A priori, there is nothing very special about shear-free or asymptotically shear-free null geodesic congruences. Surprisingly, however, they turn out to possess a large number of fascinating geometric properties and to be closely related, in the context of general relativity, to a variety of physically significant effects. It is the purpose of this paper to try to fully develop these issues. Th...

As part of their great burst of activity in the late 1970s, Cheng and Yau proved many geometric results concerning differential structures invariant under affine transformations of R n. Affine differential geometry is the study of those differential properties of hypersurfaces in R n+1 which are invariant under volume-preserving affine transformations. One way to develop this theory is to start...

We prove the reduction principle and study other attractivity properties of the center and center-unstable manifolds in the vicinity of a steady-state solution for quasilinear systems of parabolic partial differential equations with fully nonlinear boundary conditions on bounded or exterior domains.

Well-accepted hypothesis in the fluid dynamics is that if the boundary of the physical domain is impermeable then the viscous fluid adheres completely to it. Many authors recently proposed mathematical justifications for this hypothesis using the so-called rugous boundary. In this Paper we want to discuss optimality of results obtained in Bucur et al. [3], Bucur and Feireisl [4] or Dı́az et al. ...

Our goal is to investigate the asymptotic behavior of center mass elephant random walk, which a discrete-time walk on integers with complete memory its whole history. In diffusive and critical regimes, we establish almost sure convergence, law iterated logarithm quadratic strong for walk. The normality, properly normalized, also provided. Finally, prove limit theorem in superdiffusive regime. A...

We prove a definitive theorem on the asymptotic stability of point vortex solutions to full Euler equation in 2 dimensions. More precisely, we show that small, Gevrey smooth, and compactly supported perturbation leads global solution 2D, which converges weakly as $t\to\infty$ radial profile with respect vortex. The position vortex, is time dependent, stabilizes rapidly becomes center final, pro...

The interaction of two near-marginal tearing modes in the presence of shear flow is studied. To find the time asymptotic states, the resistive magnetohydrodynamic (MHD) equations are reduced to four amplitude equations, using center manifold reduction. These amplitude equations are subject to the constraints due to the symmetries of the physical problem. For the case without flow, the model tha...