Steady three-dimensional rotational flows: an approach via two stream functions and Nash-Moser iteration

We consider the stationary flow of an inviscid and incompressible fluid of constant density in the region $D=(0, L)\times \mathbb{R}^2$. We are concerned with flows that are periodic in the second and third variables and that have prescribed flux through each point of the boundary $\partial D$. The Bernoulli equation states that the "Bernoulli function" $H:= \frac 1 2 |v|^2+p$ (where $v$ is the velocity field and $p$ the pressure) is constant along stream lines, that is, each particle is associated with a particular value of $H$. We also prescribe the value of $H$ on $\partial D$. The aim of this work is to develop an existence theory near a given constant solution. It relies on writing the velocity field in the form $v=\nabla f\times \nabla g$ and deriving a degenerate nonlinear elliptic system for $f$ and $g$. This system is solved using the Nash-Moser method, as developed for the problem of isometric embeddings of Riemannian manifolds; see e.g. the book by Q. Han and J.-X. Hong (2006). Since we can allow $H$ to be non-constant on $\partial D$, our theory includes three-dimensional flows with non-vanishing vorticity

Optimal mass transportation and Mather theory

We study optimal transportation of measures on compact manifolds for costs defined from convex Lagrangians. We prove that optimal transportation can be interpolated by measured Lipschitz laminations, or geometric currents. The methods are inspired from Mather theory on Lagrangian systems. We make use of viscosity solutions of the associated Hamilton-Jacobi equation in the spirit of Fathi's approach to Mather theory

Discrete spectrum of perturbed Dirac systems with real and periodic coefficients

This paper deals with the number of eigenvalues which appear in the gaps of the spectrum of a Dirac system with real and periodic coefficients when the coefficients are perturbed. The main results provide an upper bound and a condition under which exactly one eigenvalue appears in a given ga

Generalized Flows Satisfying Spatial Boundary Conditions

In a region D in $${\mathbb{R}^2}$$ or $${\mathbb{R}^3}$$ , the classical Euler equation for the regular motion of an inviscid and incompressible fluid of constant density is given by $$\partial_t v+(v\cdot \nabla_x)v=-\nabla_x p, {\rm div}_x v=0,$$ where v(t, x) is the velocity of the particle located at $${x\in D}$$ at time t and $${p(t,x)\in\mathbb{R}}$$ is the pressure. Solutions v and p to the Euler equation can be obtained by solving $$\left\{\begin{array}{l} \nabla_x\left\{\partial_t\phi(t,x,a) + p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2 \right\}=0\,{\rm at}\,a=\kappa(t,x),\\ v(t,x)=\nabla_x \phi(t,x,a)\,{\rm at}\,a=\kappa(t,x), \\ \partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, \\ {\rm div}_x v(t,x)=0, \end{array}\right. \quad\quad\quad\quad\quad(0.1)$$ where $$\phi:\mathbb{R}\times D\times \mathbb{R}^l\rightarrow\mathbb{R}\,{\rm and}\, \kappa:\mathbb{R}\times D \rightarrow \mathbb{R}^l$$ are additional unknown mappings (l≥ 1 is prescribed). The third equation in the system says that $${\kappa\in\mathbb{R}^l}$$ is convected by the flow and the second one that $${\phi}$$ can be interpreted as some kind of velocity potential. However vorticity is not precluded thanks to the dependence on a. With the additional condition κ(0, x)=x on D (and thus l=2 or 3), this formulation was developed by Brenier (Commun Pure Appl Math 52:411-452, 1999) in his Eulerian-Lagrangian variational approach to the Euler equation. He considered generalized flows that do not cross $${\partial D}$$ and that carry each "particle” at time t=0 at a prescribed location at time t=T>0, that is, κ(T, x) is prescribed in D for all $${x\in D}$$ . We are concerned with flows that are periodic in time and with prescribed flux through each point of the boundary $${\partial D}$$ of the bounded region D (a two- or three-dimensional straight pipe). More precisely, the boundary condition is on the flux through $${\partial D}$$ of particles labelled by each value of κ at each point of $${\partial D}$$ . One of the main novelties is the introduction of a prescribed "generalized” Bernoulli's function $${H:\mathbb{R}^l\rightarrow \mathbb{R}}$$ , namely, we add to (0.1) the requirement that $$\partial_t\phi(t,x,a) +p(t,x)+(1/2)|\nabla_x\phi(t,x,a)|^2=H(a)\,{\rm at}\,a=\kappa(t,x)\quad\quad\quad\quad\quad(0.2)$$ with $${\phi,p,\kappa}$$ periodic in time of prescribed period T>0. Equations (0.1) and (0.2) have a geometrical interpretation that is related to the notions of "Lamb's surfaces” and "isotropic manifolds” in symplectic geometry. They may lead to flows with vorticity. An important advantage of Brenier's formulation and its present adaptation consists in the fact that, under natural hypotheses, a solution in some weak sense always exists (if the boundary conditions are not contradictory). It is found by considering the functional $$(\kappa,v)\rightarrow \int\limits_{0}^T \int\limits_D\left\{\frac 1 2 |v(t,x)|^2+H(\kappa(t,x))\right\}dt\, dx$$ defined for κ and v that are T-periodic in t, such that $$\partial_t\kappa(t,x)+(v\cdot\nabla_x)\kappa(t,x)=0, {\rm div}_x v(t,x)=0,$$ and such that they satisfy the boundary conditions. The domain of this functional is enlarged to some set of vector measures and then a minimizer can be obtained. For stationary planar flows, the approach is compared with the following standard minimization method: to minimize $$\int\limits_{]0,L[\times]0,1[} \{(1/2)|\nabla \psi|^2+H(\psi)\}dx\,{\rm for}\,\psi\in W^{1,2}(]0,L[\times]0,1[)$$ under appropriate boundary conditions, where ψ is the stream function. For a minimizer, corresponding functions $${\phi}$$ and κ are given in terms of the stream function

Weak KAM pairs and Monge-Kantorovich duality

The dynamics of globally minimizing orbits of Lagrangian systems can be studied using the Barrier function, as Mather first did, or using the pairs of weak KAM solutions introduced by Fathi. The central observation of the present paper is that Fathi weak KAM pairs are precisely the admissible pairs for the Kantorovich problem dual to the Monge transportation problem with the Barrier function as cost. We exploit this observation to recover several relations between the Barrier functions and the set of weak KAM pairs in an axiomatic and elementary way.Comment: Advanced Studies in Pure Mathematics 47, 2 (2007

Appunti per una semantica della rappresentanza politica: note ‘libere’ dall’incontro sassarese su “La rappresentanza nel diritto pubblico”

The paper aims to give an account of the issues of "representation in public law" treated in the Sassari’s meeting of 6 November 2014, imagining to continue the dialogue with the speakers. First, the constitutionalist Caretti has performed an attempt to outline the evolution (which proved involution) of the concept of political representation from the threshold of modernity up to the State Constitutional contemporary. Second, the Romanist Lobrano had a radical criticism of the logic of representation, that opposed the model of constitutionalism-medieval English or parliamentary-representative to the Roman-republican model, which proposes the recovery of the contract societas against the deception of the person ficta vel repraesentata. Finally, the political philosopher Mura looked to political representation as a modern form of political legitimacy of the authority and underlined the existence two conceptions of representation in the history of Western political thought: the replacement of people against the replacement of will. The author tries to rethink the representation, which also today shapes the deliberativist democratism and the Rousseau’s constitutionalism, as a force of quality 'active' than the social pluralism, which gives form to the ideas (people, nation .. ) that did not pre-exist. Il lavoro si propone di dar conto delle questioni su “La rappresentanza nel diritto pubblico” messe in gioco nell’incontro sassarese del 6 novembre 2014, immaginando di proseguire il dialogo con i relatori. Alle riflessioni del costituzionalista Caretti, che ha esperito il tentativo di delineare l’evoluzione (che si è rivelata involuzione) del concetto di rappresentanza politica dalle soglie della modernità sino allo Stato costituzionale contemporaneo, segue la critica radicale della logica della rappresentanza del romanista Lobrano, che oppone al modello del costituzionalismo medievale-inglese o parlamentare-rappresentativo il modello romano-repubblicano-municipale, del quale propone il recupero del contratto di societas contro l’inganno della persona ficta vel repraesentata. Il filosofo politico Mura, infine, ha guardato alla rappresentanza politica come forma moderna di legittimazione politica dell’autorità ed ha isolato due concezioni della rappresentanza nella storia del pensiero politico occidentale: la sostituzione di persone vs la sostituzione di volontà. L’autrice prova a ripensare la rappresentanza, della quale neppure il democraticismo deliberativista o il costituzionalismo rousseauviano paiono sino in fondo liberarsi, come una forza di qualità ‘attiva’ rispetto al pluralismo sociale, che mette in forma le idee (popolo, nazione..) che non le preesistono

Shooting methods and topological transversality

We show that shooting methods for homoclinic or heteroclinic orbits in dynamical systems may automatically guarantee the topological transversality of the stable and unstable manifolds. The interest of such results is twofold. First, these orbits persist under perturbations which destroy the structure allowing the shooting method and, second, topological transversality is often sufficient when some kind of transversality is required to obtain chaotic dynamics. We shall focus on heteroclinic solutions in the extended Fisher-Kolmogorov equatio

On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

Existence and conditional energetic stability of three-dimensional fully localised solitary gravity-capillary water waves

In this paper we show that the hydrodynamic problem for three-dimensional water waves with strong surface-tension effects admits a fully localised solitary wave which decays to the undisturbed state of the water in every horizontal direction. The proof is based upon the classical variational principle that a solitary wave of this type is a critical point of the energy subject to the constraint that the momentum is fixed. We prove the existence of a minimiser of the energy subject to the constraint that the momentum is fixed and small. The existence of a small-amplitude solitary wave is thus assured, and since the energy and momentum are both conserved quantities a standard argument may be used to establish the stability of the set of minimisers as a whole. `Stability' is however understood in a qualified sense due to the lack of a global well-posedness theory for three-dimensional water waves.Comment: 83 pages, 1 figur