53 research outputs found
Invariant Gibbs measures of the energy for shell models of turbulence; the inviscid and viscous cases
Gaussian measures of Gibbsian type are associated with some shell models of
3D turbulence; they are constructed by means of the energy, a conserved
quantity for the 3D inviscid and unforced shell model. We prove the existence
of a unique global flow for a stochastic viscous shell model and a global flow
for the deterministic inviscid shell model, with the property that these Gibbs
measures are invariant for these flows
Large deviation principle and inviscid shell models
A LDP is proved for the inviscid shell model of turbulence. As the viscosity
coefficient converges to 0 and the noise intensity is multiplied by the square
root of the viscosity, we prove that some shell models of turbulence with a
multiplicative stochastic perturbation driven by a H-valued Brownian motion
satisfy a LDP in C([0,T],V) for the topology of uniform convergence on [0,T],
but where V is endowed with a topology weaker than the natural one. The initial
condition has to belong to V and the proof is based on the weak convergence of
a family of stochastic control equations. The rate function is described in
terms of the solution to the inviscid equation
Existence and uniqueness of global solutions for the modified anisotropic 3D Navier-Stokes equations
We study a modified three-dimensional incompressible anisotropic
Navier-Stokes equations. The modification consists in the addition of a power
term to the nonlinear convective one. This modification appears naturally in
porous media when a fluid obeys the Darcy-Forchheimer law instead of the
classical Darcy law. We prove global in time existence and uniqueness of
solutions without assuming the smallness condition on the initial data. This
improves the result obtained for the classical 3D incompressible anisotropic
Navier-Stokes equations.Comment: To appear in ESAIM: Mathematical Modelling and Numerical Analysi
Mean field limit of interacting filaments and vector valued non linear PDEs
Families of interacting curves are considered, with long range, mean
field type, interaction. A family of curves defines a 1-current, concentrated
on the curves, analog of the empirical measure of interacting point particles.
This current is proved to converge, as goes to infinity, to a mean field
current, solution of a nonlinear, vector valued, partial differential equation.
In the limit, each curve interacts with the mean field current and two
different curves have an independence property if they are independent at time
zero. This set-up is inspired from vortex filaments in turbulent fluids,
although for technical reasons we have to restrict to smooth interaction,
instead of the singular Biot-Savart kernel. All these results are based on a
careful analysis of a nonlinear flow equation for 1-currents, its relation with
the vector valued PDE and the continuous dependence on the initial conditions
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