Newtonian repulsion and radial confinement: Convergence toward steady state

We investigate the large time behavior of multi-dimensional aggregation equations driven by Newtonian repulsion, and balanced by radial attraction and confinement. In case of Newton repulsion with radial confinement we quantify the algebraic convergence decay rate toward the unique steady state. To this end, we identify a one-parameter family of radial steady states, and prove dimension-dependent decay rate in energy and 2-Wassertein distance, using a comparison with properly selected radial steady states. We also study Newtonian repulsion and radial attraction. When the attraction potential is quadratic it is known to coincide with quadratic confinement. Here, we study the case of perturbed radial quadratic attraction, proving that it still leads to one-parameter family of unique steady states. It is expected that this family to serve for a corresponding comparison argument which yields algebraic convergence toward steady repulsive-attractive solutions.

The convergence rate of the fast signal diffusion limit for a Keller–Segel–Stokes system with large initial data

In this paper, we investigate the fast signal diffusion limit of solutions of the fully parabolic Keller–Segel–Stokes system to solution of the parabolic–elliptic-fluid counterpart in a two-dimensional or three-dimensional bounded domain with smooth boundary. Under the natural volume-filling assumption, we establish an algebraic convergence rate of the fast signal diffusion limit for general large initial data by developing a series of subtle bootstrap arguments for combinational functionals and using some maximal regularities. In our current setting, in particular, we can remove the restriction to asserting convergence only along some subsequence in Wang–Winkler and the second author (Cal. Var., 2019).

On the Threshold Condition for Dörfler Marking

It is an open question if the threshold condition \theta<\theta_{\star} for the Dörfler marking parameter is necessary to obtain optimal algebraic rates of adaptive finite element methods. We present a (non-PDE) example fitting into the common abstract convergence framework (axioms of adaptivity) which allows for convergence with exponential rates. However, for Dörfler marking \theta>\theta_{\star}, the algebraic convergence rate can be made arbitrarily small.

The left, the right and the sequential topology on Boolean algebras

For the algebraic convergence ?s, which generates the well known sequential topology ?s on a complete Boolean algebra B, we have ?s = ?ls ? ?li, where the convergences ?ls and ?li are defined by ?ls(x) = {lim sup x}? and ?li(x) = {lim inf x+}? (generalizing the convergence of sequences on the Alexandrov cube and its dual). We consider the minimal topology Olsi extending the (unique) sequential topologies O?s (left) and O?li (right) generated by the convergences ?ls and ?li and establish a general hierarchy between all these topologies and the corresponding a priori and a posteriori convergences. In addition, we observe some special classes of algebras and, in particular, show that in (?,2)-distributive algebras we have limOlsi = lim?s = ?s, while the equality Olsi = ?s holds in all Maharam algebras. On the other hand, in some collapsing algebras we have a maximal (possible) diversity.

Time-asymptotic convergence rates towards the discrete evolutionary stable distribution

This paper is concerned with the discrete dynamics of an integro-differential model that describes the evolution of a population structured with respect to a continuous trait. Various time-asymptotic convergence rates towards the discrete evolutionary stable distribution (ESD) are established. For some special ESD satisfying a strict sign condition, the exponential convergence rates are obtained for both semi-discrete and fully discrete schemes. Towards the general ESD, the algebraic convergence rate that we find is consistent with the known result for the continuous model.

Jørgensen’s Inequality and Algebraic Convergence Theorem in Quaternionic Hyperbolic Isometry Groups

We obtain an analogue of Jørgensen's inequality in quaternionic hyperbolic space. As an application, we prove that if ther-generator quaternionic Kleinian group satisfies I-condition, then its algebraic limit is also a quaternionic Kleinian group. Our results are generalizations of the counterparts in then-dimensional real hyperbolic space.