Nature of complex singularities for the 2D Euler equation

A detailed study of complex-space singularities of the two-dimensional incompressible Euler equation is performed in the short-time asymptotic r\'egime when such singularities are very far from the real domain; this allows an exact recursive determination of arbitrarily many spatial Fourier coefficients. Using high-precision arithmetic we find that the Fourier coefficients of the stream function are given over more than two decades of wavenumbers by \hat F(\k) = C(\theta) k^{-\alpha} \ue ^ {-k \delta(\theta)}, where \k = k(\cos \theta, \sin \theta). The prefactor exponent $\alpha$, typically between 5/2 and 8/3, is determined with an accuracy better than 0.01. It depends on the initial condition but not on $\theta$. The vorticity diverges as $s^{-\beta}$, where $\alpha+\beta= 7/2$ and $s$ is the distance to the (complex) singular manifold. This new type of non-universal singularity is permitted by the strong reduction of nonlinearity (depletion) which is associated to incompressibility. Spectral calculations show that the scaling reported above persists well beyond the time of validity of the short-time asymptotics. A simple model in which the vorticity is treated as a passive scalar is shown analytically to have universal singularities with exponent $\alpha =5/2$.Comment: 22 pages, 24 figures, published version; a version of the paper with higher-quality figures is available at http://www.obs-nice.fr/etc7/euler.pd

A Kato type Theorem for the inviscid limit of the Navier-Stokes equations with a moving rigid body

The issue of the inviscid limit for the incompressible Navier-Stokes equations when a no-slip condition is prescribed on the boundary is a famous open problem. A result by Tosio Kato says that convergence to the Euler equations holds true in the energy space if and only if the energy dissipation rate of the viscous flow in a boundary layer of width proportional to the viscosity vanishes. Of course, if one considers the motion of a solid body in an incompressible fluid, with a no-slip condition at the interface, the issue of the inviscid limit is as least as difficult. However it is not clear if the additional difficulties linked to the body's dynamic make this issue more difficult or not. In this paper we consider the motion of a rigid body in an incompressible fluid occupying the complementary set in the space and we prove that a Kato type condition implies the convergence of the fluid velocity and of the body velocity as well, what seems to indicate that an answer in the case of a fixed boundary could also bring an answer to the case where there is a moving body in the fluid

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

Existence and uniqueness for stochastic 2D Euler flows with bounded vorticity

The strong existence and the pathwise uniqueness of solutions with (Formula presented.) -vorticity of the 2D stochastic Euler equations are proved. The noise is multiplicative and it involves the first derivatives. A Lagrangian approach is implemented, where a stochastic flow solving a nonlinear flow equation is constructed. The stability under regularizations is also proved