arXiv subjects

Marc Briane

Publications and source records attributed to Marc Briane.

At least 19 recordsLinked to original sources

arXiv2023

Fine asymptotic expansion of the ODE's flow

In this paper, we study the asymptotic expansion of the flow X(t, x) solution to the nonlinear ODE: X (t, x) = b X(t, x) with X(0, x) = x $\in$ R d , where b is a regular Z dperiodic vector field in R d. More precisely, we provide various conditions on b to obtain a "fine" asymptotic expansion of X of the type: |X(t, x) -- x -- t $\zeta$(x)| $\le$ M < $\infty$, which is uniform with respect to t $\ge$ 0 and x $\in$ R d (or at least in a subset of R d), and where $\zeta$(x) for x $\in$ R d , are the rotation vectors induced by the flow X. On the one hand, we give a necessary and sufficient condition on the vector field b so that the expansion X(t, x) -- x -- t $\zeta$(x) reads as $\Phi$ X(t, x) -- $\Phi$(x), which yields immediately the desired expansion when the vector-valued function $\Phi$ is bounded. In return, we derive an admissible class of vector fields b in terms of suitable diffeomorphisms on Y d and of vector-valued functions $\Phi$. On the other hand, assuming that the two-dimensional Kolmogorov theorem and some extension in higher dimension hold, we establish different regimes depending on the commensurability of the rotation vectors of the flow X for which the fine estimate expansion of X is valid or not. It turns out that for any two-dimensional flow X associated with a non vanishing smooth vector field b and inducing a unique incommensurable rotation vector $\xi$, the fine asymptotic expansion of X holds in R 2 if, and only if, $\xi$ 1 /$\xi$ 2 is a Diophantine number. This result seems new in the setting of the ODE's flow. The case of commensurable rotation vectors $\zeta$(x) is investigated in a similar way. Finally, several examples and counterexamples illustrate the different results of the paper, including the case of a vanishing vector field b which blows up the asymptotic expansion in some direction.

math.AP↗

arXiv2021

Specific properties of the ODE's flow in dimension two versus dimension three

This paper deals with the asymptotics of the ODE's flow induced by a regular vector field b on the d-dimensional torus R d /Z d. First, we start by revisiting the Franks-Misiurewicz theorem which claims that the Herman rotation set of any two-dimensional continuous flow is a closed line segment of R 2. Various general examples illustrate this result, among which a complete study of the Stepanoff flow associated with a vector field b = a $\zeta$, where $\zeta$ is a constant vector in R 2. Furthermore, several extensions of the

math.AP↗

arXiv2021

Asymptotics of ODE's flows everywhere or almost-everywhere in the torus:from rotation sets to homogenization of transport equations

In this paper, we study various aspects of the ODE's flow $X$ solution to the equation $\partial_t X(t,x)=b(X(t,x))$, $X(0,x)=x$ in the $d$-dimensional torus $Y_d$, where $b$ is a regular $\mathbb{Z}^d$-periodic vector field from $\mathbb{R}^d$ in $\mathbb{R}^d$.We present an original and complete picture in any dimension of all logical connections between the following seven conditions involving the field $b$: (i) the everywhere asymptotics of the flow $X$, (ii) the almost-everywhere asymptotics of the flow $X$, (iii) the global rectification of the vector field $b$ in $Y_d$, (iv) the ergodicity of the flow related to an invariant probability measure which is absolutely continuous with respect to Lebesgue's measure, (v) the unit set condition for Herman's rotation set $C_b$ composed of the means of $b$ related to the invariant probability measures, (vi) the unit set condition for the subset $D_b$ of $C_b$ composed of the means of $b$ related to the invariant probability measures which are absolutely continuous with respect to Lebesgue's measure, (vii) the homogenization of the linear transport equation with oscillating data and the oscillating velocity $b(x/\varepsilon)$ when $b$ is divergence free. The main and surprising result of the paper is that the almost-everywhere asymptotics of the flow $X$ and the unit set condition for $D_b$ are equivalent when $D_b$ is assumed to be non empty, and that the two conditions turn to be equivalent to the homogenization of the transport equation when $b$ is divergence free. In contrast, using an elementary approach based on classical tools of PDE's analysis, we extend the two-dimensional results of Oxtoby and Marchetto to any $d$-dimensional Stepanoff flow: this shows that the ergodicity of the flow may hold without satisfying the everywhere asymptotics of the flow.

math.AP↗

arXiv2020

Asymptotics of ODE's flow on the torus through a singleton condition and a perturbation result. Applications

This paper deals with the long time asymptotics X(t, x)/t of the flow X solution to the autonomous vector-valued ODE: X (t, x) = b(X(t, x)) for t $\in$ R, with X(0, x) = x a point of the torus Y d := R d /Z d. We assume that the vector field b reads as the product $\rho$ $\Phi$, where $\rho$ : Y d $\rightarrow$ [0, $\infty$) is a non negative regular function and $\Phi$ : Y d $\rightarrow$ R d is a non vanishing regular vector field. In this work, the singleton condition means that the rotation set C b composed of the average values of b with respect to the invariant probability measures for the flow X is a singleton {$\zeta$}, or equivalently, that lim t$\rightarrow$$\infty$ X(t, x)/t = $\zeta$ for any x $\in$ Y d. This combined with Liouville's theorem regarded as a divergence-curl lemma, first allows us to obtain the asymptotics of the flow X when b is a current field. Then, we prove a general perturbation result assuming that $\rho$ is the uniform limit in Y d of a positive sequence ($\rho$ n) n$\in$N satisfying for any n $\in$ N, $\rho$ $\le$ $\rho$ n and C $\rho$n$\Phi$ is a singleton {$\zeta$ n }. It turns out that the limit set C b either remains a singleton, or enlarges to the closed line set [0, lim n $\zeta$ n ] of R d. We provide various corollaries of this perturbation result involving or not the classical ergodic condition, according to the positivity or not of some harmonic means of $\rho$. These results are illustrated by different examples which show that the perturbation result is limited to the scalar perturbation of $\rho$, and which highlight the alternative satisfied by the rotation set C b. Finally, we prove that the singleton condition allows us to homogenize in any dimension the linear transport equation induced by the oscillating velocity b(x/$\epsilon$) beyond any ergodic condition satisfied by the flow X.

math.AP↗

arXiv2019

Revisiting the asymptotics of the flow for some dynamical systems on the torus

In this paper we study the large time asymptotics of the flow of a dynamical system $X'=b(X)$ posed in the $d$-dimensional torus. Rather than using the classical unique ergodicity condition which is not fulfilled if $b$ vanishes at different points, we only assume that the set of the averages of $b$ with respect to the invariant probability measures for the flow is reduced to a singleton. We also rewrite the Liouville theorem which holds for any invariant probability measure $\mu$, namely $\mu\,b$ is divergence free, as a divergence-curl formula satisfied by any regular periodic function. The combination of these two tools turns out to be a new approach to get the asymptotics for some flows. This allows us to obtain the desired asymptotics in any dimension when $b = a\,\xi$ with $a$ a possibly vanishing periodic nonnegative function and $\xi$ a nonzero vector in $R^d$, or when $b = A\nabla v$ with $A$ a periodic nonnegative symmetric matrix-valued function and $v$ a periodic function.

math.DS↗

arXiv2019

Homogenization of an elastodynamics system with a strong magnetic field and soft inclusions inducing a viscoelastic effective behavior

In this paper we study the homogenization of a linear elastodynamics system in an elastic body with soft inclusions, which is embedded in a highly oscillating magnetic field. We show two limit behaviors according to the magnetic field. On the one hand, if the magnetic field has two different directions on the interface between the hard phase and the soft phase, then the limit of the displacement in the hard phase is independent of time, so that the magnetic field induces an effective infinite mass. On the other hand, if the magnetic field has a constant direction $\xi$ on the interface, then the limit of the displacement in the hard phase and in the direction $\xi$ is solution to an elastodynamics equation with a memory mass, a memory stress tensor and memory external forces depending on the initial conditions, which read as time convolutions with some kernel. When the magnetic has the same direction $\xi$ in the soft phase with smooth inclusions, we prove that the space-average of the kernel is regular and that the limit of the overall displacement in the direction $\xi$ is solution to a viscoelasticity equation.

math.AP↗

arXiv2019

Homogenization of linear transport equations. A new approach

The paper is devoted to a new approach of the homogenization of linear transport equations induced by a uniformly bounded sequence of vector fields $b_\epsilon(x)$, the solutions of which $u_\epsilon(t,x)$ agree at $t=0$ with a bounded sequence of $L^p_{\rm loc}(\mathbb{R}^N)$ for some $p\in(1,\infty)$. Assuming that the sequence $b_\epsilon\cdot\nabla w_\epsilon^1$ is compact in $L^q_{\rm loc}(\mathbb{R}^N)$ ($q$ conjugate of $p$) for some gradient field $\nabla w_\epsilon^1$ bounded in $L^N_{\rm loc}(\mathbb{R}^N)^N$, and that there exists a uniformly bounded sequence $\sigma_\epsilon>0$ such that $\sigma_\epsilon\,b_\epsilon$ is divergence free if $N\!=\!2$ or is a cross product of $(N\!-\!1)$ bounded gradients in $L^N_{\rm loc}(\mathbb{R}^N)^N$ if $N\!\geq\!3$, we prove that the sequence $\sigma_\epsilon\,u_\epsilon$ converges weakly to a solution to a linear transport equation. It turns out that the compactness of $b_\epsilon\cdot\nabla w_\epsilon^1$ is a substitute to the ergodic assumption of the classical two-dimensional periodic case, and allows us to deal with non-periodic vector fields in any dimension. The homogenization result is illustrated by various and general examples.

math.AP↗

arXiv2019

Isotropic realizability of fields and reconstruction of invariant measures under positivity properties. Asymptotics of the flow by a non-ergodic approach

The paper is devoted to the isotropic realizability of a regular gradient field u or a more general vector field b, namely the existence of a continuous positive function $\sigma$ such that $\sigma$b is divergence free in R d or in an open set of R d. First, we prove that under some suitable positivity condition satisfied by u, the isotropic realizability of u holds either in R d if u does not vanish, or in the open sets {c j < u < c j+1 } if the c j are the critical values of u (including inf R d u and sup R d u) which are assumed to be in finite number. It turns out that this positivity condition is not sufficient to ensure the existence of a continuous positive invariant measure $\sigma$ on the torus when u is periodic. Then, we establish a new criterium of the existence of an invariant measure for the flow associated with a regular periodic vector field b, which is based on the equality b $\times$ v = 1 in R d. We show that this gradient invertibility is not related to the classical ergodic assumption, but it actually appears as an alternative to get the asymptotics of the flow.

math.AP↗

arXiv2018

Increase of mass and nonlocal effects in the homogenization of magneto-elastodynamics problems

The paper deals with the homogenization of a magneto-elastodynamics equation satisfied by the displacement $u\_\varepsilon$ of an elastic body which is subjected to an oscillating magnetic field $B\_\varepsilon$ generating the Lorentz force $\partial\_t u\_\varepsilon\times B\_\varepsilon$.When the magnetic field $B\_\varepsilon$ only depends on time or on space, the oscillations of $B\_\varepsilon$ induce an increase of mass in the homogenized equation. More generally, when the magnetic field is time-space dependent through a uniformly bounded component $G\_\varepsilon(t,x)$ of $B\_\varepsilon$, besides the increase of mass the homogenized equation involves the more intricate limit $g$ of $\partial\_t u\_\varepsilon\times G\_\varepsilon$ which turns out to be decomposed in two terms. The first term of $g$ can be regarded as a nonlocal Lorentz force the range of which is limited to a light cone at each point $(t,x)$. The cone angle is determined by the maximal velocity defined as the square root of the ratio between the elasticity tensor spectral radius and the body mass. Otherwise, the second term of $g$ is locally controlled in $L^2$-norm by the compactness default measure of the oscillating initial energy.

math.AP↗

arXiv2018

Reconstruction of isotropic conductivities from non smooth electric fields

In this paper we study the isotropic realizability of a given non smooth gradient field $\nabla u$ defined in $\mathbb{R}^d$, namely when one can reconstruct an isotropic conductivity $\sigma>0$ such that $\sigma\nabla u$ is divergence free in $\mathbb{R}^d$. On the one hand, in the case where $\nabla u$ is non-vanishing, uniformly continuous in $\mathbb{R}^d$ and $\triangle u$ is a bounded function in $\mathbb{R}^d$, we prove the isotropic realizability of $\nabla u$ using the associated gradient flow combined with the DiPerna, Lions approach for solving ordinary differential equations in suitable Sobolev spaces. On the other hand, in the case where $\nabla u$ is piecewise regular, we prove roughly speaking that the isotropic realizability holds if and only if the normal derivatives of $u$ on each side of the gradient discontinuity interfaces have the same sign. Some examples of conductivity reconstruction are given.

math.AP↗

arXiv2018

A two-dimensional labile aether through homogenization

Homogenization in linear elliptic problems usually assumes coercivity of the accompanying Dirichlet form. In linear elasticity, coercivity is not ensured through mere (strong) ellipticity so that the usual estimates that render homogenization meaningful break down unless stronger assumptions, like very strong ellipticity, are put into place. Here, we demonstrate that a L^2-type homogenization process can still be performed, very strong ellipticity notwithstanding, for a specific two-phase two dimensional problem whose significance derives from prior work establishing that one can lose strong ellipticity in such a setting, provided that homogenization turns out to be meaningful.A striking consequence is that, in an elasto-dynamic setting, some two-phase homogenized laminate may support plane wave propagation in the direction of lamination on a bounded domain with Dirichlet boundary conditions, a possibility which does not exist for the associated two-phase microstructure at a fixed scale. Also, that material blocks longitudinal waves in the direction of lamination, thereby acting as a two-dimensional aether in the sense of e.g. Cauchy.

math.AP↗

arXiv2016

Homogenization of equi-coercive nonlinear energies defined on vector-valued functions, with non-uniformly bounded coefficients

The present paper deals with the asymptotic behavior of equi-coercive sequences $\{\mathcal{F}_n\}$ of nonlinear functionals defined over vector-valued functions in $W_)^{1,p}(\Omega)^M$ , where $p>1$, $M\ge1$, and $\Omega$ is a bounded open set of $\mathbb{R}^N$, $N\ge2$. The strongly local energy density $F_n({\cdot}, Du)$ of the functional $\{\mathcal{F}_n\}$ satisfies a Lipschitz condition with respect to the second variable, which is controlled by a positive sequence $\{a_n\}$ which is only bounded in some suitable space $L^r(\Omega)$. We prove that the sequence $\{\mathcal{F}_n\}$ $\Gamma$-converges for the strong topology of $L^p(\Omega)^M$ to a functional $\mathcal{F}$ which has a strongly local density $F({\cdot}, Du)$ for sufficiently regular functions $u$. This compactness result extends former results on the topic, which are based either on maximum principle arguments in the nonlinear scalar case, or adapted div-curl lemmas in the linear case. Here, the vectorial character and the nonlinearity of the problem need a new approach based on a careful analysis of the asymptotic minimizers associated with the functional $\mathcal{F}_n$. The relevance of the conditions which are imposed to the energy density $F_n({\cdot}, Du)$, is illustrated by several examples including some classical hyper-elastic energies.

math.AP↗

arXiv2016

Homogenization of weakly coercive integral functionals in three-dimensional elasticity

This paper deals with the homogenization through $\Gamma$-convergence of weakly coercive integral energies with the oscillating density $\mathbb{L}(x/\epsilon)\nabla v : \nabla v$ in three-dimensional elasticity. The energies are weakly coercive in the sense where the classical functional coercivity satisfied by the periodic tensor L (using smooth test functions v with compact support in $\mathbb{R}^3$) which reads as $\Lambda(\mathbb{L}) >0$, is replaced by the relaxed condition $\Lambda(\mathbb{L}) \ge 0$. Surprisingly, we prove that contrary to the two-dimensional case of [2] which seems a priori more constrained, the homogenized tensor $\mathbb{L}^0$ remains strongly elliptic, or equivalently $\Lambda(\mathbb{L}^0) >0$, for any tensor $\mathbb{L} = \mathbb{L}(y_1)$ satisfying $\mathbb{L}(y)M : M + D : {\rm Cof}(M) \ge 0$, a.e. $y \in \mathbb{R}^3$, $\forall M \in \mathbb{R}^{3\times 3}$, for some matrix $D \in \mathbb{R}^{3\times3}$ (which implies $\Lambda(\mathbb{L}) \ge 0$), and the periodic functional coercivity (using smooth test functions $v$ with periodic gradients) which reads as $\Lambda_{\rm per}(\mathbb{L})>0$. Moreover, we derive the loss of strong ellipticity for the homogenized tensor using a rank-two lamination, which justifies by $\Gamma$-convergence the formal procedure of [8].

math.AP↗

arXiv2016

Towards a complete characterization of the effective elasticity tensors of mixtures of an elastic phase and an almost rigid phase

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with positive definite elasticity tensors $\BC_1$ and $\BC_2=\Gd\BC_0$ comprising the set $U=\{\BC_1,\Gd\BC_0\}$ and mixed in proportions $f$ and $1-f$ is partly characterized in the limit $\Gd\to\infty$. The material with tensor $\BC_2$ corresponds to a material which (for technical reasons) is almost rigid in the limit $\Gd\to \infty$. The paper, and the underlying microgeometries, have many aspects in common with the companion paper "On the possible effective elasticity tensors of 2-dimensional printed materials". The chief difference is that one has a different algebraic problem to solve: determining the subspaces of stress fields for which the thin walled structures can be rigid, rather than determining, as in the companion paper, the subspaces of strain fields for which the thin walled structure is compliant. Recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is almost rigid to $6-p\leq 5$ independent applied stresses, yet is compliant to any strain in the orthogonal space. Thus the walls, by themselves, can support stress with almost no deformation. The region outside the walls contains "Avellaneda material" that is a hierarchical laminate which minimizes an appropriate sum of elastic energies.

cond-mat.mtrl-sci↗

arXiv2016

On the possible effective elasticity tensors of 2-dimensional and 3-dimensional printed materials

The set $GU_f$ of possible effective elastic tensors of composites built from two materials with elasticity tensors $\BC_1>0$ and $\BC_2=0$ comprising the set $U=\{\BC_1,\BC_2\}$ and mixed in proportions $f$ and $1-f$ is partly characterized. The material with tensor $\BC_2=0$ corresponds to a material which is void. (For technical reasons $\BC_2$ is actually taken to be nonzero and we take the limit $\BC_2\to 0$). Specifically, recalling that $GU_f$ is completely characterized through minimums of sums of energies, involving a set of applied strains, and complementary energies, involving a set of applied stresses, we provide descriptions of microgeometries that in appropriate limits achieve the minimums in many cases. In these cases the calculation of the minimum is reduced to a finite dimensional minimization problem that can be done numerically. Each microgeometry consists of a union of walls in appropriate directions, where the material in the wall is an appropriate $p$-mode material, that is easily compliant to $p\leq 5$ independent applied strains, yet supports any stress in the orthogonal space. Thus the material can easily slip in certain directions along the walls. The region outside the walls contains "complementary Avellaneda material" which is a hierarchical laminate which minimizes the sum of complementary energies.

cond-mat.mtrl-sci↗

arXiv2015

Isotropic realizability of a strain field for the incompressible two-dimensional Stokes equation

In the paper we study the problem of the isotropic realizability in R^2 of a regular strain field e(U)=1/2(DU+DU^T) for the incompressible Stokes equation, namely the existence of a positive viscosity mu\textgreater{}0 solving the Stokes equation in R^2 with the prescribed field e(U). We show that if e(U) does not vanish at some point, then the isotropic realizability holds in the neighborhood of that point. The global realizability in R^2 or in the torus is much more delicate, since it involves the global existence of a regular solution to a semilinear wave equation the coefficients of which depend on the derivatives of U. Using the semilinear wave equation we prove a small perturbation result: If DU is periodic and close enough to its average for the C^4-norm, then the strain field is isotropically realizable in a given disk centered at the origin. On the other hand, a counter-example shows that the global realizability in R^2 may hold without the realizability in the torus, and it is discussed in connection with the associated semilinear wave equation. The case where the strain field vanishes is illustrated by an example. The singular case of a rank-one laminate field is also investigated.

math.AP↗

arXiv2014

Isotropic realizability of current fields in R^3

This paper deals with the isotropic realizability of a given regular divergence free field j in R^3 as a current field, namely to know when j can be written as sigma Du for some isotropic conductivity sigma, and some gradient field Du. The local isotropic realizability in R^3 is obtained by Frobenius' theorem provided that j and curl j are orthogonal in R^3. A counter-example shows that Frobenius' condition is not sufficient to derive the global isotropic realizability in R^3. However, assuming that (j, curl j, j x curl j) is an orthogonal basis of R^3, an admissible conductivity sigma is constructed from a combination of the three dynamical flows along the directions j/|j|, curl j/|curl j| and (j/|j|^2) x curl j. When the field j is periodic, the isotropic realizability in the torus needs in addition a boundedness assumption satisfied by the flow along the third direction (j/|j|^2) x \curl j. Several examples illustrate the sharpness of the realizability conditions.

math.AP↗

arXiv2013

Isotropic realizability of electric fields around critical points

In this paper we study the isotropic realizability of a given regular gradient field $\nabla u$ as an electric field, namely when $\nabla u$ is solution of the equation $÷\left(\si\nabla u\right)=0$ for some isotropic conductivity $\si>0$. The case of a function $u$ without critical point was investigated in \cite{BMT} thanks to a gradient flow approach. The presence of a critical point needs a specific treatment according to the behavior of the dynamical system around the point. The case of a saddle point is the most favorable and leads us to a characterization of the local isotropic realizability through some boundedness condition involving the laplacian of $u$ along the gradient flow. The case of a sink or a source implies a strong maximum principle under the same boundedness condition. However, when the critical point is not hyperbolic the isotropic realizability is not generally satisfied even piecewisely in the neighborhood of the point. The isotropic realizability in the torus for periodic gradient fields is also discussed in particular when the trajectories of the gradient system are bounded.

math.AP↗