Università Bocconi Milano
Universidad Complutense de Madrid
Georgia Tech Atlanta
Politecnico di Milano
Università di Torino
UNC Greensboro
Sapienza Università di Roma
University of Bath
ICMAT Madrid
University of Pittsburgh
University of Bath
Universidad de Valladolid
Universidad de Granada
Martin-Luther-Universität Halle-Wittenberg
Goethe-Universität Frankfurt
Max-Planck-Institut MiS Leipzig
Politecnico di Milano
Sapienza Università di Roma
16:00 - 16:30 coffee break
11:15 - 11:45 coffee break
13:15 - 14:30 lunch
16:00 - 16:30 coffee break
11:15 - 11:45 coffee break
13:15 - 14:30 lunch
16:00 - 16:30 coffee break
19:30 social dinner
11:15 - 11:45 coffee break
13:15 - 14:30 lunch
16:00 - 16:30 coffee break
11:30 - 12:00 coffee break
13:15 - 14:30 lunch
In recent years, there has been significant progress in the study of the three-dimensional incompressible Navier–Stokes equations. One particularly interesting development concerns nonuniqueness for weak solutions in the energy class and in critical spaces. The aim of this minicourse is to present an overview of some of the main ideas behind these results. I will begin with a brief review of the classical theory, including short-time existence and the construction of Leray solutions, and then turn to more recent bifurcation-based approaches inspired by the work of Jia and Sverak, based on the stability analysis of self-similar evolutions. I will explain how this perspective leads to rigorous nonuniqueness results, including joint work with Albritton and Colombo. I will conclude with a discussion of further recent developments, including computer-assisted results and new mechanisms for nonuniqueness in critical spaces.
The main goal of the course is presenting a general overview to global bifurcation theory from the Leray-Schauder degree and its applications to the global theorem of Rabinowitz up to the most recent developments of J. Esquinas, C. Mora-Corral, J. C. Sampedro and J. Lopez-Gomez through the degrees for Fredholm operators of Fitzpatrick-Pejsachowicz-Rabier and Benevieri-Furi. The distribution of the four sessions will be the following:
Session 1: Introduction. Bifurcation from simple eigenvalues. Lyapunov--Schmidt decompositions. Bifurcation equations. A general bifurcation theorem.
Session 2: The generalized algebraic multiplicity of Esquinas and Lopez-Gomez. Properties. The uniqueness theorem of Mora-Corral.
Session 3: Leray-Schauder degree. Parity of Fitzpatrick and Pejsachowicz. The degree for Fredholm operators of Fitzpatrick, Pejsachowicz and Rabier. Axiomatization of the degree for Fredholm operators. Uniqueness theorem of Lopez-Gomez and Sampedro.
Session 4: Global bifurcation theory. Some applications.
If a variational problem has a symmetry, it is natural to ask whether an optimizer has the same symmetry. Symmetries render obvious advantages for computing the optimizers and the optimal value for the variational problem. In general, however, optimizers do not reflect the underlying symmetry. The symmetry is broken. Therefore, the mathematical analysis has to concentrate on relevant and non-trivial examples and the related mathematical techniques. Key examples will be the standard sharp Sobolev inequality and its generalization known as the Caffarelli-Kohn-Nirenberg inequalities, Sharp Hardy-Littlewood-Sobolev inequalities and Beckner's inequality. Emphasis will be placed on some of the techniques such as the use of the isoperimetric inequality and rearrangements, flows and Obata type identities as well as the analysis of the second variation operator for proving symmetry breaking. If time permits, sharp Spinor inequalities will be discussed.
We present a framework for the computer-assisted proof of bifurcations in nonlinear PDEs, combining rigorous numerics with functional analytic techniques. The method reformulates the problem as a fixed point equation in a Banach space of analytic functions and applies a Newton-Kantorovich type argument with explicit, computer-verified bounds. The approach is illustrated on models such as the Kuramoto-Sivashinsky equation and the Navier–Stokes equations with Navier boundary conditions, where we rigorously establish the existence of symmetry-breaking, pitchfork, saddle-node, and Hopf bifurcations. In particular, we prove nonuniqueness of stationary solutions and the existence of time-periodic solutions arising from Hopf bifurcation . The analysis relies on a parameterization of solution branches near critical points, reducing the problem to a finite-dimensional bifurcation equation that can be rigorously validated. These results highlight the effectiveness of computer-assisted methods for the rigorous study of bifurcations in infinite-dimensional dynamical systems.
I will discuss the existence of periodic solutions bifurcating from manifolds of periodic solutions for central force problems in classical and relativistic mechanics, in two- and three-dimensional Euclidean space. The use of suitable coordinate systems is a key step in verifying the required nondegeneracy condition. Joint works with Walter Dambrosio and Guglielmo Feltrin.
We investigate a nonlinear eigenvalue problem driven by the sum of the p- and q-Laplacians, set in the exterior of a bounded domain in a Euclidean space satisfying zero Dirichlet boundary condition at the compact boundary with decay to zero at infinity. The combination of competing nonlinearities and the lack of compactness inherent to exterior domains pose additional difficulties compared to the standard p-Laplacian in the bounded domain setting. We prove the existence of an unbounded family of principal eigenvalues and associated eigenfunctions. In addition, we establish qualitative properties of these eigenfunctions, including positivity and regularity. Further, we discuss asymptotic behaviors of the functional and of the bifurcation diagram as the eigenvalue parameter varies. Our approach relies on variational methods and the fibering method introduced by Pohozaev. This lecture is based on a recent work with P. Drabek and R. Shivaji.
We consider a fractional Lane-Emden problem, showing, via an asymptotic analysis of low energy solutions, that in Gidas-Ni-Nirenberg domains slightly sub-critical problems have a unique non-degenerate solution. The talk is based on a joint work with A. Saldaña and I. Ianni.
TBA
We will describe how bifurcation ideas, ranging from fairly sophisticated to very elementary, can be used to address interesting questions in spectral geometry. The talk is based on joint works with Antonio J. Fernández, Josef Greilhuber, and David Ruiz.
The presence of 0 in the essential spectrum of the linearized operator, and the resulting lack of a spectral gap, prevent the direct application of classical bifurcation theory to a broad class of infinite-dimensional problems. We present a general approach designed to overcome this difficulty. As a significant application, we study flow-induced oscillations in fluid–structure interaction systems and establish the existence of time-periodic motions bifurcating from steady-state configurations.
In this talk I will present some recent studies on the liquid drop model, introduced by George Gamow (1930) and Niels Bohr–John Archibald Wheeler (1939), which describes atomic nuclei through an energy that balances surface tension and nonlocal repulsion under a volume constraint. While spheres minimize the energy for small volumes, finding non-minimizing critical points at larger volumes is more challenging. We construct new large-volume solutions resembling “pearl necklaces” arranged on a circle, close to Delaunay’s unduloids. We also construct small-mass solutions shaped like two nearly equal spheres connected by a thin neck. In addition, we discuss bifurcation results that provide new solutions close to spheres near specific values of the radius. Joint work with Manuel del Pino, Andrés Zúñiga, Rupert Frank, Fabio de Regibus, and Massimo Grossi.
One of the most general approaches to the description of bifurcation diagrams for one-parameter families of scalar ordinary differential equations is based on the study of changes in the number and stability properties of critical points. The persistence of these objects under small perturbations is one of the key reasons why this analysis is possible. A natural extension of this perspective to the nonautonomous setting consists in studying the number and type of separated hyperbolic solutions, together with their possible loss of stability or disappearance as the parameter varies. When applied to third-degree polynomial ordinary differential equations, this approach leads to nontrivial extensions of the classical bifurcation diagrams. Our methods combine techniques from topological dynamics and ergodic theory with classical tools from the theory of ordinary differential equations, mainly comparison results. The analysis of these nonautonomous bifurcations is fundamental for understanding the occurrence of critical transitions in certain models arising in the applied sciences. This is joint work with Jesús Dueñas, Cinzia Elia, Roberta Fabbri, and Rafael Obaya.
In this talk we are interested in compactly supported solutions of the steady Euler equations. In 3D the existence of such solutions has been an open problem until the recent result of Gavrilov (2019). In 2D, instead, it is easy to construct solutions via radially symmetric stream functions. Low regularity solutions without radial symmetry have been found in the literature, but even the C^1 case was open. In this talk we construct this kind of solutions with C^k regularity, for any integer k given. For the proof, we look for stream functions which are solutions to a certain type of non-autonomous semilinear elliptic equations. In this framework we look for a local bifurcation around a 1-parameter family of solutions. The linearized operator turns out to be critically singular, and is defined in anisotropic Banach spaces. This is joint work with A. Enciso (ICMAT, Madrid) and Antonio J. Fernández (UAM, Madrid).
We consider bifurcation of critical points from a trivial branch for families of functionals that are invariant under the orthogonal action of a compact Lie group. Based on a previous construction of an equivariant spectral flow from a joint work with Izydorek and Janczewska, we obtain a bifurcation theorem that generalises well-known results of Smoller and Wasserman as well as Fitzpatrick, Pejsachowicz and Recht. Finally, we discuss elementary applications to strongly indefinite systems of differential equations.
For abstract semilinear equations in a Hilbert space H with a variational structure, I will present a new approach to detect prescribed norm solutions in H which does not rely on any mass-subcriticality assumptions. The solutions we obtain are detected as ground states of Nehari-Pankov type for the associated λ-dependent action functional, where λ varies in a spectral gap between sufficiently large eigenvalues of the associated linearized operator at zero. The key new observation in this abstract framework is the fact that the H-norms of these λ-dependent solution families form connected sets even though the solution families themselves may be disconnected. We shall discuss applications of this approach to semilinear Schrödinger equations on compact metric graphs and to related semilinear elliptic boundary value problems. In some situations where only the existence of solutions with small mass was known, we are able to produce multiple solutions with arbitrarily large mass by combining our abstract approach with Weyl type estimates for the length of spectral gaps, variational characterizations of eigenvalues, bounds for associated eigenfunctions and an elementary bound from analytic number theory. This is joint work with Damien Galant (Brown University).
The Couette-Taylor problem describes the motion of a viscous incompressible fluid confined between two concentric cylinders. In this talk, we will recall some classical bifurcation aspects of the problem. We will then present a new classification result for steady Navier-Stokes flows. Within a class of partially invariant solutions, we explicitly determine all admissible flows under Dirichlet boundary conditions and under boundary conditions involving the vorticity. We also prove rigidity results for small boundary data. Particular emphasis will be placed on the analytical asymmetry between the inner and outer cylinders when vorticity boundary conditions are prescribed.
We consider a partially damped and forced nonlinear nonlocal coupled system of PDEs governing the dynamics of suspension bridges. The deck is modelled as a fish-bone plate — a degenerate rectangular plate composed of a central beam moving vertically, and a continuum of cross-sections rotating around the horizontal plane. A beam-type equation dictates the vertical displacement of the beam, while a wave-type equation governs the torsional oscillations of the cross-sections. After establishing a rigorous functional framework and introducing a suitable notion of weak solution, we address the long-term behavior of the system, focusing in particular on the existence, uniqueness, and stability of periodic solutions. For purely vertical motions, we show that a unique and globally stable periodic solution exists if the (periodic) external force is sufficiently small compared to the damping. Beyond this threshold, the dynamical system may instead undergo "bifurcations" leading to multiple periodic solutions and richer dynamics, characterized by a nontrivial attractor. In contrast, we prove that for the full beam-wave system, multiple periodic solutions arise regardless of the magnitude of the forcing term. The talk is based on joint work with Maurizio Garrione and Filippo Gazzola.
We investigate the Sobolev inequality and the associated Neumann problem for the critical semilinear elliptic equation in cones. We construct a one-parameter family of domains on the sphere whose first nonzero Neumann eigenvalue of the Laplace Beltrami operator crosses the threshold at which the standard bubble loses stability. Under the assumption that this eigenvalue is simple, we prove, via the Crandall Rabinowitz bifurcation theorem, the existence of a branch of nonradial solutions bifurcating from the standard bubble. Moreover, we show that the bifurcation is global. As it is well-known, the standard bubble are known to be the only positive solutions in convex cones. Our results shows that, for nonconvex cones, symmetry breaking may occur, and it is related to the spectral properties of the Laplace Beltrami operator on the domain on the unit sphere, which spans the cone. These results are contained in a joint work with F. Pacella and L. Provenzano.