Discrete & Continuous Dynamical Systems
November 2021 , Volume 41 , Issue 11
Select all articles
We present a model of crowd motion in regions with moving obstacles, which is based on the notion of measure sweeping process. The obstacle is modeled by a set-valued map, whose values are complements to
We study circle maps with a flat interval where the critical exponents at the two boundary points of the flat spot might be different. The space of such systems is partitioned in two connected parts whose common boundary only depends on the critical exponents. At this boundary there is a phase transition in the geometry of the system. Differently from the previous approaches, this is achieved by studying the asymptotical behavior of the renormalization operator.
In this paper we consider two classes of resonant Hamiltonian PDEs on the circle with non-convex (respect to actions) first order resonant Hamiltonian. We show that, for appropriate choices of the nonlinearities we can find time-independent linear potentials that enable the construction of solutions that undergo a prescribed growth in the Sobolev norms. The solutions that we provide follow closely the orbits of a nonlinear resonant model, which is a good approximation of the full equation. The non-convexity of the resonant Hamiltonian allows the existence of fast diffusion channels along which the orbits of the resonant model experience a large drift in the actions in the optimal time. This phenomenon induces a transfer of energy among the Fourier modes of the solutions, which in turn is responsible for the growth of higher order Sobolev norms.
On a closed
as the momentum parameter
In this paper, we obtain new Carleman estimates for a class of variable coefficient degenerate elliptic operators whose constant coefficient model at one point is the so called Baouendi-Grushin operator. This generalizes the results obtained by the two of us with Garofalo in [
We study global existence and asymptotic behavior of the solutions for a chemotaxis system with chemoattractant and repellent in three dimensions. To accomplish this, we use the Fourier transform and energy method. We consider the case when the mass is conserved and we use the Lotka-Volterra type model for chemoattractant and repellent. Also, we establish
Classical KAM theory guarantees the existence of a positive measure set of invariant tori for sufficiently smooth non-degenerate near-integrable systems. When seen as a function of the frequency this invariant collection of tori is called the KAM curve of the system. Restricted to analytic regularity, we obtain strong uniqueness properties for these objects. In particular, we prove that KAM curves completely characterize the underlying systems. We also show some of the dynamical implications on systems whose KAM curves share certain common features.
In the framework of nonlinear stability of elliptic equilibria in Hamiltonian systems with
We consider the following nonlocal critical equation
The qualitative behavior of a dynamical system can be encoded in a graph. Each node of the graph is an equivalence class of chain-recurrent points and there is an edge from node
We propose and study a nonlocal Euler system with relaxation, which tends to a strictly hyperbolic system under the hyperbolic scaling limit. An independent proof of the local existence and uniqueness of this system is presented in any spatial dimension. We further derive a precise critical threshold for this system in one dimensional setting. Our result reveals that such nonlocal system admits global smooth solutions for a large class of initial data. Thus, the nonlocal velocity regularizes the generic finite-time breakdown in the pressureless Euler system.
We give a proof by foliation that the cones over
We study the stochastic stability in the zero-noise limit from a quantitative point of view.
We consider smooth expanding maps of the circle perturbed by additive noise. We show that in this case the zero-noise limit has a quadratic speed of convergence, as suggested by numerical experiments and heuristics published by Lin, in 2005 (see [
We also consider the zero-noise limit from a quantitative point of view for piecewise expanding maps showing estimates for the speed of convergence in this case.
In this paper, we study the existence and multiplicity problems for orthogonal Finsler geodesic chords in a manifold with boundary which is homeomorphic to a
In this paper we study a
Inspired by the generalized Christoffel problem, we consider a class of prescribed shifted Gauss curvature equations for horo-convex hypersurfaces in
We consider the Cauchy problem for the inhomogeneous nonlinear Schrödinger equation
This note treats several problems for the fractional perimeter or
We investigate some asymptotic properties of trigonometric skew-product maps over irrational rotations of the circle. The limits are controlled using renormalization. The maps considered here arise in connection with the self-dual Hofstadter Hamiltonian at energy zero. They are analogous to the almost Mathieu maps, but the factors commute. This allows us to construct periodic orbits under renormalization, for every quadratic irrational, and to prove that the map-pairs arising from the Hofstadter model are attracted to these periodic orbits. We believe that analogous results hold for the self-dual almost Mathieu maps, but they seem presently beyond reach.
As a consequence we prove the existence and uniqueness of solutions of Cauchy problem for the fast diffusion equation
We also prove the asymptotic large time behaviour of such singular solution
Add your name and e-mail address to receive news of forthcoming issues of this journal:
[Back to Top]