43 research outputs found
Euler equations and turbulence: analytical approach to intermittency
Physical models of intermittency in fully developed turbulence employ many
phenomenological concepts such as active volume, region, eddy, energy
accumulation set, etc, used to describe non-uniformity of the energy cascade.
In this paper we give those notions a precise mathematical meaning in the
language of the Littlewood-Paley analysis. We further use our definitions to
recover scaling laws for the energy spectrum and second order structure
function with proper intermittency correction.Comment: This is an updated version. Published in SIMA, 201
A study of energy concentration and drain in incompressible fluids
In this paper we examine two opposite scenarios of energy behavior for
solutions of the Euler equation. We show that if is a regular solution on a
time interval and if for some , where is the dimension of the fluid, then the energy at the
time cannot concentrate on a set of Hausdorff dimension samller than . The same holds for solutions of the three-dimensional
Navier-Stokes equation in the range . Oppositely, if the energy
vanishes on a subregion of a fluid domain, it must vanish faster than
(T-t)^{1-\d}, for any \d>0. The results are applied to find new exclusions
of locally self-similar blow-up in cases not covered previously in the
literature.Comment: an update of the previous versio
Linear instability criteria for ideal fluid flows subject to two subclasses of perturbations
In this paper we examine the linear stability of equilibrium solutions to
incompressible Euler's equation in 2- and 3-dimensions. The space of
perturbations is split into two classes - those that preserve the topology of
vortex lines and those in the corresponding factor space. This classification
of perturbations arises naturally from the geometric structure of
hydrodynamics; our first class of perturbations is the tangent space to the
co-adjoint orbit. Instability criteria for equilibrium solutions are
established in the form of lower bounds for the essential spectral radius of
the linear evolution operator restricted to each class of perturbation.Comment: 29 page
A geometric condition implying energy equality for solutions of 3D Navier-Stokes equation
We prove that every weak solution to the 3D Navier-Stokes equation that
belongs to the class and \n u belongs to localy
away from a 1/2-H\"{o}lder continuous curve in time satisfies the generalized
energy equality. In particular every such solution is suitable.Comment: 10 page
On the ill/well-posedness and nonlinear instability of the magneto-geostrophic equations
We consider an active scalar equation that is motivated by a model for
magneto-geostrophic dynamics and the geodynamo. We prove that the non-diffusive
equation is ill-posed in the sense of Hadamard in Sobolev spaces. In contrast,
the critically diffusive equation is well-posed. In this case we give an
example of a steady state that is nonlinearly unstable, and hence produces a
dynamo effect in the sense of an exponentially growing magnetic field.Comment: We have modified the definition of Lipschitz well-posedness, in order
to allow for a possible loss in regularity of the solution ma
On formation of a locally self-similar collapse in the incompressible Euler equations
The paper addresses the question of existence of a locally self-similar
blow-up for the incompressible Euler equations. Several exclusion results are
proved based on the -condition for velocity or vorticity and for a range
of scaling exponents. In particular, in dimensions if in self-similar
variables and u \sim \frac{1}{t^{\a/(1+\a)}}, then the blow-up
does not occur provided \a >N/2 or -1<\a\leq N/p. This includes the
case natural for the Navier-Stokes equations. For \a = N/2 we exclude
profiles with an asymptotic power bounds of the form |y|^{-N-1+\d} \lesssim
|u(y)| \lesssim |y|^{1-\d}. Homogeneous near infinity solutions are eliminated
as well except when homogeneity is scaling invariant.Comment: A revised version with improved notation, proofs, etc. 19 page