arXiv subjects

Giuseppe La Scala

Publications and source records attributed to Giuseppe La Scala.

6 recordsLinked to original sources

arXiv2026

Rigidity for capillary liquid drops of nearly circular section with constant vorticity

We consider time-independent solutions with constant vorticity of the free boundary Euler equations for a 3D liquid drop with capillarity. A rigidity result for the solutions of this problem has been recently proved with variational methods: if a certain quantity involving the vorticity parameter, the capillarity coefficient and the area of the equatorial section of the drop is below a certain value, then the solution has necessarily cylindrical symmetry, the shape of the drop is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. In this paper we develop a perturbation analysis of the problem for fluid domains whose equatorial section is close in C2 norm to a disc, and we show that a rigidity result holds also above the threshold obtained with variational methods.

math.AP↗

arXiv2026

A rigidity result for the 3D capillary liquid drop with constant vorticity

We consider the free boundary problem for the Euler equations of fluid dynamics governing the motion of a 3D liquid drop with capillarity $σ_0$ and nearly spherical shape, under the assumption of constant vorticity $(0, 0, α_0)$. First we study the compatibility of the constant vorticity condition with the evolution in time of the system, showing that, for $α_0 \neq 0$, any smooth solution with convex domain must satisfy a strong geometrical constraint on the shape of the fluid domain, and that the constant vorticity condition (unlike in the irrotational case $α_0 = 0$) does not define an invariant set for the time evolution of the system. Then we focus on the time-independent solutions of the problem and we prove a new rigidity result: starting without assuming any symmetry condition, we show that, if the ratio $α_0^2/σ_0$ is not too large, then any nearly spherical solution has necessarily cylindrical symmetry, and therefore it is the unique axisymmetric solution already known in literature, the fluid domain is close, but not equal, to a ball, more precisely it is an oblate spheroid, flattened at the poles and bulged at the equator, and each fluid particle moves along a horizontal, circular trajectory with constant angular velocity. To the best of our knowledge, this is the first result for the capillary liquid drop with constant vorticity obtained without assuming cylindrical symmetry.

math.AP↗

arXiv2026

2D capillary liquid drops with constant vorticity: rotating waves existence and a conditional energetic stability result for rotating circles

We consider a two-dimensional, pure capillary drop of nearly-circular shape, having constant vorticity. We write the Craig-Sulem equations on the unit circle, then on the flat torus. We show their Hamiltonian structure and we then observe symmetries and we derive constants of motions. After showing linear stability for rotating circles, we prove the existence of rotating waves by combining a bifurcation-theoretical approach together with critical point theory. Finally, by exploiting the Hamiltonian structure, we show that whenever volume and barycenter are fixed to be the same as those of rotating circle, this solution is also conditionally energetically stable. This holds in the irrotational case as well, in agreement with the stability analysis of rotating cylinder jets in Rayleigh [25].

math.AP↗

arXiv2025

Optimal control of mean-field limit of multiagent systems with and without common noise

We consider a generic, suitable class of optimal control problems under a constraint given by a finite-dimensional SDE-ODE system, describing a system of two interacting species of particles: the herd, described by SDEs, and the herders, described by ODEs with the addition of a control function. In particular, we firstly show that for a low number of herders and for the limit of large number of herd individuals, the SDE-ODE system can be approximated by an infinite-dimensional system given by a McKean-Vlasov single SDE coupled with ODEs. Then, thanks to this we show the $Γ-$convergence of the optimal control problem for the finite-dimensional system to a certain optimal control problem for the mean-field system. Differently from Ascione-Castorina-Solombrino [9] (SIAM J. Math. Anal., Vol. 55, No. 6, pp. 6965-6990 (2023)), we do not consider an additive noise for the herd, but a more general class, given by idiosyncratic noises (due to a single herd individual) together with common noise (due to how the environment affects the whole herd), and they are independent one from another. As well as this, we consider a more general class of control functions in the ODEs for herders, where the control is applied not only on the herd dynamics, but also on the herd one.

math.OC↗

arXiv2025

Two-dimensional capillary liquid drop: Craig-Sulem formulation on $\mathbb{T}^1$ and bifurcations from multiple eigenvalues of rotating waves

We consider the free boundary problem for a two-dimensional, incompressible, perfect, irrotational liquid drop of nearly circular shape with capillarity: that is, we consider the 2D version of the 3D capillary drop problem treated in Baldi-Julin-La Manna [11] and Baldi-La Manna-La Scala [12]. In particular, we derive its Craig-Sulem formulation firstly over the circle, then over the one-dimensional flat torus; the arising equations are similar to the pure capillary Water Waves for the ocean problem, apart from conformal factors and additional terms due to curvature terms. Then, we show its Hamiltonian structure and we derive constants of motions from symmetries, one of which is the invariance by the torus action. Thanks to this invariance, we show the existence of orbits of rotating wave solutions (which are the analogous of travelling waves of the ocean problem) by bifurcation from multiple eigenvalues in the spirit of Moser-Weinstein [44, 56] and Craig-Nicholls [22] variational approaches; in particular, we can parametrize such orbits by the angular momentum, and for each value of it they are unique. This will imply that each orbit is generated by symmetric rotating waves.

math.AP↗

arXiv2025

Bifurcation from multiple eigenvalues of rotating traveling waves on a capillary liquid drop

We consider the free boundary problem for a liquid drop of nearly spherical shape with capillarity, and we study the existence of nontrivial (i.e., non spherical) rotating traveling profiles bifurcating from the spherical shape, where the bifurcation parameter is the angular velocity. We prove that every eigenvalue of the linearized problem is a bifurcation point, extending the known result for simple eigenvalues to the general case of eigenvalues of any multiplicity. We also obtain a lower bound on the number of bifurcating solutions. The proof is based on the Hamiltonian structure of the problem and on the variational argument of constrained critical points for traveling waves of Craig and Nicholls (2000, SIAM J. Math. Anal. 32, 323-359), adapted to the nearly spherical geometry; in particular, the role of the action functional is played here by the angular momentum with respect to the rotation axis. Moreover, the bifurcation equation presents a 2-dimensional degeneration, related to some symmetries of the physical problem. This additional difficulty is overcome thanks to a crucial transversality property, obtained by using the Hamiltonian structure and the prime integrals corresponding to those symmetries by Noether theorem, which are the fluid mass and the component along the rotation axis of the velocity of the fluid barycenter.

math.AP↗