168 research outputs found
Universality in Blow-Up for Nonlinear Heat Equations
We consider the classical problem of the blowing-up of solutions of the
nonlinear heat equation. We show that there exist infinitely many profiles
around the blow-up point, and for each integer , we construct a set of
codimension in the space of initial data giving rise to solutions that
blow-up according to the given profile.Comment: 38 page
Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor
In this paper we provide a sufficient condition, in terms of only one of the
nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity
vector field, for the global regularity of strong solutions to the
three-dimensional Navier-Stokes equations in the whole space, as well as for
the case of periodic boundary conditions
Three-dimensional stability of Burgers vortices
Burgers vortices are explicit stationary solutions of the Navier-Stokes
equations which are often used to describe the vortex tubes observed in
numerical simulations of three-dimensional turbulence. In this model, the
velocity field is a two-dimensional perturbation of a linear straining flow
with axial symmetry. The only free parameter is the Reynolds number , where is the total circulation of the vortex and is
the kinematic viscosity. The purpose of this paper is to show that Burgers
vortex is asymptotically stable with respect to general three-dimensional
perturbations, for all values of the Reynolds number. This definitive result
subsumes earlier studies by various authors, which were either restricted to
small Reynolds numbers or to two-dimensional perturbations. Our proof relies on
the crucial observation that the linearized operator at Burgers vortex has a
simple and very specific dependence upon the axial variable. This allows to
reduce the full linearized equations to a vectorial two-dimensional problem,
which can be treated using an extension of the techniques developped in earlier
works. Although Burgers vortices are found to be stable for all Reynolds
numbers, the proof indicates that perturbations may undergo an important
transient amplification if is large, a phenomenon that was indeed observed
in numerical simulations.Comment: 31 pages, no figur
Pearling and Pinching: Propagation of Rayleigh Instabilities
A new category of front propagation problems is proposed in which a spreading
instability evolves through a singular configuration before saturating. We
examine the nature of this front for the viscous Rayleigh instability of a
column of one fluid immersed in another, using the marginal stability criterion
to estimate the front velocity, front width, and the selected wavelength in
terms of the surface tension and viscosity contrast. Experiments are suggested
on systems that may display this phenomenon, including droplets elongated in
extensional flows, capillary bridges, liquid crystal tethers, and viscoelastic
fluids. The related problem of propagation in Rayleigh-like systems that do not
fission is also considered.Comment: Revtex, 7 pages, 4 ps figs, PR
Particle approximation of the one dimensional Keller-Segel equation, stability and rigidity of the blow-up
We investigate a particle system which is a discrete and deterministic
approximation of the one-dimensional Keller-Segel equation with a logarithmic
potential. The particle system is derived from the gradient flow of the
homogeneous free energy written in Lagrangian coordinates. We focus on the
description of the blow-up of the particle system, namely: the number of
particles involved in the first aggregate, and the limiting profile of the
rescaled system. We exhibit basins of stability for which the number of
particles is critical, and we prove a weak rigidity result concerning the
rescaled dynamics. This work is complemented with a detailed analysis of the
case where only three particles interact
Interaction of vortices in viscous planar flows
We consider the inviscid limit for the two-dimensional incompressible
Navier-Stokes equation in the particular case where the initial flow is a
finite collection of point vortices. We suppose that the initial positions and
the circulations of the vortices do not depend on the viscosity parameter \nu,
and we choose a time T > 0 such that the Helmholtz-Kirchhoff point vortex
system is well-posed on the interval [0,T]. Under these assumptions, we prove
that the solution of the Navier-Stokes equation converges, as \nu -> 0, to a
superposition of Lamb-Oseen vortices whose centers evolve according to a
viscous regularization of the point vortex system. Convergence holds uniformly
in time, in a strong topology which allows to give an accurate description of
the asymptotic profile of each individual vortex. In particular, we compute to
leading order the deformations of the vortices due to mutual interactions. This
allows to estimate the self-interactions, which play an important role in the
convergence proof.Comment: 39 pages, 1 figur
Stochastic Reaction-diffusion Equations Driven by Jump Processes
We establish the existence of weak martingale solutions to a class of second
order parabolic stochastic partial differential equations. The equations are
driven by multiplicative jump type noise, with a non-Lipschitz multiplicative
functional. The drift in the equations contains a dissipative nonlinearity of
polynomial growth.Comment: See journal reference for teh final published versio
The Inviscid Limit and Boundary Layers for Navier-Stokes Flows
The validity of the vanishing viscosity limit, that is, whether solutions of
the Navier-Stokes equations modeling viscous incompressible flows converge to
solutions of the Euler equations modeling inviscid incompressible flows as
viscosity approaches zero, is one of the most fundamental issues in
mathematical fluid mechanics. The problem is classified into two categories:
the case when the physical boundary is absent, and the case when the physical
boundary is present and the effect of the boundary layer becomes significant.
The aim of this article is to review recent progress on the mathematical
analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of
Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final
publication is available at http://www.springerlink.co
Multi-model seascape genomics identifies distinct environmental drivers of selection among sympatric marine species
Background
As global change and anthropogenic pressures continue to increase, conservation and management increasingly needs to consider speciesâ potential to adapt to novel environmental conditions. Therefore, it is imperative to characterise the main selective forces acting on ecosystems, and how these may influence the evolutionary potential of populations and species. Using a multi-model seascape genomics approach, we compare putative environmental drivers of selection in three sympatric southern African marine invertebrates with contrasting ecology and life histories: Cape urchin (Parechinus angulosus), Common shore crab (Cyclograpsus punctatus), and Granular limpet (Scutellastra granularis).
Results
Using pooled (Pool-seq), restriction-site associated DNA sequencing (RAD-seq), and seven outlier detection methods, we characterise genomic variation between populations along a strong biogeographical gradient. Of the three species, only S. granularis showed significant isolation-by-distance, and isolation-by-environment driven by sea surface temperatures (SST). In contrast, sea surface salinity (SSS) and range in air temperature correlated more strongly with genomic variation in C. punctatus and P. angulosus. Differences were also found in genomic structuring between the three species, with outlier loci contributing to two clusters in the East and West Coasts for S. granularis and P. angulosus, but not for C. punctatus.
Conclusion
The findings illustrate distinct evolutionary potential across species, suggesting that species-specific habitat requirements and responses to environmental stresses may be better predictors of evolutionary patterns than the strong environmental gradients within the region. We also found large discrepancies between outlier detection methodologies, and thus offer a novel multi-model approach to identifying the principal environmental selection forces acting on species. Overall, this work highlights how adding a comparative approach to seascape genomics (both with multiple models and species) can elucidate the intricate evolutionary responses of ecosystems to global change
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