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Pietro Baldi

Publications and source records attributed to Pietro Baldi.

At least 19 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.

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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 $\sigma_0$ and nearly spherical shape, under the assumption of constant vorticity $(0, 0, \alpha_0)$. First we study the compatibility of the constant vorticity condition with the evolution in time of the system, showing that, for $\alpha_0 \neq 0$, 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 $\alpha_0 = 0$) 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 $\alpha_0^2/\sigma_0$ 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.

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arXiv2026

On the Dirichlet-Neumann operator for nearly spherical domains

We consider the Dirichlet-Neumann operator for a nearly spherical domain in R^n, and prove sharp analytic and tame estimates in Sobolev class. The novelty of this paper concerns technical improvements, the most important of which are the independence of the analyticity radius on the high norms and the regularity loss of one in the elevation function. These properties are expectable but nontrivial to prove. The result is obtained by introducing local charts and a convenient class of non-isotropic Sobolev spaces of high, possibly fractional tangential regularity and integer, limited regularity in the normal direction.

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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.

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arXiv2024

Liquid drop with capillarity and rotating traveling waves

We consider the free boundary problem for a 3-dimensional, incompressible, irrotational liquid drop of nearly spherical shape with capillarity. We study the problem from the beginning, extending some classical results from the flat case (capillary water waves) to the spherical geometry: the reduction to a problem on the boundary, its Hamiltonian structure, the analyticity and tame estimates for the Dirichlet-Neumann operator in Sobolev class, and a linearization formula for it, both with the method of the good unknown of Alinhac and by a differential geometry approach. Then we prove the bifurcation of traveling waves, which are nontrivial (i.e., nonspherical) fixed profiles rotating with constant angular velocity.

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arXiv2023

Effective chaos for the Kirchhoff equation on tori

We consider the Kirchhoff equation on tori of any dimension and we construct solutions whose Sobolev norms oscillates in a chaotic way on certain long time scales. The chaoticity is encoded in the time between oscillations of the norm, which can be chosen in any prescribed way. This phenomenon, that we name as effective chaos (it occurs over a long, but finite, time scale), is consequence of the existence of symbolic dynamics for an effective system. Since the first order resonant dynamics has been proved to be essentially stable, we need to perform a second order analysis to find an effective model displaying chaotic dynamics. More precisely, after some reductions, this model behaves as two weakly coupled pendulums.

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arXiv2020

A Whitney extension theorem for functions taking values in scales of Banach spaces

We introduce a modified version of the Whitney extension operators for collections of functions from a closed subset of $\mathbb{R}^n$ into scales of Banach spaces with smoothing operators. We prove an extension theorem for collections whose elements take values in different spaces of the scale. A motivation for considering this kind of collections comes from very basic observations on the composition of functions of more than one real variable. The idea at the base of the proof is rather natural in the context of scales of Banach spaces, and consists in introducing smoothing operators in the construction of the extension, with smoothing parameters related to the diameter of each Whitney dyadic cube. Classical examples of scales of Banach spaces with smoothing operators are also given, and some new related observations are proved.

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arXiv2020

Longer lifespan for many solutions of the Kirchhoff equation

We consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 + \int_{\mathbb T^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb T^d$, and its Cauchy problem with initial data $u(0,x)$, $\partial_t u(0,x)$ of size $\varepsilon$ in Sobolev class. The effective equation for the dynamics at the quintic order, obtained in previous papers by quasilinear normal form, contains resonances corresponding to nontrivial terms in the energy estimates. Such resonances cannot be avoided by tuning external parameters (simply because the Kirchhoff equation does not contain parameters). In this paper we introduce nonresonance conditions on the initial data of the Cauchy problem and prove a lower bound $\varepsilon^{-6}$ for the lifespan of the corresponding solutions (the standard local theory gives $\varepsilon^{-2}$, and the normal form for the cubic terms gives $\varepsilon^{-4}$). The proof relies on the fact that, under these nonresonance conditions, the growth rate of the "superactions" of the effective equations on large time intervals is smaller (by a factor $\varepsilon^2$) than its a priori estimate based on the normal form for the cubic terms. The set of initial data satisfying such nonresonance conditions contains several nontrivial examples that are discussed in the paper.

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arXiv2020

On the normal form of the Kirchhoff equation

Consider the Kirchhoff equation $$ \partial_{tt} u - \Delta u \Big( 1 + \int_{\mathbb{T}^d} |\nabla u|^2 \Big) = 0 $$ on the $d$-dimensional torus $\mathbb{T}^d$. In a previous paper we proved that, after a first step of quasilinear normal form, the resonant cubic terms show an integrable behavior, namely they give no contribution to the energy estimates. This leads to the question whether the same structure also emerges at the next steps of normal form. In this paper, we perform the second step and give a negative answer to the previous question: the quintic resonant terms give a nonzero contribution to the energy estimates. This is not only a formal calculation, as we prove that the normal form transformation is bounded between Sobolev spaces.

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arXiv2020

Quasi-periodic incompressible Euler flows in 3D

We prove the existence of time-quasi-periodic solutions of the incompressible Euler equation on the three-dimensional torus $\T^3$, with a small time-quasi-periodic external force. The solutions are perturbations of constant (Diophantine) vector fields, and they are constructed by means of normal forms and KAM techniques for reversible quasilinear PDEs.

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arXiv2019

Size of data in implicit function problems and singular perturbations for nonlinear Schr\"odinger systems

We investigate a general question about the size and regularity of the data and the solutions in implicit function problems with loss of regularity. First, we give a heuristic explanation of the fact that the optimal data size found by Ekeland and S\'er\'e with their recent non-quadratic version of the Nash-Moser theorem can also be recovered, for a large class of nonlinear problems, with quadratic schemes. Then we prove that this heuristic observation applies to the singular perturbation Cauchy problem for the nonlinear Schr\"odinger system studied by M\'etivier, Rauch, Texier, Zumbrun, Ekeland, S\'er\'e. Using a "free flow component" decomposition and applying an abstract Nash-Moser-H\"ormander theorem, we improve the existing results regarding both the size of the data and the regularity of the solutions.

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arXiv2018

On the existence time for the Kirchhoff equation with periodic boundary conditions

We consider the Cauchy problem for the Kirchhoff equation on $\mathbb{T}^d$ with initial data of small amplitude $\varepsilon$ in Sobolev class. We prove a lower bound $\varepsilon^{-4}$ for the existence time, which improves the bound $\varepsilon^{-2}$ given by the standard local theory. The proof relies on a normal form transformation, preceded by a nonlinear transformation that diagonalizes the operator at the highest order, which is needed because of the quasilinear nature of the equation.

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arXiv2017

Time quasi-periodic gravity water waves in finite depth

We prove the existence and the linear stability of Cantor families of small amplitude time quasi-periodic standing water wave solutions - namely periodic and even in the space variable x - of a bi-dimensional ocean with finite depth under the action of pure gravity. Such a result holds for all the values of the depth parameter in a Borel set of asymptotically full measure. This is a small divisor problem. The main difficulties are the quasi-linear nature of the gravity water waves equations and the fact that the linear frequencies grow just in a sublinear way at infinity. We overcome these problems by first reducing the linearized operators obtained at each approximate quasi-periodic solution along the Nash-Moser iteration to constant coefficients up to smoothing operators, using pseudo-differential changes of variables that are quasi-periodic in time. Then we apply a KAM reducibility scheme which requires very weak Melnikov non-resonance conditions (losing derivatives both in time and space), which we are able to verify for most values of the depth parameter using degenerate KAM theory arguments.

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arXiv2016

Controllability of quasi-linear Hamiltonian NLS equations

We prove internal controllability in arbitrary time, for small data, for quasi-linear Hamiltonian NLS equations on the circle. We use a procedure of reduction to constant coefficients up to order zero and HUM method to prove the controllability of the linearized problem. Then we apply a Nash-Moser-H\"ormander implicit function theorem as a black box.

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arXiv2016

A Nash-Moser-H\"ormander implicit function theorem with applications to control and Cauchy problems for PDEs

We prove an abstract Nash-Moser implicit function theorem which, when applied to control and Cauchy problems for PDEs in Sobolev class, is sharp in terms of the loss of regularity of the solution of the problem with respect to the data. The proof is a combination of: (i) the iteration scheme by H\"ormander (ARMA 1976), based on telescoping series, and very close to the original one by Nash; (ii) a suitable way of splitting series in scales of Banach spaces, inspired by a simple, clever trick used in paradifferential calculus (for example, by M\'etivier). As an example of application, we apply our theorem to a control and a Cauchy problem for quasi-linear perturbations of KdV equations, improving the regularity of a previous result. With respect to other approaches to control and Cauchy problems, the application of our theorem requires lighter assumptions to be verified.

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arXiv2016

Non-Existence of Theta-Shaped Self-Similarly Shrinking Networks Moving by Curvature

We prove that there are no networks homeomorphic to the Greek "theta" letter (a double cell) embedded in the plane with two triple junctions with angles of $120$ degrees, such that under the motion by curvature they are self-similarly shrinking. This fact completes the classification of the self-similarly shrinking networks in the plane with at most two triple junctions.

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