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The behavior of an ordinary differential equation for the low wave number velocity
mode is analyzed. This equation was derived in [5]
by an iterative process on the two-dimensional Navier-Stokes equations (NSE). It
resembles the NSE in form, except
that the kinematic viscosity is replaced by an iterated viscosity
which is a partial sum, dependent on the low-mode velocity. The convergence of
this sum as the number of iterations is taken to be arbitrarily large is explored.
This leads to a limiting...
The paper analyzes the issue of existence of solutions to linear problems in two dimensional exterior domains, linearizations of the Navier-Stokes equations. The systems are studied with a slip boundary condition. The main results prove the existence of distributional solutions for arbitrary data.
This paper is divided into two parts and focuses on the linear independence of boundary traces of eigenfunctions of boundary value problems. Part I deals with second-order elliptic operators, and Part II with Stokes (and Oseen) operators.
Part I: Let be an eigenvalue of a second-order elliptic operator defined on an open, sufficiently smooth, bounded domain Ω in ℝⁿ, with Neumann homogeneous boundary conditions on Γ = tial Ω. Let be the corresponding linearly independent (normalized) eigenfunctions...
The free oscillations of the gate system proposed [1,2] to defend the Venice Lagoon from the phenomenon of high water are analyzed. Free transverse modes of oscillations exist which may be either subharmonic or synchronous with respect to typical waves in the Adriatic sea. This result points out the need to examine whether such modes may be excited as a result of a Mathieu type resonance occurring when the gate system is forced by incident waves. The latter investigation is performed in part 2 of...
We show that the transverse subharmonic modes characterizing the free oscillations of the gate system proposed to defend the Venice Lagoon from the phenomenon of high water (see Note I[1]) can be excited when the gate system is forced by plane monochromatic waves orthogonal to the gates with the typical characteristics of large amplitude waves in the Adriatic sea close to the lagoon inlets. A linear stability analysis of the coupled motion of the system sea-gates-lagoon reveals that for typical...
For robust discretizations of the Navier-Stokes equations with small viscosity, standard Galerkin schemes have to be augmented by stabilization terms due to the indefinite convective terms and due to a possible lost of a discrete inf-sup condition. For optimal control problems for fluids such stabilization have in general an undesired effect in the sense that optimization and discretization do not commute. This is the case for the combination of streamline upwind Petrov-Galerkin (SUPG) and pressure...
We consider a 1-D tank containing an inviscid incompressible irrotational fluid. The tank is subject to the control which consists of horizontal moves. We assume that the motion of the fluid is well-described by the Saint–Venant equations (also called the shallow water equations). We prove the local controllability of this nonlinear control system around any steady state. As a corollary we get that one can move from any steady state to any other steady state.
We consider a 1-D tank containing an inviscid incompressible
irrotational fluid. The tank is subject to the control which consists
of horizontal moves. We assume that the motion of the fluid
is well-described by the Saint–Venant equations (also
called the
shallow water equations).
We prove the local
controllability of this nonlinear control
system around any steady state.
As a corollary we get that one can move from any steady state to any
other steady state.
In this talk, I will present a recent result obtained in [6] with O. Glass, S. Guerrero and J.-P. Puel on the local exact controllability of the -d compressible Navier-Stokes equations. The goal of these notes is to give an informal presentation of this article and we refer the reader to it for extensive details.
In this paper we deal with the local exact controllability of the
Navier-Stokes system with nonlinear Navier-slip boundary
conditions and distributed controls supported in small sets. In a
first step, we prove a Carleman inequality for the linearized
Navier-Stokes system, which leads to null controllability of this
system at any time T>0. Then, fixed point arguments lead to the
deduction of a local result concerning the exact controllability
to the trajectories of the Navier-Stokes system.
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