arXiv subjects

Jean-Yves Chemin

Publications and source records attributed to Jean-Yves Chemin.

At least 19 recordsLinked to original sources

arXiv2024

Ekman boundary layers in a domain with topography

We investigate the asymptotic behaviour of fast rotating incompressible fluids with vanishing viscosity, in a {three dimensional} domain with topography including the case of land area. Assuming the initial data is well-prepared, we prove a convergence theorem of the velocity fields to a two-dimensional vector field solving a linear, damped ordinary differential equation.The proof is based on a weak-strong uniqueness argument, combinedwith an abstract result implying that the weak convergence of a familyof weak solutions to the Navier-Stokes-Coriolis system can be translated into a form of uniform-in-time convergence.This argument yields strong convergence of the velocity fields, without a precise rate though.

math.AP↗

arXiv2020

On the radius of analyticity of solutions to semi-linear parabolic systems

We study the radius of analyticity~$R(t)$ in space, of strong solutions to systems of scale-invariant semi-linear parabolic equations. It is well-known that near the initial time,~$R(t)t^{-\frac12}$ is bounded from below by a positive constant. In this paper we prove that~$\displaystyle\liminf_{t\rightarrow 0} R(t)t^{-\frac12}= \infty$, and assuming higher regularity for the initial data, we obtain an improved lower bound near time zero. As an application, we prove that for any global solution~$u\in C([0,\infty); H^{\frac12}(\R^3))$ of the Navier-Stokes equations, there holds~$\displaystyle\liminf_{t\rightarrow \infty} R(t)t^{-\frac12}= \infty$.

math.AP↗

arXiv2018

Some remarks about the possible blow-up for the Navier-Stokes equations

In this work we investigate the question of preventing the three-dimensional, incompressible Navier-Stokes equations from developing singularities, by controlling one component of the velocity field only, in space-time scale invariant norms. In particular we prove that it is not possible for one component of the velocity field to tend to~$0$ too fast near blow up. We also introduce a space "almost" invariant under the action of the scaling such that if one component of the velocity field measured in this space remains small enough, then there is no blow up.

math.AP↗

arXiv2018

A non linear estimate on the life span of solutions of the three dimensional Navier-Stokes equations

The purpose of this article is to establish bounds from below for the life span of regular solutions to the incompressible Navier-Stokes system, whichinvolve norms not only of the initial data, but also of nonlinear functions of the initial data. We provide examples showing that those bounds are significant improvements to the one provided by the classical fixed point argument. One of the important ingredients is the use of a scale-invariant energy estimate.

math.AP↗

arXiv2017

Tempered distributions and Fourier transform on the Heisenberg group

The final goal of the present work is to extend the Fourier transform on the Heisenberg group $\H^d,$ to tempered distributions. As in the Euclidean setting, the strategy is to first show that the Fourier transform is an isomorphism on the Schwartz space, then to define the extension by duality. The difficulty that is here encountered is that the Fourier transform of an integrable function on $\H^d$is no longer a function on $\H^d$ : according to the standard definition, it is a family of bounded operators on $L^2(\R^d).$ Following our new approach in\ccite{bcdFHspace}, we here define the Fourier transform of an integrable functionto be a mapping on the set~$\wt\H^d=\N^d\times\N^d\times\R\setminus\{0\}$endowed with a suitable distance $\wh d$.This viewpoint turns out to provide a user friendly description of the range of the Schwartz space on $\H^d$ by the Fourier transform, which makes the extension to the whole set of tempered distributions straightforward. As a first application, we give an explicit formula for the Fourier transform of smooth functions on $\H^d$ that are independent of the vertical variable. We also provide other examples.

math.FA↗

arXiv2016

A Frequency Space for the Heisenberg Group

We here revisit Fourier analysis on the Heisenberg group H^d. Whereas, according to the standard definition, the Fourier transform of an integrable function f on H^d is a one parameter family of bounded operators on L 2 (R^d), we define (by taking advantage of basic properties of Hermite functions) the Fourier transform f\_H of f to be a uniformly continuous mapping on the set N^d x N^d xR \ {0} endowed with a suitable distance. This enables us to extend f\_H to the completion of that space, and to get an explicit asymptotic description of the Fourier transform when the 'vertical' frequency tends to 0. We expect our approach to be relevant for adapting to the Heisenberg framework a number of classical results for the Euclidean case that are based on Fourier analysis. As an example, we here establish an explicit extension of the Fourier transform for smooth functions on H^d that are independent of the vertical variable.

math.CA↗

arXiv2015

On the critical one component regularity for 3-D Navier-Stokes system: general case

Let us consider an initial data $v_0$ for the homogeneous incompressible 3D Navier-Stokes equation with vorticity belonging to $L^{\frac 32}\cap L^2$. We prove that if the solution associated with $v_0$ blows up at a finite time $T^\star$, then for any $p$ in $]4,\infty[$, and any unit vector $e$ of $\R^3$, the $L^p$ norm in time with value in $\dot{H}^{\frac 12+\frac 2 p }$ of $(v|e)_{\R^3}$ blows up at $T^\star$

math.AP↗

arXiv2015

Inhomogeneous incompressible viscous flows with slowly varying initial data

The purpose of this paper is to provide a large class of initial data which generates global smooth solution of the 3-D inhomogeneous incompressible Navier-Stokes system in the whole space~$\R^3$. This class of data is based on functions which vary slowly in one direction. The idea is that 2-D inhomogeneous Navier-Stokes system with large data is globally well-posedness and we construct the 3-D approximate solutions by the 2-D solutions with a parameter. One of the key point of this study is the investigation of the time decay properties of the solutions to the 2-D inhomogeneous Navier-Stokes system. We obtained the same optimal decay estimates as the solutions of 2-D homogeneous Navier-Stokes system.

math.AP↗

arXiv2015

Local existence for the non-resistive MHD equations in Besov spaces

In this paper we prove the existence of solutions to the viscous, non-resistive magnetohydrodynamics (MHD) equations on the whole of $\mathbb{R}^{n}$, $n=2,3$, for divergence-free initial data in certain Besov spaces, namely $\boldsymbol{u}_{0} \in B^{n/2-1}_{2,1}$ and $\boldsymbol{B}_{0} \in B^{n/2}_{2,1}$. The a priori estimates include the term $\int_{0}^{t} \| \boldsymbol{u}(s) \|_{H^{n/2}}^{2} \, \mathrm{d} s$ on the right-hand side, which thus requires an auxiliary bound in $H^{n/2-1}$. In 2D, this is simply achieved using the standard energy inequality; but in 3D an auxiliary estimate in $H^{1/2}$ is required, which we prove using the splitting method of Calderón (Trans. Amer. Math. Soc. 318(1), 179--200, 1990). By contrast, we prove that such solutions are unique in 3D, but the proof of uniqueness in 2D is more difficult and remains open.

math.AP↗

arXiv2014

Remarks on the global solutions of 3-D Navier-Stokes system with one slow variable

By applying Wiegner' method in \cite{Wiegner}, we first prove the large time decay estimate for the global solutions of a 2.5 dimensional Navier-Stokes system, which is a sort of singular perturbed 2-D Navier-Stokes system in three space dimension. As an application of this decay estimate, we give a simplified proof for the global wellposedness result in \cite{cg3} for 3-D Navier-Stokes system with one slow variable. Let us also mention that compared with the assumptions for the initial data in \cite{cg3}, here the assumptions in Theorem \ref{slowvarsimplifie} are weaker.

math.AP↗

arXiv2013

On the critical one component regularity for 3-D Navier-Stokes system

Given an initial data $v_0$ with vorticity $\Om_0=\na\times v_0$ in $L^{\frac 3 2},$ (which implies that $v_0$ belongs to the Sobolev space $H^{\frac12}$), we prove that the solution $v$ given by the classical Fujita-Kato theorem blows up in a finite time $T^\star$ only if, for any $p$ in $ ]4,6[$ and any unit vector $e$ in $\R^3,$ there holds $ \int_0^{T^\star}\|v(t)\cdot e\|_{\dH^{\f12+\f2p}}^p\,dt=\infty.$ We remark that all these quantities are scaling invariant under the scaling transformation of Navier-Stokes system.

math.AP↗

arXiv2013

Stability by rescaled weak convergence for the Navier-Stokes equations

We prove a weak stability result for the three-dimensional homogeneous incompressible Navier-Stokes system. More precisely, we investigate the following problem : if a sequence $(u_{0, n})_{n\in \N}$ of initial data, bounded in some scaling invariant space, converges weakly to an initial data $u_0$ which generates a global regular solution, does $u_{0, n}$ generate a global regular solution ? A positive answer in general to this question would imply global regularity for any data, through the following examples $u_{0,n} = n \vf_0(n\cdot)$ or $u_{0,n} = \vf_0(\cdot-x_n)$ with $|x_n|\to \infty$. We therefore introduce a new concept of weak convergence (rescaled weak convergence) under which we are able to give a positive answer. The proof relies on profile decompositions in anisotropic spaces and their propagation by the Navier-Stokes equations.

math.AP↗

arXiv2013

Global large solutions to 3-D inhomogeneous Navier-Stokes system with one slow variable

In this paper, we are concerned with the global wellposedness of 3-D inhomogeneous incompressible Navier-Stokes equations \eqref{1.3} in the critical Besov spaces with the norm of which are invariant by the scaling of the equations and under a nonlinear smallness condition on the isentropic critical Besov norm to the fluctuation of the initial density and the critical anisotropic Besov norm of the horizontal components of the initial velocity which have to be exponentially small compared with the critical anisotropic Besov norm to the third component of the initial velocity. The novelty of this results is that the isentropic space structure to the homogeneity of the initial density function is consistent with the propagation of anisotropic regularity for the velocity field. In the second part, we apply the same idea to prove the global wellposedness of \eqref{1.3} with some large data which are slowly varying in one direction.

math.AP↗

arXiv2013

Self-improving bounds for the Navier-Stokes equations

We consider regular solutions to the Navier-Stokes equation and provide an extension to the Escauriaza-Seregin-Sverak blow-up criterion in the negative regularity Besov scale, with regularity arbitrarly close to -1. Our results rely on turning a priori bounds for the solution in negative Besov spaces into bounds in the positive regularity scale.

math.AP↗

arXiv2012

The role of spectral anisotropy in the resolution of the three-dimensional Navier-Stokes equations

We present different classes of initial data to the three-dimensional, incompressible Navier-Stokes equations, which generate a global in time, unique solution though they may be arbitrarily large in the end-point function space in which a fixed-point argument may be used to solve the equation locally in time. The main feature of these initial data is an anisotropic distribution of their frequencies. One of those classes is taken from previous papers by two of the authors and collaborators, and another one is new.

math.AP↗

arXiv2010

Sums of large global solutions to the incompressible Navier-Stokes equations

Let G be the (open) set of~$\dot H^{\frac 1 2}$ divergence free vector fields generating a global smooth solution to the three dimensional incompressible Navier-Stokes equations. We prove that any element of G can be perturbed by an arbitrarily large, smooth divergence free vector field which varies slowly in one direction, and the resulting vector field (which remains arbitrarily large) is an element of G if the variation is slow enough. This result implies that through any point in G passes an uncountable number of arbitrarily long segments included in G.

math.AP↗

arXiv2008

Global regularity for some classes of large solutions to the Navier-Stokes equations

In three previous papers by the two first authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The main feature of the initial data considered in the last paper is that it varies slowly in one direction, though in some sense it is ``well prepared'' (its norm is large but does not depend on the slow parameter). The aim of this article is to generalize the setting of that last paper to an ``ill prepared'' situation (the norm blows up as the small parameter goes to zero).The proof uses the special structure of the nonlinear term of the equation.

math.AP↗

arXiv2007

Large, global solutions to the Navier-Stokes equations, slowly varying in one direction

In to previous papers by the authors, classes of initial data to the three dimensional, incompressible Navier-Stokes equations were presented, generating a global smooth solution although the norm of the initial data may be chosen arbitrarily large. The aim of this article is to provide new examples of arbitrarily large initial data giving rise to global solutions, in the whole space. Contrary to the previous examples, the initial data has no particular oscillatory properties, but varies slowly in one direction. The proof uses the special structure of the nonlinear term of the equation.

math.AP↗