相关论文: Sturm 3-ball global attractors 1: Thom-Smale compl…
This is the second of three papers on the geometric and combinatorial characterization of global Sturm attractors which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$ on the…
Examples complete our trilogy on the geometric and combinatorial characterization of global Sturm attractors $\mathcal{A}$ which consist of a single closed 3-ball. The underlying scalar PDE is parabolic, $$ u_t = u_{xx} + f(x,u,u_x)\,, $$…
We embark on a detailed analysis of the close relations between combinatorial and geometric aspects of the scalar parabolic PDE \begin{equation}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) \tag{$*$} \end{equation} on the unit interval $0 < x<1$…
We systematically explore a simple class of global attractors, called Sturm due to nodal properties, for the semilinear scalar parabolic PDE \begin{equation*}\label{eq:*} u_t = u_{xx} + f(x,u,u_x) %\tag{$*$} \end{equation*} on the unit…
We study global attractors $\mathcal{A}=\mathcal{A}_f$ of semiflows generated by semilinear partial parabolic differential equations of the form $u_t = u_{xx} + f(x,u,u_x), 0<x<1$, satisfying Neumann boundary conditions. The equilibria…
This sequel continues our exploration arxiv:2302.12531 of a deceptively ``simple'' class of global attractors, called Sturm due to nodal properties. They arise for the semilinear scalar parabolic PDE \begin{equation}\label{eq:*} u_t =…
We survey the global dynamics of semiflows generated by scalar semilinear parabolic equations which are $\mathbb{SO}(2)$ equivariant under spatial shifts of $x\in \mathbb{S}^1=\mathbb{R}/2\pi\mathbb{Z}$, i.e. $$ u_t = u_{xx} +…
The Birman-Williams theorem gives a connection between the collection of unstable periodic orbits (UPOs) contained within a chaotic attractor and the topology of that attractor, for three-dimensional systems. In certain cases, the fractal…
The goal of this paper is to construct explicitly the global attractors of parabolic equations with singular diffusion coefficients on the boundary, as it was done without the singular term for the semilinear case by Brunovsk'y and Fiedler…
We explicitly construct global attractors of fully nonlinear parabolic equations. The attractors are decomposed as equilibria (time independent solutions) and heteroclinic orbits (solutions that converge to distinct equilibria backwards and…
About a decade ago Thurston proved that a vast collection of 3-manifolds carry metrics of constant negative curvature. These manifolds are thus elements of {\em hyperbolic geometry}, as natural as Euclid's regular polyhedra. For a closed…
In this work we derive a lower bounds for the Hausdorff and fractal dimensions of the global attractor of the Sabra shell model of turbulence in different regimes of parameters. We show that for a particular choice of the forcing and for…
We show here the global, in time, regularity of the three dimensional viscous Camassa-Holm (Lagrangian Averaged Navier-Stokes-alpha) equations. We also provide estimates, in terms of the physical parameters of the equations, for the…
The goal of this paper is to construct explicitly the global attractors of quasilinear parabolic equations, as it was done for the semilinear case by Brunovsk\'y and Fiedler (1986), and generalized by Fiedler and Rocha (1996). In…
The Thurston norm is a seminorm on the second real homology group of a compact orientable 3-manifold. The unit ball of this norm is a convex polyhedron, whose shape's data (e.g. number of vertices, regularity) measures the complexity of the…
We discuss a theoretical framework to define an optimal sub-grid closure for shell models of turbulence. The closure is based on the ansatz that consecutive shell multipliers are short-range correlated, following the third hypothesis of…
We develop a three-dimensional dynamical-systems framework for passive gliding and identify the global phase-space structures that organize its motion. Extending previous two-dimensional models of non-equilibrium gliding, we show that the…
We study the asymptotic behavior of solutions of one coupled PDE-ODE system arising in mathematical biology as a model for the development of a forest ecosystem. In the case where the ODE-component of the system is monotone, we establish…
We present a new three-parameter family of self-consistent equilibrium models for quasi-relaxed stellar systems that are subject to the combined action of external tides and rigid internal rotation. These models provide an idealised…
The purpose of this paper is to give a characterization of the structure of non-autonomous attractors of the problem $u_t= u_{xx} + \lambda u - \beta(t)u^3$ when the parameter $\lambda > 0$ varies. Also, we answer a question proposed in…