Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equations

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Solutions in the case of a pair of vortices

The conservation of H, M and I allows us to determine explicitly all the solutions to the pointvortex equations (2.4) in the particular case where N = 2, which corresponds to a pair of vortices. The hamitonian writes then as a function of the distance between the two vortices. Since the hamiltonian is conserved, this implies that the distance r0 between the two vortices remains constant. The speed at which every vortex moves is also constant and proportional to jdG=drj evaluated at r0 and to the intensity ai of the other vortex.
Figure 2.1 { The dierent solutions of the point-vortex problem in the case of 2 vortices a1x1 and a2x2. The center of vorticity, that is immobile, is represented by b. Left : the case where a1 and a2 have the same sign. Right : the case where a1 and a2 have opposite sign but ai + aj 6= 0. Bottom : the case ai + aj = 0. In the case where ai +aj 6= 0, the center of vorticity is well-dened and left invariant by the dynamic.
This one is at a xed distance to respectively x1 and x2, and therefore we can deduce that the trajectories are circles. The sense in which a vortex xi moves along its circle is given by the sign of ai and by the sign of the derivative of G in r0. On the contrary, if a1 + a2 = 0, the center of vorticity is no longer dened (or is at innity). The vortices are translating at same speed along two parallel lines. See Figure 2.1 for an illustration.

An example of collision

To study the well-posedness of this system of dierential equations, the natural tool is the Cauchy- Lipschitz theorem. Nevertheless, the term at the right-hand side of (2.4) may be not locally Lipschitz
because this term may blow up if the system comes close to a collision situation, meaning xi = xj for
some i 6= j. By regularity assumption on G, the system is well-posed if and only if no collision occur.
Unfortunately, there exist initial conditions leading to a collision of vortices in nite time and therefore the system in this case is not well-dened for all time. For the cases where G is the Green kernel of the Laplace operator (2.3), which is 􀀀log, a construction of such a collision is provided by [56, §4]. We refer to it for more details about this construction and give here the main ideas (see Figure 2.2 for an illustration).
At time t = 0, the three vortices x1, x2 and x3 are at positions respectively (􀀀1; 0), and (1; 0) and at (1; p 2). Their respective intensities are a1 = 2, a2 = 2 and a3 = 􀀀1. This conguration leads to a collision in nite time and the three vortices collapse on their center of vorticity B. This collision is auto-similar in the sense that in this particular conguration the angles between the the vertices of Figure 2.2 { Initial datum of the point-vortex model (2.4) with G = 􀀀 1 2 log leading to a collision in nite time. the triangle are preserved by the ow. This triangle at time t is, up to a rotation-dilatation, equal to the initial one. Moreover, the distances jxi(t) 􀀀 xj(t)j for i 6= j behaves like t 7! Cij p T 􀀀 t, where T > 0 is the time of collision. This means that these distances behave like the uni-dimensional system dened by dx=dt = 􀀀=x for > 0. This scale of convergence in square root, since G = 􀀀log, is conform to the natural intuition. This intuition is that, seen the form of system (2.4), the collisions of vortices in the case where dG=dr = r􀀀a􀀀1 with a 0 have an auto-similar behavior and the distance between vortices behaves like the uni-dimensional system given by dx=dt = 􀀀x􀀀a􀀀1. Nevertheless, to our knowledge there exist no study of the dierent types of collisions existing for the point-vortex system.

Convergence result for Euler point-vortices with non-neutral clusters hypothesis

The systems of vortices that are studied with more details in [56, §4] are the systems for which the non-neutral clusters hypothesis (2.16) holds. Concerning this kind of systems, in the Euler case, we are also able to write a convergence result. When vortices come to collision, their speed may become innite as a consequence of the kernel prole singularity in 0+. But if their speed blows up, then any pathological behavior near the time of collision TX is a priori possible. Only the continuity of the trajectories on [0; TX) is ensured. In the situation of the non-neutral clusters hypothesis, Theorem 2.2 implies that the trajectories remain bounded. We proved here after that the trajectories are actually convergent in the Euler case.

Mono and multi-scales collisions for Euler point-vortices

The idea behind this notion of mono-scale collisions consists in comparing the Euler point-vortex model, where the speed behaves like the inverse of the distances with the uni-dimensional dierential equation given by x_ = 􀀀 x . In this situation, the solutions are x(t) = p 2(T 􀀀 t). Such a solution \collides » with 0 at time T. It is natural to expect that the distances jxi(t) 􀀀 xj(t)j are also expected to behave in time like the square-root function or at least some power function. In the Euler case it is possible to study with more details the collisions in some special cases where the collision shows a nice multi-scale structure.


Energy estimate of the clusters of collision

The proof starts by recalling the multi-scale clusterization lemma (Lemma 2.2). We underline the particular role of the rst scale of collision m = 1. We call the associated clusters P1 l the clusters of collision since their respective size vanishes as a consequence of (2.34) but the distance between two remains bounded below by a positive constant, consequence of (2.35) with f0(z) = 1. Now that the dynamics of the vortices is separated into clusters, it is possible to study the vorticity energy associated to a given cluster.

Reduction of the maximization problem to a compact setting

The energy Es is well-dened on _H 􀀀s(R2), while the circulation C and the impulse L make sense in the (weighted) Lebesgue spaces L1(R2), respectively L1(R2; jxj2dx). This non-trivial functional framework is a rst obstacle to solve this problem. A second one originates in the non-compact nature of the setting. In order to by-pass these diculties, we follow the arguments developed by Turkington in [82]. We consider the section of annulus S := n (r cos(); r sin()) : a0 r a1 and 􀀀 2N 2N o .

Table of contents :

1 Introduction de la these 
1 Elements de mecanique des uides
1.1 Equations de la mecanique des uides
1.2 Equations d’Euler incompressibles homogenes
1.3 Equations d’Euler incompressibles bi-dimensionnelles
2 Modeliser l’atmosphere terrestre
2.1 Mecanique des uides geophysiques
2.2 Approximations et modelisation : vers le modele geostrophique
2.3 In uence de la thermodynamique et equations quasi-geostrophiques
3 Etat de l’art sur les equations quasi-geostrophiques surfaciques non visqueuses (SQG)
3.1 Presentation des equations
3.2 Liens avec les equations d’Euler
3.3 Existence et unicite des solutions
3.4 Formation de front de discontinuite
3.5 Vortex et vorticite pour (SQG)
4 Plan de la these
4.1 Collision de vortex pour les modeles de Euler et quasi-geostrophique
4.2 Desingularisation du collier de vortex pour (SQG)
4.3 La paire de vortex C1 en translation
4.4 Le tassement, un rearrangement un dimension 1
2 Collisions of vortices for Euler and Quasi-Geostrophic models 
1 The point-vortex system
1.1 Inviscid surface quasi-geostrophic equation
1.2 Presentation of the point-vortex model
2 Main results
2.1 The point-vortex system in a general framework
2.2 Improbability of collisions result for point-vortex
2.3 Convergence result for Euler point-vortices with non-neutral clusters hypothesis
2.4 Mono and multi-scales collisions for Euler point-vortices
3 Sketch of the proof for Theorem 2.4
3.1 The modied system
3.2 Regularization of the kernel Gs
3.3 Estimate the collisions
4 Sketch of the proof for Theorem 2.5
4.1 The vortices are Dirac masses
4.2 Convergence of the vortices
5 Sketch of the proof for Theorem 2.7
5.1 Energy estimate of the clusters of collision
5.2 Algebraic condition associated to a multi-scale convergence
5.3 Sketch of the proof for point (iii)
6 Proofs of the main lemmas and theorems
6.1 Proof of Theorem 2.2
6.2 Proofs of the lemmas for Theorem 2.5
6.3 Proofs of the lemmas for Theorem 2.4
6.4 Proofs of the lemmas for Theorem 2.7
3 Co-rotating vortices with N fold symmetry for the inviscid surface quasi-geostrophic equations 
1 Introduction
2 Outline of the proof of Theorem 1.1
2.1 Construction of the co-rotating vortex patches
2.2 Description of the vortex patch support
2.3 Desingularization of a point vortex pair
3 Details of the proofs
3.1 Proofs for Section 2.1
3.2 Proofs for Section 2.2
3.3 Proofs for Section 2.3
4 The angular Steiner symmetrization
4 Smooth traveling-wave solutions to the generalized inviscid surface quasi-geostrophic equation 
1 Presentiation of the problem
1.1 The quasi-geostrophic equations
1.2 Variational formulation
1.3 Nehari Manifold and presentation of the main result
2 Strategy of proof and main lemmas
2.1 Properties of the Nehari Manifold and minimizing sequences
2.2 Existence of the solution of the minimizaing problem
2.3 Properties of the solution
3 Proofs of the lemmas
3.1 Proofs of the lemmas of section 2.1
3.2 Proof of Lemma 2.2
3.3 Proof of Lemma 2.3
3.4 Proofs of the lemmas of section 2.2
3.5 Proofs of the lemmas of section 2.3
5 Tamped functions, A rearrangement in dimension 1 
1 Presentation of the problem
1.1 Layer-cake representation
1.2 The Schwarz non-increasing rearrangement
1.3 Rearrangement inequalities
1.4 Limitation of the Schwarz rearrangement, preserving Dirichlet boundary condition
2 Denition of the rearrangement by tamping
2.1 Denition of the tamping on voxel functions
2.2 Denition of the tamping in M+(R+).
2.3 Best non-decreasing upper bound
3 Main results about the tamping
3.1 Functional analysis
3.2 Topological results in Lp +(R+).
3.3 Polya{Szeg}o inequality for the tamping
3.4 About a Riesz inequality for the tamping
4 Proofs of the main results
4.1 Proof of Property 2.5
4.2 Proof of Property 2.6
4.3 Proof of Lemma 2.4
4.4 Proof of Lemma 3.1
4.5 Proof of Lemma 3.2
4.6 Proof of Lemma 3.3
4.7 Proof of Lemma 3.4 for p = 1
4.8 Proof of Lemma 3.5
4.9 Proof of Theorem 3.1
4.10 Proof of Theorem 3.2
4.11 Proof of the counter-example for the Riesz inequality


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