dH^{1} − d_{i}η_{x_{l}}(a_{i})𝕁_{ln} ∫_{∂B_{ρ}} F_{x_{j}}[G_{x_{j}}(x) − G_{x_{j}}(a_{i})]ν^{n}dH^{1}. (5.22)

Since F ∼ log |x − a_{i}|, |F_{x_{j}}| ∼ \frac{1}{ρ} on ∂B_{ρ} and G is C^{1} on B_{ρ}, the second integral contributes O(ρ). The G_{x_{j}}G_{x_{j}} term contributes O(ρ) as well.

Next, we calculate

II_{ρ} = ∫_{∂B_{ρ}} η_{x_{l}}𝕁_{ln}Φ_{x_{n}}Φ_{x_{j}}ν^{j}dH^{1}

by expanding using (5.19). The F_{x_{n}}F_{x_{j}} term again vanishes by symmetry. The G_{x_{n}}G_{x_{j}} term contributes O(ρ) and the cross terms remain to be estimated. The first cross term gives

d_{i}η_{x_{l}}(a_{i})G_{x_{j}}(a_{i})𝕁_{ln} ∫_{∂B_{ρ}} F_{x_{n}}ν^{j}dH^{1} + O(ρ). (5.23)

The second cross term contributes

d_{i}η_{x_{l}}(a_{i})G_{x_{n}}(a_{i})𝕁_{ln} ∫_{∂B_{ρ}} F_{x_{j}}ν^{j}dH^{1} + O(ρ). (5.24)

3. Since _{∫∂B_{ρ}}F_{x_{n}}ν^{j}dH^{1} = _{∫∂B_{ρ}}F_{x_{j}}ν^{n}dH^{1}, the first term in (5.22) and (5.23) cancel and the only remaining contribution is (5.24). Finally, observe that

∫_{∂B_{ρ}} F_{x_{j}}ν^{j}dH^{1} = ∫_{B_{ρ}} ΔF dx = 2π − 2π^{2}ρ^{2}, (5.25)

using (2.11). Therefore

∫_{𝕋^{2}} η_{x_{j}x_{l}}𝕁_{jk}j^{k}(H)j^{l}(H)dx = 2πd_{i}η_{x_{l}}(a_{i})𝕁_{ln}G_{x_{n}}(a_{i}) + O(ρ). (5.26)

Since ρ can be taken arbitrarily small, we have proved the lemma.

6. Acknowledgments

Many of the ideas in this paper grow out of the second author’s joint work with H.M. Soner and reflect the influence of numerous conversations over a period of years. The first author was partially supported by a grant from the University of Illinois Research Board and by a NSF Postdoctoral Fellowship. The second author was partially supported by NSF grant # DMS 96-00080.

References

[1] F. Bethuel, H. Brezis, and F Hélein. Ginzburg-Landau Vortices. Birkhäuser, Boston, 1994.

[2] J. Bourgain. Fourier transform restriction phenomena for certain lattice subsets and applications to nonlinear evolution equations I,II. Geom. Funct. Anal., 3:107–156, 209–262, 1993.

[3] H. Brézis, J. M. Coron, and E. Lieb. Harmonic maps with defects. Comm. Math. Phys., 107(4):649–705, 1986.

[4] J. E. Colliander and R. L. Jerrard. Ginzburg-Landau Vortex Dynamics: weak stability and Schrödinger equation dynamics. preprint, 1997.

[5] W. E. Dynamics of vortices in Ginzburg-Landau theories with applications to superconductivity. Physica D, 77:383–404, 1994.

[6] L.C. Evans and W. Gangbo. Differential equations methods for the Monge- Kantorovitch mass transfer problem. to appear.

[7] A.L. Fetter. Vortices in an Imperfect Bose Gas. I. the Condensate. Physical Review, 138(2A):A429–A437, 1965.

[8] V.L. Ginzburg and L.P. Pitaevskii. On the theory of superfluidity. Soviet Physics JETP, 34(7)(5):858–861, 1958.

[9] E.P. Gross. Dynamics of interacting bosons. In E. Meeron, editor, Physics of many particle systems, pages 231–406. Gordon and Breech, New York, 1966.

[10] R.L. Jerrard. Lower bounds for generalized Ginzburg-Landau functionals. preprint, 1997.

[11] R.L. Jerrard and H.M. Soner. Dynamics of Ginzburg-Landau vortices. to appear, Arch. Rat. Mech. Anal., 1998.

[12] L.D. Landau. Journal of Experimental and Theoretical Physics (U.S.S.R.), 11, 1941.

[13] F.H. Lin. Some dynamical properties of Ginzburg-Landau vortices. Comm. Pure. Appl. Math., 49(4), 1996.

[14] C. Marchioro and M. Pulvirenti. Mathematical theory of incompressible nonviscous fluids. Springer, 1994.

[15] J.C. Neu. Vortices in complex scalar fields. Physica D, 43:385–406, 1990.

[16] L.P. Pitaevskii. Vortex lines in an imperfect Bose gas. Soviet Physics JETP, 13(2):451 –454, 1961.

[17] E. Sandier. Lower bounds for the energy of unit vector fields and applications. preprint, 1997.

J. E. Colliander, Mathematical Sciences Research Institute, Berkeley, CA 94720

R. L. Jerrard, University of Illinois, Urbana, IL 61801