583 research outputs found
The role of BKM-type theorems in Euler, Navier-Stokes and Cahn-Hilliard-Navier-Stokes analysis
The Beale-Kato-Majda theorem contains a single criterion that controls the
behaviour of solutions of the incompressible Euler equations. Versions of
this theorem are discussed in terms of the regularity issues surrounding the
incompressible Euler and Navier-Stokes equations together with a
phase-field model for the statistical mechanics of binary mixtures called the
Cahn-Hilliard-Navier-Stokes (CHNS) equations. A theorem of BKM-type is
established for the CHNS equations for the full parameter range. Moreover, for
this latter set, it is shown that there exists a Reynolds number and a bound on
the energy-dissipation rate that, remarkably, reproduces the upper
bound on the inverse Kolmogorov length normally associated with the
Navier-Stokes equations alone. An alternative length-scale is introduced and
discussed, together with a set of pseudo-spectral computations on a
grid.Comment: 3 figures and 3 table
Mathematics for 2d Interfaces
We present here a survey of recent results concerning the mathematical
analysis of instabilities of the interface between two incompressible, non
viscous, fluids of constant density and vorticity concentrated on the
interface. This configuration includes the so-called Kelvin-Helmholtz (the two
densities are equal), Rayleigh-Taylor (two different, nonzero, densities) and
the water waves (one of the densities is zero) problems. After a brief review
of results concerning strong and weak solutions of the Euler equation, we
derive interface equations (such as the Birkhoff-Rott equation) that describe
the motion of the interface. A linear analysis allows us to exhibit the main
features of these equations (such as ellipticity properties); the consequences
for the full, non linear, equations are then described. In particular, the
solutions of the Kelvin-Helmholtz and Rayleigh-Taylor problems are necessarily
analytic if they are above a certain threshold of regularity (a consequence is
the illposedness of the initial value problem in a non analytic framework). We
also say a few words on the phenomena that may occur below this regularity
threshold. Finally, special attention is given to the water waves problem,
which is much more stable than the Kelvin-Helmholtz and Rayleigh-Taylor
configurations. Most of the results presented here are in 2d (the interface has
dimension one), but we give a brief description of similarities and differences
in the 3d case.Comment: Survey. To appear in Panorama et Synth\`ese
Exact two-dimensionalization of rapidly rotating large-Reynolds-number flows
We consider the flow of a Newtonian fluid in a three-dimensional domain,
rotating about a vertical axis and driven by a vertically invariant horizontal
body-force. This system admits vertically invariant solutions that satisfy the
2D Navier-Stokes equation. At high Reynolds number and without global rotation,
such solutions are usually unstable to three-dimensional perturbations. By
contrast, for strong enough global rotation, we prove rigorously that the 2D
(and possibly turbulent) solutions are stable to vertically dependent
perturbations: the flow becomes 2D in the long-time limit.
These results shed some light on several fundamental questions of rotating
turbulence: for arbitrary Reynolds number and small enough Rossby number, the
system is attracted towards purely 2D flow solutions, which display no energy
dissipation anomaly and no cyclone-anticyclone asymmetry. Finally, these
results challenge the applicability of wave turbulence theory to describe
stationary rotating turbulence in bounded domains.Comment: To be published in Journal of Fluid Mechanic
Finite-time blowup for a Navier-Stokes model equation for the self-amplification of strain
In this paper we consider a model equation for the Navier-Stokes strain
equation. This model equation has the same identity for enstrophy growth and a
number of the same regularity criteria as the full Navier-Stokes strain
equation, and is also an evolution equation on the same constraint space. We
prove finite-time blowup for this model equation, which shows that the identity
for enstrophy growth and the strain constraint space are not sufficient on
their own to guarantee global regularity. The mechanism for the finite-time
blowup of this model equation is the self-amplification of strain, which is
consistent with recent research suggesting that strain self-amplification, not
vortex stretching, is the main mechanism behind the turbulent energy cascade.
Because the strain self-amplification model equation is obtained by dropping
certain terms from the full Navier-Stokes strain equation, we will also prove a
conditional blowup result for the full Navier-Stokes equation involving a
perturbative condition on the terms neglected in the model equation
Mathematical Aspects of Hydrodynamics
The workshop dealt with the partial differential equations that describe fluid motion, namely the Euler equations and the Navier-Stokes equations. This included topics in both inviscid and viscous fluids in two and three dimensions. A number of the talks were connected with issues of turbulence. Some talks addressed aspects of fluid dynamics such as magnetohydrodynamics, quantum and high energy physics, liquid crystals and the particle limit governed by the Boltzmann equations
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