348 research outputs found
Turbulence properties and global regularity of a modified Navier-Stokes equation
We introduce a modification of the Navier-Stokes equation that has the
remarkable property of possessing an infinite number of conserved quantities in
the inviscid limit. This new equation is studied numerically and turbulence
properties are analyzed concerning energy spectra and scaling of structure
functions. The dissipative structures arising in this new equation are curled
vortex sheets contrary to vortex tubes arising in Navier-Stokes turbulence. The
numerically calculated scaling of structure functions is compared with a
phenomenological model based on the She-L\'ev\^eque approach. Finally, for this
equation we demonstrate global well-posedness for sufficiently smooth initial
conditions in the periodic case and in . The key feature is the
availability of an additional estimate which shows that the -norm of the
velocity field remains finite
On Recent Progress for the Stochastic Navier Stokes Equations
We give an overview of the ideas central to some recent developments in the
ergodic theory of the stochastically forced Navier Stokes equations and other
dissipative stochastic partial differential equations. Since our desire is to
make the core ideas clear, we will mostly work with a specific example: the
stochastically forced Navier Stokes equations. To further clarify ideas, we
will also examine in detail a toy problem. A few general theorems are given.
Spatial regularity, ergodicity, exponential mixing, coupling for a SPDE, and
hypoellipticity are all discussed.Comment: Corrected version of Journees Equations aux derivees partielles
paper(June 2003). Original at
http://www.math.sciences.univ-nantes.fr/edpa/2003
Stability and nonlinear adjustment of vortices in Keplerian flows
We investigate the stability, nonlinear development and equilibrium structure
of vortices in a background shearing Keplerian flow. We make use of
high-resolution global two-dimensional compressible hydrodynamic simulations.
We introduce the concept of nonlinear adjustment to describe the transition of
unbalanced vortical fields to a long-lived configuration. We discuss the
conditions under which vortical perturbations evolve into long-lived persistent
structures and we describe the properties of these equilibrium vortices. The
properties of equilibrium vortices appear to be independent from the initial
conditions and depend only on the local disk parameters. In particular we find
that the ratio of the vortex size to the local disk scale height increases with
the decrease of the sound speed, reaching values well above the unity. The
process of spiral density wave generation by the vortex, discussed in our
previous work, appear to maintain its efficiency also at nonlinear amplitudes
and we observe the formation of spiral shocks attached to the vortex. The
shocks may have important consequences on the long term vortex evolution and
possibly on the global disk dynamics. Our study strengthens the arguments in
favor of anticyclonic vortices as the candidates for the promotion of planetary
formation. Hydrodynamic shocks that are an intrinsic property of persistent
vortices in compressible Keplerian flows are an important contributor to the
overall balance. These shocks support vortices against viscous dissipation by
generating local potential vorticity and should be responsible for the eventual
fate of the persistent anticyclonic vortices. Numerical codes have be able to
resolve shock waves to describe the vortex dynamics correctly.Comment: 12 pages, 10 figure
Stress relaxation models with polyconvex entropy in Lagrangean and Eulerian coordinates
The embedding of the equations of polyconvex elastodynamics to an augmented
symmetric hyperbolic system provides in conjunction with the relative entropy method
a robust stability framework for approximate solutions \cite{LT06}.
We devise here a model of stress relaxation motivated by the
format of the enlargement process which formally approximates
the equations of polyconvex elastodynamics. The model is endowed with
an entropy function which is not convex but rather of polyconvex type.
Using the relative entropy we prove a stability estimate and convergence
of the stress relaxation model to polyconvex elastodynamics in the
smooth regime. As an application, we show that models of pressure relaxation for
real gases in Eulerian coordinates fit into the proposed framework
Vanishing diffusion limit and boundary layers for a nonlinear hyperbolic system with damping and diffusion
We consider an initial and boundary value problem for a nonlinear hyperbolic system with damping and diffusion. This system was derived from the Rayleigh–Benard equation. Based on a new observation of the structure of the system, two identities are found; then, the following results are proved by using the energy method. First, the well-posedness of the global large solution is established; then, the limit with a boundary layer as some diffusion coefficient tending to zero is justified. In addition, the convergence rate in terms of the diffusion coefficient is obtained together with the estimation of the thickness of the boundary layer
Transient times, resonances and drifts of attractors in dissipative rotational dynamics
In a dissipative system the time to reach an attractor is often influenced by
the peculiarities of the model and in particular by the strength of the
dissipation. In particular, as a dissipative model we consider the spin-orbit
problem providing the dynamics of a triaxial satellite orbiting around a
central planet and affected by tidal torques. The model is ruled by the
oblateness parameter of the satellite, the orbital eccentricity, the
dissipative parameter and the drift term. We devise a method which provides a
reliable indication on the transient time which is needed to reach an attractor
in the spin-orbit model; the method is based on an analytical result, precisely
a suitable normal form construction. This method provides also information
about the frequency of motion. A variant of such normal form used to
parametrize invariant attractors provides a specific formula for the drift
parameter, which in turn yields a constraint - which might be of interest in
astronomical problems - between the oblateness of the satellite and its orbital
eccentricity.Comment: 21 pages, 7 figures, colo
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