58 research outputs found
Invariant manifolds and the long-time asymptotics of the Navier-Stokes and vorticity equations on R^2
We construct finite-dimensional invariant manifolds in the phase space of the
Navier-Stokes equation on R^2 and show that these manifolds control the
long-time behavior of the solutions. This gives geometric insight into the
existing results on the asymptotics of such solutions and also allows one to
extend those results in a number of ways.Comment: 46 pages, 3 figure
Spectral asymptotics for large skew-symmetric perturbations of the harmonic oscillator
Originally motivated by a stability problem in Fluid Mechanics, we study the
spectral and pseudospectral properties of the differential operator on , where is a
real-valued function and a small parameter. We define
as the infimum of the real part of the spectrum of
, and as the supremum of the norm of the
resolvent of along the imaginary axis. Under appropriate
conditions on , we show that both quantities ,
go to infinity as , and we give precise
estimates of the growth rate of . We also provide an example
where is much larger than if is
small. Our main results are established using variational "hypocoercive"
methods, localization techniques and semiclassical subelliptic estimates.Comment: 38 pages, 4 figure
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
Existence and Stability of Propagating Fronts for an Autocatalytic Reaction-Diffusion System
We study a one-dimensional reaction-diffusion system which describes an
isothermal autocatalytic chemical reaction involving both a quadratic (A + B ->
2B) and a cubic (A + 2B -> 3B) autocatalysis. The parameters of this system are
the ratio D = D_B/D_A of the diffusion constants of the reactant A and the
autocatalyst B, and the relative activity k of the cubic reaction. First, for
all values of D > 0 and k >= 0, we prove the existence of a family of
propagating fronts (or travelling waves) describing the advance of the
reaction. In particular, in the quadratic case k=0, we recover the results of
Billingham and Needham [BN]. Then, if D is close to 1 and k is sufficiently
small, we prove using energy functionals that these propagating fronts are
stable against small perturbations in exponentially weighted Sobolev spaces.
This extends to our system part of the stability results which are known for
the scalar Fisher equation.Comment: 32 pages, 1 Postscript figur
Orbital stability of periodic waves for the nonlinear Schroedinger equation
The nonlinear Schroedinger equation has several families of quasi-periodic
travelling waves, each of which can be parametrized up to symmetries by two
real numbers: the period of the modulus of the wave profile, and the variation
of its phase over a period (Floquet exponent). In the defocusing case, we show
that these travelling waves are orbitally stable within the class of solutions
having the same period and the same Floquet exponent. This generalizes a
previous work where only small amplitude solutions were considered. A similar
result is obtained in the focusing case, under a non-degeneracy condition which
can be checked numerically. The proof relies on the general approach to orbital
stability as developed by Grillakis, Shatah, and Strauss, and requires a
detailed analysis of the Hamiltonian system satisfied by the wave profile.Comment: 34 pages, 7 figure
Phase Slips and the Eckhaus Instability
We consider the Ginzburg-Landau equation, , with complex amplitude . We first analyze the phenomenon of
phase slips as a consequence of the {\it local} shape of . We next prove a
{\it global} theorem about evolution from an Eckhaus unstable state, all the
way to the limiting stable finite state, for periodic perturbations of Eckhaus
unstable periodic initial data. Equipped with these results, we proceed to
prove the corresponding phenomena for the fourth order Swift-Hohenberg
equation, of which the Ginzburg-Landau equation is the amplitude approximation.
This sheds light on how one should deal with local and global aspects of phase
slips for this and many other similar systems.Comment: 22 pages, Postscript, A
- …