arXiv subjects

Christophe Prange

Publications and source records attributed to Christophe Prange.

26 records · Page 2Linked to original sources

arXiv2019

Localized smoothing for the Navier-Stokes equations and concentration of critical norms near singularities

This paper is concerned with two dual aspects of the regularity question of the Navier-Stokes equations. First, we prove a local in time localized smoothing effect for local energy solutions. More precisely, if the initial data restricted to the unit ball belongs to the scale-critical space $L^3$, then the solution is locally smooth in space for some short time, which is quantified. This builds upon the work of Jia and Šverák, who considered the subcritical case. Second, we apply these localized smoothing estimates to prove a concentration phenomenon near a possible Type I blow-up. Namely, we show if $(0, T^*)$ is a singular point then $$\|u(\cdot,t)\|_{L^{3}(B_{R}(0))}\geq γ_{univ},\qquad R=O(\sqrt{T^*-t}).$$ This result is inspired by and improves concentration results established by Li, Ozawa, and Wang and Maekawa, Miura, and Prange. We also extend our results to other critical spaces, namely $L^{3,\infty}$ and the Besov space $\dot B^{-1+\frac3p}_{p,\infty}$, $p\in(3,\infty)$.

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arXiv2018

Infinite energy solutions to the Navier-Stokes equations in the half-space and applications

This short note serves as an introduction to the papers arXiv:1711.01651 and arXiv:1711.04486 by Maekawa, Miura and Prange. These two works deal with the existence of mild solutions on the one hand and local energy weak solutions on the other hand to the Navier-Stokes equations in the half-space $\R^3_+$. We emphasize a concentration result for (sub)critical norms near a potential singularity. The contents of these notes were presented during the X-EDP seminar at IHÉS in October 2017.

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arXiv2016

Quantitative analysis of boundary layers in periodic homogenization

We prove quantitative estimates on the rate of convergence for the oscillating Dirichlet problem in periodic homogenization of divergence-form uniformly elliptic systems. The estimates are optimal in dimensions larger than three and new in every dimension. We also prove a regularity estimate on the homogenized boundary condition.

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arXiv2015

Improved Regularity in Bumpy Lipschitz Domains

This paper is devoted to the proof of Lipschitz regularity, down to the microscopic scale, for solutions of an elliptic system with highly oscillating coefficients, over a highly oscillating Lipschitz boundary. The originality of this result is that it does not assume more than Lipschitz regularity on the boundary. Our Theorem, which is a significant improvement of our previous work on Lipschitz estimates in bumpy domains, should be read as an improved regularity result for an elliptic system over a Lipschitz boundary. Our progress in this direction is made possible by an estimate for a boundary layer corrector. We believe that this estimate in the Sobolev-Kato class is of independent interest.

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arXiv2014

Uniform Lipschitz Estimates in Bumpy Half-Spaces

This paper is devoted to the proof of uniform Hölder and Lipschitz estimates close to oscillating boundaries, for divergence form elliptic systems with periodically oscillating coefficients. Our main point is that no structure is assumed on the oscillations of the boundary. In particular, those are neither periodic, nor quasiperiodic, nor stationary ergodic. We investigate the consequences of our estimates on the large scales of Green and Poisson kernels. Our work opens the door to the use of potential theoretic methods in problems concerned with oscillating boundaries, which is an area of active research.

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arXiv2014

Well-posedness of the Stokes-Coriolis system in the half-space over a rough surface

This paper is devoted to the well-posedness of the stationary $3$d Stokes-Coriolis system set in a half-space with rough bottom and Dirichlet data which does not decrease at space infinity. Our system is a linearized version of the Ekman boundary layer system. We look for a solution of infinite energy in a space of Sobolev regularity. Following an idea of Gérard-Varet and Masmoudi, the general strategy is to reduce the problem to a bumpy channel bounded in the vertical direction thanks a transparent boundary condition involving a Dirichlet to Neumann operator. Our analysis emphasizes some strong singularities of the Stokes-Coriolis operator at low tangential frequencies. One of the main features of our work lies in the definition of a Dirichlet to Neumann operator for the Stokes-Coriolis system with data in the Kato space $H^{1/2}_{uloc}$.

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arXiv2012

Asymptotic analysis of boundary layer correctors in periodic homogenization

This paper is devoted to the asymptotic analysis of boundary layers in periodic homogenization. We investigate the behaviour of the boundary layer corrector, defined in the half-space $Ω_{n,a}:=\{y\cdot n-a>0\}$, far away from the boundary and prove the convergence towards a constant vector field, the boundary layer tail. This problem happens to depend strongly on the way the boundary $\partialΩ_{n,a}$ intersects the underlying microstructure. Our study complements the previous results obtained on the one hand for $n\in\mathbb R\mathbb Q^d$, and on the other hand for $n\notin\mathbb R\mathbb Q^d$ satisfying a small divisors assumption. We tackle the case of arbitrary $n\notin\mathbb R\mathbb Q^d$ using ergodicity of the boundary layer along $\partialΩ_{n,a}$. Moreover, we get an asymptotic expansion of Poisson's kernel $P=P(y,\tilde{y})$, associated to the elliptic operator $-\nabla\cdot A(y)\nabla\cdot$ and $Ω_{n,a}$, for $|y-\tilde{y}|\rightarrow\infty$. Finally, we show that, in general, convergence towards the boundary layer tail can be arbitrarily slow, which makes the general case very different from the rational or the small divisors one.

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arXiv2011

First-order expansion for the Dirichlet eigenvalues of an elliptic system with oscillating coefficients

This paper is concerned with the homogenization of the Dirichlet eigenvalue problem, posed in a bounded domain $Ω\subset\mathbb R^2$, for a vectorial elliptic operator $-\nabla\cdot A^ε(\cdot)\nabla$ with $ε$-periodic coefficients. We analyse the asymptotics of the eigenvalues $λ^{ε,k}$ when $ε\rightarrow 0$, the mode $k$ being fixed. A first-order asymptotic expansion is proved for $λ^{ε,k}$ in the case when $Ω$ is either a smooth uniformly convex domain, or a convex polygonal domain with sides of slopes satisfying a small divisors assumption. Our results extend those of Moskow and Vogelius restricted to scalar operators and convex polygonal domains with sides of rational slopes. We take advantage of the recent progress due to Gérard-Varet and Masmoudi in the homogenization of boundary layer type systems.

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