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Non-autonomous 2D Navier–Stokes system with a simple global attractor and some averaging problems

V. V. Chepyzhov, M. I. Vishik (2010)

ESAIM: Control, Optimisation and Calculus of Variations

We study the global attractor of the non-autonomous 2D Navier–Stokes system with time-dependent external force g(x,t). We assume that g(x,t) is a translation compact function and the corresponding Grashof number is small. Then the global attractor has a simple structure: it is the closure of all the values of the unique bounded complete trajectory of the Navier–Stokes system. In particular, if g(x,t) is a quasiperiodic function with respect to t, then the attractor is a continuous image...

Nonlinear compressible vortex sheets in two space dimensions

Jean-François Coulombel, Paolo Secchi (2008)

Annales scientifiques de l'École Normale Supérieure

We consider supersonic compressible vortex sheets for the isentropic Euler equations of gas dynamics in two space dimensions. The problem is a free boundary nonlinear hyperbolic problem with two main difficulties: the free boundary is characteristic, and the so-called Lopatinskii condition holds only in a weak sense, which yields losses of derivatives. Nevertheless, we prove the local existence of such piecewise smooth solutions to the Euler equations. Since the a priori estimates for the linearized...

Nonlinear elliptic problems with incomplete Dirichlet conditions and the stream function solution of subsonic rotational flows past profiles or cascades of profiles

Miloslav Feistauer (1989)

Aplikace matematiky

The paper is devoted to the solvability of a nonlinear elliptic problem in a plane multiply connected domain. On the inner components of its boundary Dirichlet conditions are known up to additive constants which have to be determined together with the sought solution so that the so-called trailing stagnation conditions are satisfied. The results have applications in the stream function solution of subsonic flows past groups of profiles or cascades of profiles.

Nonlinear filtering for observations on a random vector field along a random path. Application to atmospheric turbulent velocities

Christophe Baehr (2010)

ESAIM: Mathematical Modelling and Numerical Analysis

To filter perturbed local measurements on a random medium, a dynamic model jointly with an observation transfer equation are needed. Some media given by PDE could have a local probabilistic representation by a Lagrangian stochastic process with mean-field interactions. In this case, we define the acquisition process of locally homogeneous medium along a random path by a Lagrangian Markov process conditioned to be in a domain following the path and conditioned to the observations. The nonlinear...

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