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The concern of this paper is the Cauchy problem for the Prandtl equation. This problem is known to be well-posed for analytic data [13, 10], or for data with monotonicity properties [11, 15]. We prove here that it is linearly ill-posed in Sobolev type spaces. The key of the analysis is the construction, at high tangential frequencies, of unstable quasimodes… (More)

We consider the homogenization of elliptic systems with ε-periodic coefficients. Classical two-scale approximation yields a O(ε) error inside the domain. We discuss here the existence of higher order corrections, in the case of general polygonal domains. The corrector depends in a non-trivial way on the boundary. Our analysis extends substantially previous… (More)

This paper deals with the homogenization of elliptic systems with Dirichlet boundary condition, when the coefficients of both the system and the boundary data are ε-periodic. We show that, as ε → 0, the solutions converge in L2 with a power rate in ε, and identify the homogenized limit system. Due to a boundary layer phenomenon, this homogenized system… (More)

- David Gérard-Varet, Toàn Nguyên
- Asymptotic Analysis
- 2012

In the lines of the recent paper [5], we establish various ill-posedness results for the Prandtl equation. By considering perturbations of stationary shear flows, we show that for some linearizations of the Prandtl equation and some C∞ initial data, local in time C∞ solutions do not exist. At the nonlinear level, we prove that if a flow exists in the… (More)

We consider the Navier-Stokes equation in a domain with irregular boundaries. The irregularity is modeled by a spatially homogeneous random process, with typical size ε≪ 1. In the parent paper [8], we derived a homogenized boundary condition of Navier type as ε→ 0. We show here that for a large class of boundaries, this Navier condition provides a O(ε3/2|… (More)

We investigate the evolution of rigid bodies in a viscous incompressible fluid. The flow is governed by the 2D Navier-Stokes equations, set in a bounded domain with Dirichlet boundary conditions. The boundaries of the solids and the domain have Hölder regularity C, 0 < α ≤ 1. First, we show the existence and uniqueness of strong solutions up to collision. A… (More)

3 Boundary layer sizes and equations 17 3.1 Definitions and hypothesis (H4) . . . . . . . . . . . . . . . . . . . . . . . . . . 17 3.2 Solvability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18 3.2.1 Order zero operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 3.2.2 Operators with constant coefficients . .… (More)

We consider the homogenization of the Navier-Stokes equation, set in a channel with a rough boundary, of small amplitude and wavelength ǫ. It was shown recently that, for any non-degenerate roughness pattern, and for any reasonable condition imposed at the rough boundary, the homogenized boundary condition in the limit ε = 0 is always no-slip. We give in… (More)

- David Gérard-Varet
- SIAM J. Math. Analysis
- 2005

<lb>We consider the free fall of a sphere above a wall in a viscous incompressible fluid. We<lb>investigate the influence of boundary conditions on the finite-time occurrence of contact<lb>between the sphere and the wall. We prove that slip boundary conditions enable to<lb>circumvent the ”no-collision” paradox associated with no-slip boundary conditions.… (More)