Let $X$ be a compact subset of a separable Hilbert space $H$ with finite fractal dimension $d_F\left(X\right)$, and $P_0$ an orthogonal projection in $H$ of rank greater than or equal to $2d_F\left(X\right)+1$. For every $\delta >0$, there exists an orthogonal projection $P$ in $H$ of the same rank as $P_0$, which is injective when restricted to $X$ and such that $\parallel P-P_0\parallel <\delta$. This result follows from Mañé’s paper. Thus the inverse ${\left(P{|}_X\right)}^{-1}$ of the restricted mapping ${P|}_{X}X\to PX$ is well defined. It is natural to ask whether there exists a universal modulus of continuity for the inverse of Mañé’s...

We study a model of motion of a passive tracer particle in a turbulent flow that is strongly mixing in time variable. In [8] we have shown that there exists a probability measure equivalent to the underlying physical probability under which the quasi-Lagrangian velocity process, i.e. the velocity of the flow observed from the vintage point of the moving particle, is stationary and ergodic. As a consequence, we proved the existence of the mean of the quasi-Lagrangian velocity, the so-called Stokes...

The aim of this work is to establish, from a mathematical point of view, the limit α → +∞ in the system $i{\partial }_{t}E+\nabla (\nabla .E)-{\alpha }^2\nabla \times \nabla \times E=-{\left|E\right|}^{2\sigma }E,$ where $E:{ℝ}^3\to {ℂ}^3$. This corresponds to an approximation which is made in the context of Langmuir turbulence in plasma Physics. The L2-subcritical σ (that is σ ≤ 2/3) and the H1-subcritical σ (that is σ ≤ 2) are studied. In the physical case σ = 1, the limit is then studied for the $H^1\left({ℝ}^3\right)$ norm.

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE’s, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite...

The paper deals with the numerical resolution of the convection-diffusion system which arises when modeling combustion for turbulent flow. The considered model is of compressible turbulent reacting type where the turbulence-chemistry interactions are governed by additional balance equations. The system of PDE's, that governs such a model, turns out to be in non-conservation form and usual numerical approaches grossly fail in the capture of viscous shock layers. Put in other words, classical finite...

We consider a free boundary problem of a two-dimensional Navier-Stokes shear flow. There exist a unique global in time solution of the considered problem as well as the global attractor for the associated semigroup. As in [1] and [2], we estimate from above the dimension of the attractor in terms of given data and the geometry of the domain of the flow. This research is motivated by a free boundary problem from lubrication theory where the domain of the flow is usually very thin and the roughness...