507 research outputs found
On the stability in weak topology of the set of global solutions to the Navier-Stokes equations
Let be a suitable function space and let \cG \subset X be the set of
divergence free vector fields generating a global, smooth solution to the
incompressible, homogeneous three dimensional Navier-Stokes equations. We prove
that a sequence of divergence free vector fields converging in the sense of
distributions to an element of \cG belongs to \cG if is large enough,
provided the convergence holds "anisotropically" in frequency space. Typically
that excludes self-similar type convergence. Anisotropy appears as an important
qualitative feature in the analysis of the Navier-Stokes equations; it is also
shown that initial data which does not belong to \cG (hence which produces a
solution blowing up in finite time) cannot have a strong anisotropy in its
frequency support.Comment: To appear in Archive for Rational and Mechanical Analysi
Stochastic representation of the Reynolds transport theorem: revisiting large-scale modeling
We explore the potential of a formulation of the Navier-Stokes equations
incorporating a random description of the small-scale velocity component. This
model, established from a version of the Reynolds transport theorem adapted to
a stochastic representation of the flow, gives rise to a large-scale
description of the flow dynamics in which emerges an anisotropic subgrid
tensor, reminiscent to the Reynolds stress tensor, together with a drift
correction due to an inhomogeneous turbulence. The corresponding subgrid model,
which depends on the small scales velocity variance, generalizes the Boussinesq
eddy viscosity assumption. However, it is not anymore obtained from an analogy
with molecular dissipation but ensues rigorously from the random modeling of
the flow. This principle allows us to propose several subgrid models defined
directly on the resolved flow component. We assess and compare numerically
those models on a standard Green-Taylor vortex flow at Reynolds 1600. The
numerical simulations, carried out with an accurate divergence-free scheme,
outperform classical large-eddies formulations and provides a simple
demonstration of the pertinence of the proposed large-scale modeling
Sums of large global solutions to the incompressible Navier-Stokes equations
Let G be the (open) set of~ divergence free vector fields
generating a global smooth solution to the three dimensional incompressible
Navier-Stokes equations. We prove that any element of G can be perturbed by an
arbitrarily large, smooth divergence free vector field which varies slowly in
one direction, and the resulting vector field (which remains arbitrarily large)
is an element of G if the variation is slow enough. This result implies that
through any point in G passes an uncountable number of arbitrarily long
segments included in G.Comment: Accepted for publication in Journal f\"ur die reine und angewandte
Mathemati
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