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Dynamics of helical vortex laments in non viscous
incompressible ows
Martin Donati, Christophe Lacave, Evelyne Miot
To cite this version:
Martin Donati, Christophe Lacave, Evelyne Miot. Dynamics of helical vortex laments in non viscous
incompressible ows. 2024. �hal-04484560�
Dynamics of helical vortex filaments in non viscous incompressible
flows
Martin Donati, Christophe Lacave, Evelyne Miot
February 29, 2024
Abstract
In this paper we study concentrated solutions of the three-dimensional Euler equations in helical
symmetry without swirl. We prove that any helical vorticity solution initially concentrated around
helices of pairwise distinct radii remains concentrated close to filaments. As suggested by the vortex
filament conjecture, we prove that those filaments are translating and rotating helices. Similarly
to what is obtained in other frameworks, the localization is weak in the direction of the movement
but strong in its normal direction, and holds on an arbitrary long time interval in the naturally
rescaled time scale. In order to prove this result, we derive a new explicit formula for the singular
part of the Biot-Savart kernel in a two-dimensional reformulation of the problem. This allows us
to obtain an appropriate decomposition of the velocity field to reproduce recent methods used to
describe the dynamics of vortex rings or point-vortices for the lake equation.
1 Introduction
The purpose of this paper is to study the time evolution for 3D inviscid flows for which the vorticity
is initially concentrated around helical curves.
We consider the Euler equations governing the dynamics of a three-dimensional inviscid, incom-
pressible fluid in a domain Ω:
∂tU+ (U· ∇)U=−∇Pin Ω ×R∗
+,
div(U) = 0 in Ω ×R+,
U·n= 0 on ∂Ω×R+,
(E)
where nis the outward normal vector, U: Ω ×R∗
+→R3denotes the velocity of the fluid and Pthe
pressure. We shall focus on particular flows, called vortex filaments, for which the vorticity curl(U)
is sharply concentrated in a thin tube around a curve in R3. Understanding the stability (namely,
whether the concentration of the filament around a curve persists in time) and the dynamics of
vortex filaments in three-dimensional flows are a longstanding issue in mathematical physics. Da Rios
formally derived in [8] that, to leading order, the asymptotic motion law for one single vortex filament
in a tube of size εaround a curve parametrized by χ(·, t), with arc-length parameter σ, is governed
by the binormal curvature flow:
∂tχ=c|ln ε|(∂σχ×∂σσχ) (BF)
where cis the curvature. Note that (BF) exhibits some trivial solutions: the stationary vortex line,
the uniformly translating circle (known as “vortex ring”) and the translating-rotating helix (referred
to as “helical filaments” in the remaining of the paper). We refer e.g. to [22] and to references therein
for a general introduction on the subject. The “vortex filament conjecture” is the conjecture that
vorticity initially concentrated around a curve remains close to a curve evolving according to (BF)
to leading order for at least a certain interval of time. While it is completely settled in the 2D case
(where vortex filaments reduce to point vortices), see [23], it is open in general. Jerrard and Seis [20]
provided a rigorous derivation of (BF)assuming the vorticity remains concentrated around the curve.
Without assuming a priori concentration, further results have been obtained under supplementary
symmetry assumptions. For axisymmetric flows without swirl, Butt`a, Cavallaro and Marchioro [4]
1
recently rigorously justified the dynamics of several vortex rings of different radii. They also established
a “semi-strong” localization result: the filaments remains for all time sharply localized in the radial
direction (namely with respect to the distance to the symmetry axis). Their approach inspired a
recent work by Hientzsch, Lacave and Miot [19] in the setting of point vortices for the lake equations,
which is a 2D model for incompressible flows inheriting an anelastic constraint from the 3D case.
In the special case of vortex rings, Fraenkel [15] exhibited a family of solutions of (E) such that
the corresponding vorticity concentrates for all time on a curve solution of (BF). In [9] Davila, Pino,
Musso and Wei constructed such a family of solutions that do not change form, concentrating to one
or several polygonally distributed rotating-translating helical filaments, by means of elliptic singular
perturbation techniques. Another kind of construction was obtained by Cao and Wan in [6]. We
also mention the recent work by Guerra and Musso [17], that constructs a special family of solutions
concentrating to a collapsing configuration of helical filaments. Nevertheless, all these particular
solutions are related to a certain class of initial data.
Our objective here is to establish the dynamics of helical filaments starting from generic initial
data, with very few and natural assumptions relating only to the initial concentration, and with other
techniques.
The helical symmetry is a physically relevant framework since this symmetry is obtained in many
different contexts, in particular in the wake of rotors. This covers a wide range of situations from the
study of wind or water turbine to vertical flight of helicopters for instance. Moreover, in the wake of
each wing of an airplane, vortices of the same sign are created on straight lines, but immediately start
interacting with each other, inducing a rotation that creates a local helical symmetry. Recent papers
(see for instance [2,1,7]) study theoretically, numerically and experimentally physical properties of
such flows and in particular instabilities due to non helical perturbations and viscosity.
We now introduce with some more details our working framework. We focus on flows with helical
symmetry and without helical swirl. More precisely, following [13,14], for some fixed h > 0 we define
the following operators for all θ∈R:
Rθ=
cos θ−sin θ0
sin θcos θ0
0 0 1
and Sθ,hx=Rθx+h
0
0
θ
, x ∈Ω
and we say that Ω is a helical domain if it satisfies Sθ,hΩ = Ω, for all θ∈R. We say that (U, P ) is a
helical solution to (E) on the helical domain Ω if:
U(Sθ,hx) = RθU(x), P (Sθ,hx) = P(x),∀x∈Ω,∀θ∈R.
Finally, we say that (U, P ) is helical without swirl if (U, P ) is helical and Uis orthogonal to the helices,
namely:
U(x)·ξ(x) = 0,∀x=
x1
x2
x3
∈Ω,where ξ(x) =
−x2
x1
h
.(1.1)
Such properties are formally preserved by (E).
Global existence and uniqueness of weak solutions to (E) that are helical without swirl have been
proved in [13,14,3,21,18]. Such solutions are also Lagrangian, see Section 2.1 hereafter for more
details. It turns out that the “no swirl” condition implies that the vorticity is parallel to ξwhich
allows us to define a scalar quantity ω:
curl U(x, t) = 1
hω(e
R−x3
h(x1, x2), t)ξ(x),where e
Rθ=cos θ−sin θ
sin θcos θ.(1.2)
From this observation, it was proved in [14] that, for helical flows without swirl, Equation (E) reduces to
a two-dimensional system for the vorticity posed on the 2D cross-section U={(x1, x2)|(x1, x2,0) ∈Ω}
2
of Ω:
∂tω+v· ∇ω= 0 in U × R∗
+,
v=∇⊥Ψ in U × R+,
div K∇Ψ=ωin U × R+,Ψ = 0 on ∂U × R+,
ω(·,0) = ω0in U,
(1.3)
where Kis a symmetric positive-definite matrix defined by
K(x) = 1
x2
1+x2
2+h2h2+x2
2−x1x2
−x1x2h2+x2
1.(1.4)
We have used the notation, ∇⊥Ψ = (∇Ψ)⊥with the convention (a, b)⊥= (−b, a). For this result
and for the rest of this article, we assume that Uis a bounded, simply connected domain, with C1,1
boundary for simplicity reasons. More details on weak solutions to (1.3) will be given in Section 2.1.
In the 2D reduction of the 3D system, vortex filaments reduce to point vortices, that correspond
to the 2D projection of the filaments. So we are left to investigating the persistence and dynamics of
point vortices for Equation (1.3). Note that in view of (BF), it is more judicious to consider another
time-scale in order to obtain a velocity of order one when considering concentrated vortices, thus we
are led to consider a rescaled system:
∂tωε+1
|ln ε|vε· ∇ωε= 0 in U × R∗
+,
vε=∇⊥Ψεin U × R+,
div K∇Ψε=ωεin U × R+,Ψε= 0 on ∂U × R+,
ωε(·,0) = ωε
0in U.
(1.5)
One of the contributions of this paper is that in Proposition 2.5 we obtain an important decomposition
of the Green’s function of the operator L= div(K∇·) as
GK,U:= GK+SK,U,
where SK,U∈W1,∞and GKis explicitly given at (2.5). A similar decomposition was obtained in [6]
exhibiting a different singular term, but the remainder did not have the regularity W1,∞which we
crucially need in the following. From this Green’s function, we obtain a Biot-Savart law that gives in
Proposition 2.6 a sharp decomposition of the velocity field vε.
We may now state our main result as follows:
Theorem 1.1. Assume that there exists RU>0such that B(0, RU)⊂ U. Let (zi,0)16i6Nbe Npoints
in B(0, RU)such that |zi,0| 6=|zj,0|for every i6=j. Let γi∈R∗.
For every ε > 0such that ε < RU−maxi|zi,0|, let ωε
0∈L∞(U)such that
ωε
0=
N
X
i=1
ωε
i,0,
supp ωε
i,0⊂Bzi,0, ε,
ωε
i,0has a definite sign and ZU
ωε
i,0(x)dx=γi,
|ωε
0|6M0
ε2,for some M0>0.
(1.6)
For T > 0, let (vε, ωε)be the unique weak solution of (1.5)on [0, T ](in the sense of Definition 2.1
below). Let zi(t) = e
Rtνizi,0with
νi=−γi
4πhp|zi,0|2+h2.
3
Then, there exists a decomposition
ωε=
N
X
i=1
ωε
i, ωε
i∈L∞(U)
which satisfies:
(i)A weak localization property: there are CT, εT>0such that, for any ε∈(0, εT], we have
sup
t∈[0,T ]γi−ZB(zi(t),rε)
ωε
i(x, t)dx6CT
ln |ln ε|,where rε=ln |ln ε|
|ln ε|1/2
,
and
sup
t∈[0,T ]
1
γiZU
xωε
i(x, t)dx−zi(t)6CT
p|ln ε|.
(ii)A strong localization property in the radial direction: for every κ∈(0,1/4), there is Cκ,T and
εκ,T >0such that, for every ε∈(0, εκ,T ], we have
supp ωε
i(·, t)⊂x∈ U ,|x| − |zi,0|6Cκ,T
|ln ε|κ,for all t∈[0, T ].
From Theorem 1.1, after reconstructing the 3D flow from the 2D solution, we indeed see that for
any concentrated initial vorticity, the solution remains close to the translating rotating helices given
by (BF). The velocities νiare coherent with this model and with other results on the subject (see
[6, Equation (3.1)] for instance, where their νhas the opposite sign but their matrix of rotation is
clockwise).
We recall that the main difference of Theorem 1.1 with the results obtained in [9] is that we prove
the localization for a much larger class of initial data. However the localization in Theorem 1.1 is
only weak in the direction of the movement, while [9] obtains a strong confinement in both directions
due to the choice of a well prepared initial data. Due to this weak confinement, we are restricted to
study helices of different radii which excludes in particular the case of polygonally distributed helices,
covered in [9].
This weak confinement in the direction of the movement is a quite natural limitation. Indeed, for
general initial data, even in the usual planar 2D case, some filamentation may happen in bounded
time, namely some vorticity may be driven away from the core of the vortex. In 3D, this vorticity
would then slow down compared to the core of the filament (recall that the leading order movement
is due to the self interaction of the filament due to concentration and his curvature), which in turns
drives the lost vorticity further away.
Our strategy is to follow the techniques of [4,5] and developed by [19]. The plan of the paper is
the following. In Section 2, we recall and establish properties of the reduction to the 2D problem. In
particular, in Proposition 2.5 we derive an explicit formula for the singular part of the Biot-Savart
kernel for this problem. In Section 3, we set up the proof of Theorem 1.1 and introduce an other
reduced model, considering a single filament in an exterior field. Section 4is dedicated to this reduced
model. We control different quantities such as the energy and vorticity moments of the solution to
derive the dynamics and obtain the localization results. Section 5concludes the proof of Theorem 1.1.
Remarks on notation. Except in Subsection 2.1,xwill from now on denote a point (x1, x2) in R2,
and we save then the notation Xfor the following definition: |X|:= p|x|2+h2. Similarly, y,z, and
bε(t) will be points in R2and |Y|,|Z|, and |Bε(t)|should be understood according to the previous
definition. Unless specified otherwise, the integrals are always on U. The values of the constants
named Care always irrelevant and may change from line to line. The constants Care allowed to
depend on Uand hwithout further mention since those objects are fixed once and for all.
4
2 The helical symmetry framework
2.1 Some known facts on the 2D reduction and the well-posedness of the 3D Euler
in helical symmetry
Let Ω be a helical domain with 2D cross-section U. In order to use known results on well-posedness
for the three-dimensional Euler equation (E), we shall always assume in this section that Uis simply
connected, bounded and has C1,1boundary. Let an initial vector field U0be smooth and without
helical swirl (i.e. satisfying (U0·ξ)(x) = 0, for all x∈Ω with ξdefined in (1.1)). It was proved in [14]
that any smooth helical solution of (E) remains without helical swirl for positive times. In this case,
the three-dimensional vorticity curl Uis related to a scalar quantity ωvia (1.2), and (E) reduces to
System (1.3) for ω, with Kdefined by (1.4).
In domains with bounded cross-section, global well-posedness for smooth helical solutions of (E)
was obtained by Dutrifoy [13], whereas global well-posedness of weak (bounded) solutions of (1.3) was
proved by Ettinger and Titi [14]. For the whole space R3, Bronzi, Lopes and Lopes [3] established
the global existence of a weak solution for curl U0∈Lp
c(R3). Such a result was extended in [21] for
curl U0∈L1∩Lp(R3) (for p∈(1,∞]).
Finally, Guo and Zhao used a Lagrangian method to establish in [18] global existence and unique-
ness of the weak helical solution without swirl for (E) when the initial vorticity curl U0∈L1
1∩L∞
1(R3).
They also proved that the solution to the corresponding 2D reduction is a Lagrangian solution, namely
it is constant along the characteristics of the flow associated to the velocity field.
We shall consider here the following definition of a weak solution to (1.3), which is mainly inspired1
by [14, Definition 3.10]. Before stating our definition, we need to introduce the following operator
LΨ := div(K∇Ψ) (2.1)
for Ψ an integrable function in U.
Definition 2.1. Let ω0∈L∞(U). Let Ψ0∈W2,1∩H1
0(U)be the unique solution of LΨ0=ω0. We
set v0=∇⊥Ψ0. We say that (v, ω)is a weak bounded solution of (1.3)on [0, T ]with initial condition
(v0, ω0)if:
1. There exists Ψ∈L∞([0, T ], W 2,1(U)) with Ψ = 0 a.e. on ∂U × [0, T ]such that v=∇⊥Ψand
ω=LΨa.e. in U × [0, T ];
2. We have ω∈L∞(U × [0, T ]);
3. For all test function Φin C∞
c(U ×[0, T )), we have
−ZU
Φ(x, 0)ω0(x) dx=ZT
0ZU
ω(x, s) (∂tΦ + v· ∇Φ) (x, s) dxds.
As already mentioned, existence and uniqueness of the weak bounded solution as in Defini-
tion 2.1 for all T > 0 is proved in [14, Theorem 3.11]. Moreover, the velocity field v=∇⊥Ψ
satisfies the Calder´on-Zygmund inequality [14, Corollary 3.8]: we have k∇v(·, t)kLp6Cpkω(·, t)kLp6
Cpkω(·, t)kL∞for all 2 6p < ∞. Observing that div v= 0 a.e. on U ×[0, T ] and that v·n= 0 a.e. on
∂U × [0, T ], we may apply classical results by DiPerna and Lions [12, p. 546] on the theory of linear
transport equations and Lagrangian flows, see also [11, Theorem 1, Theorem 2]. We infer that there
exists a unique measure-preserving Lagrangian flow X: (x, t, t0)∈ U × [0, T ]×[0, T ]7→ X(x, t, t0)∈ U
associated to v. Moreover, denoting further X(x, t) = X(x, t, 0) for simplicity, the unique weak solu-
tion ω∈L∞(U × [0, T ]) to the transport equation ∂tω+v· ∇ω= 0 satisfies
ω(·, t) = X(·, t)#ω0,∀t∈[0, T ]
in the sense that for all ϕ∈Cc(U) we have RUω(x, t)ϕ(x) dx=RUω0(x)ϕ(X(x, t)) dx.
1The definition of weak solution in [14] slightly differs from the one of the present paper, because it is given only in
terms of Ψ, but it can be straightforwardly proved that it coincides with the one given below for a weak solution.
5
By Morrey’s inequality, Cald´eron-Zygmund’s estimate implies that |v(x, t)−v(y, t)|6CTp|x−
y|1−2/p, for all p > 1. Setting p=|ln |x−y|| for |x−y|< e, we then get that vis log-lipschitz
locally uniformly in time: |v(x, t)−v(y, t)|6CT|x−y|(1 + |ln |x−y|) for t∈[0, T ) and x, y ∈ U. By
Cauchy-Lipschitz theorem, we infer that for a.e x∈ U the curve t7→ X(x, t) is the unique solution in
C1([0, T ); U) to the ODE
dX(x, t)
dt=v(X(x, t), t), X(x, 0) = x.
Let us note that the divergence free condition immediately implies the conservation of the Lpnorm:
kω(·, t)kLp=kω0kLp, for t∈[0, T ], for p∈[1,∞].
In the setting of (1.6), the fact that the log-lipschitz constant of the velocity field vεdepends on
kωε(·, t)kL∞=kωε
0kL∞, diverging possibly as ε−2, constitutes a major difficulty. Indeed, this does not
allow to control the distance between the supports of the components ωε
iand ωε
juniformly in ε. Such
a uniform control will be included in the forthcoming definition of Tε, see (3.5), and one of the main
consequences of the strong localization will be to state that Tε=T.
Remark 2.2. In view of the relation in terms of the flow map, it is natural to define the decomposition
of ωεin Theorem 1.1 as the transport of the decomposition of the initial data:
ωε
i(·, t) := Xε(·, t)#ωε
i,0=ωε
i,0◦Xε(·, t)−1.
2.2 Some useful properties of the matrix K
We recall that the definition of Kis given at (1.4), that h > 0 is given and that for every x∈R2we
denote by |X|:= p|x|2+h2. We observe that we have the following decomposition:
∀x∈R2, K(x) = I2−1
h2+|x|2N(x) with N(x) = x2
1x1x2
x1x2x2
2.
The eigenvalues of N(x) are 0 and |x|2associated respectively to the eigenvectors x⊥and x. Moreover,
N(x)2=|x|2N(x).
Consequently, we have the following lemma.
Lemma 2.3. For every x∈R2,K(x)is a symmetric positive-definite matrix, with eigenvalues 1 and
h2/|X|2so that
det K(x) = h2
|X|2.
In particular, Kis uniformly elliptic in the bounded domain U.
Setting
Λ(x) = I2+1
h|X|+h2N(x) = I2+1
h|X|+h2x2
1x1x2
x1x2x2
2,
we can check that its inverse is given by
Λ(x)−1=I2−1
h|X|+|X|2N(x).
and that
(Λ(x)−1)2=K(x),∀x∈R2.
In order to study the elliptic problem LΨ = ωon U, it will be useful to introduce a C1-
diffeomorphism Tsuch that DTis proportional to K−1/2.
Lemma 2.4. There exists a radial function f∈C∞R2,[1,+∞)and a C1-diffeomorphism T:R2→
R2such that DT(x) = f(x)K(x)−1/2=f(x)Λ(x).
6
Proof. Let f(x) = ρ(x2
1+x2
2) where ρ∈C1(R+,R∗
+) will be chosen later. Now let
∀x∈R2,T(x) := f(x)x=ρ(x2
1+x2
2)x1
x2.
We compute:
DT(x) = ρ(x2
1+x2
2)I2+ 2ρ′(x2
1+x2
2)x2
1x1x2
x1x2x2
2=ρ(x2
1+x2
2)I2+ 2ρ′(x2
1+x2
2)N(x)
=ρ(x2
1+x2
2)I2+2ρ′(x2
1+x2
2)
ρ(x2
1+x2
2)N(x).
This means that DT(x) = f(x)Λ(x) if and only if
2ρ′(x2
1+x2
2)
ρ(x2
1+x2
2)=1
h|X|+h2,
which also writes ρ′(s)
ρ(s)=g(s),∀s>0,(2.2)
with
g(s) = 1
2(h√s+h2+h2).
We then take
ρ(s) = exp Zs
0
g(u)du>1,∀s>0,
and check that (2.2) holds true, hence that DT(x) = f(x)Λ(x).
Now we need to prove that Tis a C1-diffeomorphism. By construction, it is clear that it is a C1
map. Now we assume that for some x, y ∈R2we have T(x) = T(y). By the definition of Tthis
implies first that |x|ρ(|x|2) = |y|ρ(|y|2), with s7→ sρ(s2) strictly increasing, thus |x|=|y|. Finally,
we get x=y, therefore Tis injective. Moreover, by construction, the matrix DT(x) = f(x)Λ(x) is
invertible for every x∈R2, so by the global inverse function theorem, Tis a C1-diffeomorphism from
R2to T(R2). Finally, g(s)∼1
2h√sas s→ ∞ so ρ(s)→+∞, and therefore, T(R2) = R2.
The function Tconstructed in the previous proof belongs to C2(R2). Using also T−1∈C1(R2),
we infer from the mean value theorem that there exists C > 0 such that
|DT(x)−DT(y)|6C|x−y|6C2|T (x)− T (y)|6C3|x−y|,∀x, y ∈B(0, RU).(2.3)
2.3 Expansion of the Biot-Savart law
This section is devoted to the construction of a suitable decomposition of the solution of the problem
LΨ = ω. Indeed, we expect that vε=∇⊥Ψεdiverges as O(ε−1) (at least pointwise) when the vorticity
ωεis concentrated. Thus it will be crucial to obtain an explicit formula for the most singular term to
find some cancellations by symmetry properties. This kind of expansion will also be the key to find
the term of order |ln ε|, which will not give symmetry cancellation and will give rise to the motion of
the vortex filament. Such an expansion is one of the main tool in the studies of the vortex rings (see
[5,4]) and of the lake vortices (see [19]). For helical flows, an expansion was derived by Cao and Wan
in [6] such that the remainder term is bounded in C1which is not enough for our study. Therefore,
we propose an independent proof where we find a different leading term, such that the remainder will
be bounded in C2uniformly in ε.
As we expect that such an explicit expansion can be useful for other problems, we establish the
following proposition with a general matrix Ksatisfying only its relevant properties: those established
in our case in Lemmas 2.3 and 2.4.
Proposition 2.5. Let U ⊂ R2be a bounded simply-connected domain with C1,1boundary. Let K∈
C1(R2;M2(R)) such that
7
(i) For every x∈ U, K(x)is a symmetric positive-definite matrix.
(ii) There exists a C1-diffeomorphism T:R2→R2and a function f:R2→R∗
+such that DT(x) =
f(x)K−1/2(x).
Then, for every ω∈L∞
c(U)the unique solution Ψ∈H1
0(U)of LΨ = ωbelongs to T16p<∞W2,p(U)
and is given by the expression
Ψ(x) = ZUGK,U(x, y)ω(y)dywhere GK,U:= GK+SK,U,(2.4)
with
∀x, y ∈ U, x 6=y, GK(x, y) = 1
2πdet K(x) det K(y)−1/4ln |T (x)− T (y)|,(2.5)
and for all y∈ U,x7→ SK,U(x, y) = SK,U(y, x)belongs to T16p<∞W2,p(U)and is the unique solution
in H1(U)of
LSK,U(x, y) = −pdet DT(y)
2πf (y)ln |T (x)− T (y)|∇ · K(x)∇ pdet DT(x)
f(x)!! ∀x∈ U,
SK,U(x, y) = −GK(x, y)∀x∈∂U.
(2.6)
Before proving Proposition 2.5, let us observe that the boundary condition for SK,Uis imposed in
order to comply with the boundary condition GK,U(x, y ) = 0 if x∈∂U, so that eventually Ψ = 0 on
∂U. Moreover, by hypothesis (ii), GKis also given by the expression
GK(x, y) = 1
2πpdet DT(x)pdet DT(y)
f(x)f(y)ln |T (x)− T (y)|.
In addition, we observe that if K(x) = 1
b(x)I2, with b>c > 0 then it satisfies the hypotheses of
Proposition 2.5 with T(x) = xand f(x) = 1/pb(x). The expression we obtain for the Green’s kernel
coincide with the one obtained for the lake equation as in [10, Proposition 3.1]. Taking K=I2gives
the usual 2D Green’s kernel.
Proof. Let ω∈L∞
c(U). By hypothesis (i), the operator Lgiven in (2.1) is uniformly elliptic with
coefficients belonging to C0(U). As ω∈ ∩p>1Lp(U), we know from [16, Theorem 9.15] that the
unique variational solution belongs to W2,p(U) for every p∈[1,+∞).
Noticing that for every V⋐Uand y∈Vfixed, the right hand side term of the first equality in
(2.6) belongs to ∩p>1Lp(U) whereas the boundary condition is C∞(∂U) (both uniformly with respect
to y∈V), we also deduce by [16, Theorem 9.15], that the unique solution x7→ SK,U(x, y) to (2.6)
belongs to W2,p(U) for every p>1, uniformly with respect to y∈V. In particular, the function
x7→ SK,U(x, y) is C1and bounded in W1,∞(U) uniformly with respect to y∈V.
In order to justify many computations in the following, we consider first the case of a smooth
source term which we denote by w. More precisely, we let w∈C∞
c(U) be fixed and let
Ψ1(x) := ZU
GK(x, y)w(y)dy, Ψ(x) = Ψ[w](x) := ZU
(GK+SK,U)(x, y)w(y)dy.
We compute in the sense of distribution the quantity hLΨ1, ϕi. Let ϕ∈C∞
c(U) be a test function,
then the functions are regular enough to differentiate under the integral sign and to apply Fubini’s
theorem so that
hLΨ1, ϕi=−ZU
[K(x)∇Ψ1(x)] · ∇ϕ(x)dx=−ZUZU
[K(x)∇xGK(x, y)] · ∇ϕ(x)w(y)dxdy
=−ZUpdet DT(y)
2πf (y)ZU"K(x)∇x pdet DT(x)
f(x)ln |T (x)− T (y)|!#· ∇ϕ(x)dx w(y)dy
8
=−ZUpdet DT(y)
2πf (y)ZU"K(x)pdet DT(x)
f(x)DTT(x)T(x)− T (y)
|T (x)− T (y)|2#· ∇ϕ(x)dx w(y)dy
−ZUpdet DT(y)
2πf (y)ZU
K(x)∇ pdet DT(x)
f(x)!ln |T (x)− T (y)|·∇ϕ(x)dx w(y)dy
:= −ZUpdet DT(y)
2πf (y)(A1(y) + A2(y))w(y)dy.
For the term A1, let us notice first that Kand DTare symmetric matrices so that
∇ϕ(x) = ∇(ϕ◦ T −1◦ T )(x) = DT(x)∇(ϕ◦ T −1)(T(x)) (2.7)
and thus
A1(y) = ZU
f(x)pdet DT(x)T(x)− T (y)
|T (x)− T (y)|2· ∇(ϕ◦ T −1)(T(x))dx
=ZT(U)
f(T−1(z))pdet DT(T−1(z)) z− T (y)
|z− T (y)|2· ∇(ϕ◦ T −1)(z)|det DT−1(z)|dz
where we have made the change of variable z=T(x). Since
det DT−1(z) = 1
det DT(T−1(z)) (2.8)
we obtain that
A1(y) = ZT(U)
f(T−1(z)) z− T (y)
|z− T (y)|2· ∇(ϕ◦ T −1)(z)pdet DT−1(z)dz
=ZT(U)
z− T (y)
|z− T (y)|2· ∇ ϕ◦ T −1×√det DT−1×f◦ T −1(z)dz
−ZT(U)
f(T−1(z)) z− T (y)
|z− T (y)|2ϕ(T−1(z)) · ∇ √det DT−1(z)dz
−ZT(U)
z− T (y)
|z− T (y)|2ϕ(T−1(z))pdet DT−1(z)· ∇ f◦ T −1(z)dz
:=A11(y) + A12(y) + A13(y).
Identifying that z−T (y)
|z−T (y)|2= 2π∇GR2(z− T (y)), where GR2(ξ) = 1
2πln |ξ|denotes the fundamental
solution of the Laplacian on R2, we may write
A11(y) = −2πϕ◦ T −1×√det DT−1×f◦ T −1(T(y)) = −2πϕ(y)f(y)pdet DT−1(T(y)).
In conclusion, we have obtained that
hLΨ1, ϕi=ZU
ϕ(y)f(y)dy−ZUpdet DT(y)
2πf (y)(A12(y) + A13(y) + A2(y))w(y)dy, (2.9)
where the second right hand side integral motivates our definition of SK,U. To recognize SK,U, we
come back to the variable x=T−1(z):
A12(y) + A13(y) = −ZU
f(x)T(x)− T (y)
|T (x)− T (y)|2ϕ(x)· ∇ √det DT−1(T(x))|det DT(x)|dx
−ZU
T(x)− T (y)
|T (x)− T (y)|2ϕ(x)pdet DT−1(T(x)) · ∇f◦ T −1(T(x))|det DT(x)|dx
We now observe using again relations (2.7) and (2.8) that
∇√det DT−1(T(x)) = (DT(x))−1∇√det DT−1◦ T (x)
9
=1
(f(x))2K(x)DT(x)∇1
√det DT(x)
=−1
(f(x))2K(x)DT(x)∇√det DT(x)
det DT(x)
and similarly,
∇(f◦ T −1)(T(x)) = (DT(x))−1∇(f◦ T −1◦ T )(x) = 1
(f(x))2K(x)DT(x)∇f(x).
Therefore, using once again that DT(x) is symmetric, that it commutes with K(x), and recalling that
det DT>0,
A12(y) + A13 (y) = ZU
DT(x)T(x)− T (y)
|T (x)− T (y)|2ϕ(x)·K(x)∇√det DT(x)
f(x)dx
+ZU
DT(x)T(x)− T (y)
|T (x)− T (y)|2ϕ(x)·K(x)−∇f(x)
(f(x))2pdet DT(x)dx
=ZU
DT(x)T(x)− T (y)
|T (x)− T (y)|2ϕ(x)·K(x)∇ pdet DT(x)
f(x)!dx
and thus recalling that
A2(y) = ZU
K(x)∇ pdet DT(x)
f(x)!ln |T (x)− T (y)| · ∇ϕ(x)dx
we obtain, by an integration by parts, that
A12(y) + A13 (y) + A2(y) = ZU
K(x)∇ pdet DT(x)
f(x)!· ∇ln |T (x)− T (y)|ϕ(x)dx
=−ZU
ln |T (x)− T (y)|ϕ(x)∇ · K(x)∇ pdet DT(x)
f(x)!dx.
By the definition of LSK,Uin (2.6) this allows us to conclude that
−ZUpdet DT(y)
2πf (y)(A12(y) + A13(y) + A2(y))w(y)dy=−LZU
SK,U(·, y)w(y)dy, ϕ.
Finally, recalling (2.9), we have proved that
hLΨ[w], ϕi=ZU
w(x)ϕ(x)dx(2.10)
for all wand ϕbelonging to C∞
c(U). Actually, the regularity of GK,Udiscussed at the beginning of
the proof allows us to state that Ψ[w]∈H1
0(U). Then, passing to the limit in the test functions, we
infer that (2.10) holds also true for w∈C∞
c(U) and ϕ∈H1
0(U).
Thanks to this property, we are in position to establish the symmetry for SK,U. Indeed, for any
ϕ, w ∈C∞
c(U), we use (2.10) with the test function Ψ[ϕ]∈H1
0(U) and the symmetry of the matrix
K(x) for all x∈ U to infer that
ZwΨ[ϕ] = hLΨ[w],Ψ[ϕ]i=−ZK(x)∇Ψ[w]· ∇Ψ[ϕ] = ZϕΨ[w].
Thus, by the symmetry of GK(i.e. GK(x, y) = GK(y, x)) we obtain
ZUZU
SK,U(x, y)w(x)ϕ(y)dydx=ZUZU
SK,U(x, y)ϕ(x)w(y)dydx=ZUZU
SK,U(y, x)w(x)ϕ(y)dydx
10
and finally SK,U(x, y) = SK,U(y, x).
Finally, we consider ω∈L∞
c(U) and (wn) a sequence of functions belonging to C∞
c(U) which
converges to ωin Lp(U) for some p > 2. Without any loss of generality, we can assume that ωand
wnfor all nare compactly supported in some V⋐U. Due to the C1regularity of x7→ SK,U(x, y)
uniformly in y∈V, we deduce that ∇Ψ[ω] = R∇xGK,Uω(y)dyand that Ψ[wn]→Ψ[ω] when n→ ∞
in W1,∞(U). In particular, we may pass at the limit in (2.10) to conclude that Ψ[ω]∈H1
0(U) is
solution of LΨ = ω.
2.4 Decomposition of the velocity
We now come back to our problem with Kbeing the matrix given by (1.4), which satisfies the
hypotheses of Proposition 2.5. Let us give some details on the singular part GKof the Biot-Savart
kernel.
We now denote by Hthe kernel
H(x, y) := 1
2πdet K(x) det K(y)−1/4=p|X||Y|
2πh ,(2.11)
where the last equality comes from Lemma 2.3, so that
GK(x, y) = H(x, y) ln |T (x)− T (y)|.(2.12)
We now compute:
∇⊥
xGK(x, y) = ∇⊥
xH(x, y) ln |T (x)− T (y)|+H(x, y) DT(x)T(x)− T (y)
T(x)− T (y)2!⊥
.(2.13)
Moreover,
∇⊥
xH(x, y) = 1
4πh
x⊥
|X|p|Y|
p|X|=H(x, y)
2
x⊥
|X|2.
A consequence of Lemma 2.4, Proposition 2.5 (in particular (2.4)), and of the latter computations
is that the velocity field vεin System (1.5) decomposes as follows.
Proposition 2.6. Let (vε, ωε)be a weak bounded solution of (1.5)in the sense of Definition 2.1.
Then,
vε=vε
K+vε
L+vε
R,(2.14)
with vε
Kdefined as
vε
K(x, t) = ZU
H(x, y)DT(x)T(x)− T (y)
T(x)− T (y)2ωε(y, t)dy!⊥
,(2.15)
vε
Ldefined as
vε
L(x, t) = x⊥
2|X|2ZU
GK(x, y)ωε(y, t)dy, (2.16)
and
vε
R(x, t) = ZU∇⊥
xSK,U(x, y)ωε(y, t)dy. (2.17)
When ωεis close to a Dirac mass, as in Theorem 1.1, the part vε
Kof the velocity is the most
singular, of order 1/ε, however as usual in the study of 2D point-vortices, has a symmetric structure
and will give the standard spinning around the filament which will not contribute to the displacement
of the vortex core. The part vε
L, also singular as it is of order |ln ε|near the singularity, induces
a rotation (and a vertical translation) of the helix around the origin at speed of order 1 thanks to
the rescaling in System (1.5). Finally, the part vε
Rof the velocity is a bounded remainder, whose
contribution to the movement goes to 0 as ε→0 under the rescaling.
We conclude with an estimate on ∇GK.
11
Lemma 2.7. For every R > 0, there exists a constant CRsuch that for every x, y ∈B(0, R),x6=y,
there holds that
|∇xGK(x, y)|6CR
|x−y|.
Proof. We use the following decomposition :
∇xGK(x, y) = H(x, y)
2
x
|X|2ln |T (x)− T (y)|+H(x, y)DT(x)T(x)− T (y)
T(x)− T (y)2
:= A1(x, y) + A2(x, y).
Let R > 0. Let us notice that for every x∈B(0, R),
h6|X|6ph2+R2(2.18)
hence His a bounded map on B(0, R). So are DTand T−1, so (2.3) gives a constant Csuch that
|A2(x, y)|6C
|x−y|.
Using again (2.3), we have a constant Csuch that
ln |T (x)− T (y)|6ln |x−y|+C
so we have for every x, y ∈B(0, R) that
|x−y|ln |T (x)− T (y)|6C.
By (2.18) and since His bounded, we conclude that
|A1(x, y)|6C
|x−y|.
The constants involved only depend on Rso our proof is complete.
3 Reduction of the problem to a single vortex
In order to prove Theorem 1.1, we introduce a reduced problem that focuses on the dynamics of a
single blob of vorticity, by considering the effect of the other blobs as an exterior field.
To proceed, we place ourselves in the framework of Theorem 1.1, namely we consider ωε(·, t) =
PN
i=1 ωε
i(·, t) the solution of (1.5) with initial data ωε(·,0) = PN
i=1 ωε
i,0satisfying (1.6). As in Re-
mark 2.2,ωε
icorresponds to the transport of ωi,0.
For each i∈ {1,...,N}, we introduce the exterior field Fε
i:U × [0,+∞)→R2given by
Fε
i(x, t) = X
j6=iZU∇⊥
xGK,U(x, y)ωε
j(y, t)dy, (3.1)
so that the i-th blob ωε
isatisfies the following equations:
∂tωε
i+1
|ln ε|(vε
i+Fε
i)· ∇ωε
i= 0 in U × R∗
+
vε
i=∇⊥Ψε
iin U × R+
div K(x)∇Ψε
i=ωε
iin U × R+
ωε
i(·,0) = ωε
i,0in U.
(3.2)
12
For any r>0 and η > 0 we define the annulus Ar
ηas
Ar
η=x∈R2,|x| − r< η.(3.3)
Now recall from the hypotheses of Theorem 1.1 that Ucontains a ball B(0, RU), and that for ever
i∈ {1,...,N},|zi,0|< RU. Let
η0=1
4min n|zi,0| − |zj,0|, i 6=jo[nRU− |zi,0|, i ∈ {1,...,N}o!,
and let T > 0. From the definition of η0, we infer in particular that A|zi,0|
η0⊂ U and more precisely
that
dist N
[
i=1 A|zi,0|
η0, ∂U!>η0.(3.4)
This gives the following proposition.
Proposition 3.1. There exists a constant Csuch that for every x, y ∈SN
j=1 A|zj,0|
η× U,
|SK,U(x, y)|6C
|∇xSK,U(x, y)|6C.
Proof. This is a direct consequence of relation (3.4) and the fact exposed in the proof of Proposition 2.5
that for every V⋐U,x7→ SK,U(x, y) is bounded in W1,∞(U) uniformly in yon V.
We introduce
Tε= sup nt∈[0, T ],∀s∈[0, t],∀i∈ {1,...,N},supp ωε
i(·, s)⊂ A|zi,0|
η0o,(3.5)
which is a time during which every blob of vorticity is localized in a fixed annulus. By continuity of
the trajectories, we know that Tε>0 for every ε > 0.
On time interval [0, Tε], we have the following useful estimates on Fε
i.
Lemma 3.2. There exists a constant Csuch that for every ε > 0small enough, for every i∈
{1,...,N}, for every x, y ∈ A|zi,0|
η0and for every t∈[0, Tε],
Fε
i(x, t)−Fε
i(y, t)6C|x−y|
|Fε
i(x, t)|6C,
div Fε
i(x, t) = 0.
Proof. Let i∈ {1,...,N}and t∈[0, Tε]. By definition of Fε
i(given in (3.1)), div Fε
i= 0, for every
x∈ U.
Now let x, y ∈ A|zi,0|
η0. Then
Fε
i(x, t)−Fε
i(y, t) = X
j6=iZU∇⊥
xGK,U(x, z)− ∇⊥
xGK,U(y, z )ωε
j(z, t)dz.
From the definition of GK(2.12), we note that GK(·, z)∈C2A|zi,0|
η0uniformly to z∈Sj6=iA|zj,0|
η0.
In the same way, the right hand side term of (2.6) has the same regularity, hence by elliptic regularity,
SK,Ubelongs also to C2A|zi,0|
η0uniformly to z∈Sj6=iA|zj,0|
η0. Therefore, drawing a curve included
in A|zi,0|
η0between any x, y ∈ A|zi,0|
η0, we conclude that there exists a constant Csuch that for every
z∈Sj6=iA|zj,0|
η0∇⊥
xGK,U(x, z)− ∇⊥
xGK,U(y, z )6C|x−y|,
and thus we have that Fε
i(x, t)−Fε
i(y, t)6C|x−y|kωεkL1.
13
Recalling that kωεkL1does not depend on ε, we have that
Fε
i(x, t)−Fε
i(y, t)6C|x−y|.
Using only the C1regularity of GK,Uuniformly in z, we have in the same way
|Fε
i(x, t)|6C.
4 Single vortex in an exterior field
In this section we turn to the study of the reduced problem with a single blob of vorticity. Let
i∈ {1,...,N}. We denote by z0=zi,0,γ=γi,r0=|z0|,
ν=−γ
4πhph2+r2
0
,
and define
z(t) = e
Rtν z0.
This way, we drop completely the index iand simply consider a solution ωεof (3.2) with an exterior
field Fε. Without loss of generality, one can assume that γ > 0 so that the hypotheses on ωε
0now
become, for every ε > 0,
supp ωε
0⊂B(z0, ε)
06ωε
06M0ε−2
Zωε
0(x)dx=γ.
(4.1)
For the sake of clarity, we also denote by Aηthe annulus Ar0
ηin this section since all of our study will
take place near r0. By construction, ωεsatisfies
∀t∈[0, Tε],supp ωε(·, t)⊂ Aη0(4.2)
and the exterior field Fεsatisfies by Lemma 3.2 that for every x, y ∈ Aη0and for every t∈[0, Tε],
Fε(x, t)−Fε(y, t)6C|x−y|
|Fε(x, t)|6C,
∇ · Fε(x, t) = 0.
(4.3)
Remark 4.1. We recall that Tεgiven by (3.5)takes into account all the index i. Therefore, every
estimate obtained in this section holds on the time interval [0, Tε]for every blob simultaneously.
This section is devoted to the proof of the following intermediate result describing how supp ωε(·, t)
“mostly” shrinks to z(t) as ε→0 at least on the time interval Tε.
Theorem 4.2. The following properties hold true.
(i)There exists constants CTand εTsuch that for every ε∈(0, εT], by letting rε=ln |ln ε|
|ln ε|1/2, we
have
sup
t∈[0,Tε]γ−ZB(z(t),rε)
ωε(x, t)dx6CT
ln |ln ε|,
and
sup
t∈[0,Tε]
1
γZxωε(x, t)dx−z(t)6CT
p|ln ε|.
14
(ii)For every κ∈(0,1
4), there exists constants Cκ,T and εκ,T >0, such that for every ε∈(0, εκ,T ]
and for every t∈[0, Tε], we have
supp ωε(·, t)⊂x∈R2,|x| − |z|6Cκ,T
|ln ε|κ.
The plan of the proof of Theorem 4.2 is the following. We derive precise estimates on energy and
vorticity moments of ωεto obtain the weak localization (i). Then we reproduce the now classical
arguments (see [4,19]) to obtain the strong localization (ii). Even though these properties are only
obtained on the time interval [0, Tε], the fact that the localization from (ii) is better that the a priori
localization will yield in the end that Tε=Tfor every εsmall enough.
4.1 Preliminary computations
Let us introduce a few preliminary technical lemmas. We start with a useful formula, which is a
consequence of Remark 2.2 (recalling that the flow Xεassociated to the velocity field vε+Fεis
divergence free):
Lemma 4.3. Let α∈C1(U × [0, Tε],R). Then for every t∈[0, Tε],
d
dtZα(x, t)ωε(x, t)dx=Z∂tα(x, t)ωε(x, t) + 1
|ln ε|Z∇α(x, t)·vε(x, t) + Fε(x, t)ωε(x, t)dx.
Lemma 4.4. There exists a constant Cindependent of εsuch that for εsmall enough and for every
t∈[0, Tε], we have ZZ ln |T (x)− T (y)|ωε(x, t)ωε(y, t)dxdy6C|ln ε|.
Moreover, at time 0, we have
ZZ ln |T (x)− T (y)|ωε
0(x)ωε
0(y)dxdy=−γ2|ln ε|+O(1)
as ε→0.
Proof. Notice first that
ZZ ln |T (x)− T (y)|ωε(x, t)ωε(y, t)dxdy=ZZ ln |T (x)− T (y)|
|x−y|ωε(x, t)ωε(y, t)dxdy
+ZZ ln |x−y|ωε(x, t)ωε(y, t)dxdy. (4.4)
By (2.3), there exists a constant Csuch that for every x, y ∈ Aη0,
ln |T (x)− T (y)|
|x−y|6C,
hence, recalling that kωε(·, t)kL1=γis independent of ε,
ZZ ln |T (x)− T (y)|
|x−y|ωε(x, t)ωε(y, t)dxdy=O(1) (4.5)
as ε→0, namely is bounded uniformly in ε.
We now observe that there exists a constant Csuch that for every x, y ∈ Aη0,
ln |x−y|6C,
and thus, recalling that ωε>0,
ZZ ln |x−y|ωε(x, t)ωε(y, t)dxdy6C. (4.6)
15
We now apply Lemma B.1 from [19], which is recalled as Lemma A.1, to g(s) = −ln(s)1(0,1),M=
M0ε−2. Indeed by (4.1) and since the L1and L∞norms of ωεare conserved, then ωε(·, t)∈ EM,γ for
every t∈[0, Tε]. Therefore, by letting r=εqγ
πM0, we obtain that
−Zln |x−y|ωε(y, t)dy6−2πM0ε−2Zr
0
sln(s)ds
=−2πM0ε−2r2ln r
2+O(ε2)=−γln ε+O(1) = γ|ln ε|+O(1)
as ε→0, which yields that
ZZ ln |x−y|ωε(x, t)ωε(y, t)dxdy>−γ2|ln ε|+O(1).(4.7)
Gathering (4.4), (4.5), (4.6) and (4.7), we conclude that for εsmall enough,
ZZ ln |T (x)− T (y)|ωε(x, t)ωε(y, t)dxdy6C|ln ε|.
At time 0, we improve the upper-bound obtained in (4.6) by using the strong localization hypothesis
(4.1). Indeed we have that
∀x, y ∈supp ωε
0,|x−y|62ε,
therefore, we get that
ZZ ln |x−y|ωε
0(x)ωε
0(y)dxdy6ZZ ln(2ε)ωε
0(x)ωε
0(y)dxdy6−γ2|ln ε|+O(1),
which, combined with (4.4), (4.5) and (4.7) gives that
ZZ ln |T (x)− T (y)|ωε
0(x, t)ωε
0(y, t)dxdy=−γ2|ln ε|+O(1).
Remark 4.5. We also have that
ZZ ln |T (x)− T (y)|ωε(x, t)ωε(y, t)dxdy6C|ln ε|
and ZZ ln |T (x)− T (y)|ωε
0(x)ωε
0(y)dxdy=γ2|ln ε|+O(1)
with very few adaptations to the proof of Lemma 4.4 since the for every t∈[0, Tε],ωε(·, t)is supported
in Aη0which is bounded.
4.2 Estimates on the local energy
We introduce the local energy
ψε(x, t) := ZGK(x, y)ωε(y, t)dy. (4.8)
In particular, we have from (2.16) that
vε
L(x, t) = x⊥
2|X|2ψε(x, t).(4.9)
We establish an important lemma on the local energy ψεdefined at (4.8).
16
Lemma 4.6. There exists a constant Csuch that for every t6Tεand x∈ Aη0,
−C6−ψε(x, t)6γ|X|
2πh |ln ε|+O(1),
as ε→0.
Proof. We recall that
ψε(x, t) = ZGK(x, y)ωε(y, t)dy=Zp|X||Y|
2πh ln |T (x)− T (y)|ωε(y, t)dy.
Since for every x∈ Aη0,|x|6r0+η0, and since Tis bounded on B(0, r0+η0), there exists a constant
Cindependent of εsuch that
ψε(x, t)6C.
Now we write that
ψε(x, t) = |X|
2πh Zln |T (x)− T (y)|ωε(y, t)dy+p|X|
2πh Zp|Y| − p|X|ln |T (x)− T (y)|ωε(y, t)dy.
Since the map x7→ p|X|is smooth on the set Aη0, there exists a constant Csuch that for all
x, y ∈ Aη0,p|Y| − p|X|6C|x−y|and therefore
p|X|
2πh Zp|Y| − p|X|ln |T (x)− T (y)|ωε(y, t)dy=O(1),
so that
−ψε(x, t) = −|X|
2πh Zln |T (x)− T (y)|ωε(y, t)dy+O(1).
Reproducing the arguments of Lemma 4.4, we obtain that
−ψε(x, t)6γ|X|
2πh |ln ε|+O(1).
4.3 Estimate on radial vorticity moments
For every k>1, for every t6Tε, let
Jε
k(t) = Z|x|kωε(x, t)dx. (4.10)
Following the observation in [19], we obtain sharp estimates on Jε
kusing the fact that vε
L(x, t)·x= 0.
Lemma 4.7. Assume that either k>2, or k= 1 and r06= 0. Then for every t6Tε, there exists a
constant Cksuch that Jε
k(t)−γrk
06Ck
|ln ε|.
Proof. Let k>2, or k= 1 and r06= 0. We compute using Lemma 4.3:
d
dtJε
k(t) = k
|ln ε|Z|x|k−2x·(vε+Fε)(x, t)ωε(x, t)dx.
Then, using the decomposition (2.14) and (2.16), we have that
d
dtJε
k(t) = k
|ln ε|Z|x|k−2x·vε
K(x, t)ωε(x, t)dx+k
|ln ε|Z|x|k−2x·(vε
R+Fε)(x, t)ωε(x, t)dx
17
:= k
|ln ε|A1+A2.
We start with A2. Recalling that Fεis bounded from (4.3), and observing that vε
R, defined at
relation (2.17), is bounded by Proposition 3.1, we obtain that
A2=O(1).
We then turn to A1. We compute using the definition (2.15) of vε
Kand the symmetry of DT:
A1=ZZ H(x, y)|x|k−2x· DT(x)T(x)− T (y)
T(x)− T (y)2!⊥
ωε(y, t)ωε(x, t)dxdy
=−ZZ H(x, y)|x|k−2x⊥·DT(x)T(x)− T (y)
T(x)− T (y)2ωε(y, t)ωε(x, t)dxdy
=−1
2ZZ H(x, y)|x|k−2DT(x)x⊥− |y|k−2DT(y)y⊥·T(x)− T (y)
T(x)− T (y)2ωε(y, t)ωε(x, t)dxdy.
Now we observe that since k>2 or r06= 0, the map x7→ |x|k−2DT(x)x⊥is smooth on Aη0so for
every x, y ∈supp ωε(·, t),
|x|k−2DT(x)x⊥− |y|k−2DT(y)y·T(x