51 research outputs found
Controllability of 2D Euler and Navier-Stokes equations by degenerate forcing
We study controllability issues for the 2D Euler and Navier-
Stokes (NS) systems under periodic boundary conditions. These systems
describe motion of homogeneous ideal or viscous incompressible fluid on
a two-dimensional torus T^2. We assume the system to be controlled by
a degenerate forcing applied to fixed number of modes.
In our previous work [3, 5, 4] we studied global controllability by
means of degenerate forcing for Navier-Stokes (NS) systems with nonvanishing
viscosity (\nu > 0). Methods of dfferential geometric/Lie algebraic
control theory have been used for that study. In [3] criteria for
global controllability of nite-dimensional Galerkin approximations of
2D and 3D NS systems have been established. It is almost immediate
to see that these criteria are also valid for the Galerkin approximations
of the Euler systems. In [5, 4] we established a much more intricate suf-
cient criteria for global controllability in finite-dimensional observed
component and for L2-approximate controllability for 2D NS system.
The justication of these criteria was based on a Lyapunov-Schmidt
reduction to a finite-dimensional system. Possibility of such a reduction
rested upon the dissipativity of NS system, and hence the previous
approach can not be adapted for Euler system.
In the present contribution we improve and extend the controllability
results in several aspects: 1) we obtain a stronger sufficient condition for
controllability of 2D NS system in an observed component and for L2-
approximate controllability; 2) we prove that these criteria are valid for
the case of ideal incompressible uid (\nu = 0); 3) we study solid controllability
in projection on any finite-dimensional subspace and establish a
sufficient criterion for such controllability
Enhanced four-wave mixing via elimination of inhomogeneous broadening by coherent driving of quantum transition with control fields
We show that atoms from wide velocity interval can be concurrently involved
in Doppler-free two-photon resonant far from frequency degenerate four-wave
mixing with the aid of auxiliary electromagnetic field. This gives rise to
substantial enhancement of the output radiation generated in optically thick
medium. Numerical illustrations addressed to typical experimental conditions
are given.Comment: LaTeX2e, hyperref, 7 pages, 5 figures, to appear in PRA 1 august 200
Some qualitative properties of the solutions of the Magnetohydrodynamic equations for nonlinear bipolar fluids
In this article we study the long-time behaviour of a system of nonlinear
Partial Differential Equations (PDEs) modelling the motion of incompressible,
isothermal and conducting modified bipolar fluids in presence of magnetic
field. We mainly prove the existence of a global attractor denoted by \A for
the nonlinear semigroup associated to the aforementioned systems of nonlinear
PDEs. We also show that this nonlinear semigroup is uniformly differentiable on
\A. This fact enables us to go further and prove that the attractor \A is
of finite-dimensional and we give an explicit bounds for its Hausdorff and
fractal dimensions.Comment: The final publication is available at Springer via
http://dx.doi.org/10.1007/s10440-014-9964-
The finite dimension behavior of the Schrödinger equation with nonlocal integral nonlinearity
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