arXiv subjects
Publications and source records attributed to Xavier Lamy.
We consider the Ginzburg-Landau heat flow on the two-dimensional flat torus, starting from an initial data with a finite number of nondegenerate zeros -- but possibly very high initial energy. We show that the initial zeros are conserved and the flow rapidly enters a logarithmic energy regime, from which the evolution of vortices can be described by the works of Bethuel, Orlandi and Smets.
We consider line-energy models of Ginzburg-Landau type in a two-dimensional simply-connected bounded domain. Configurations of vanishing energy have been characterized by Jabin, Otto and Perthame: the domain must be a disk, and the configuration a vortex. We prove a quantitative version of this statement in the class of $C^{1,1}$ domains, improving on previous results by Lorent. In particular, the deviation of the domain from a disk is controlled by a power of the energy, and that power is optimal. The main tool is a Lagrangian representation introduced by the second author, which allows to decompose the energy along characteristic curves.
For any compact connected one-dimensional submanifold $K\subset \mathbb R^{2\times 2}$ which has no rank-one connection and is elliptic, we prove the quantitative rigidity estimate \[ \inf_{M\in K}\int_{B_{1/2}}| Du -M |^2\,dx \leq C \int_{B_1} \mathrm{dist}^2(Du, K)\, dx, \qquad\forall u\in H^1(B_1;\mathbb R^2). \] This is an optimal generalization, for compact connected submanifolds of $\mathbb R^{2\times 2}$, of the celebrated quantitative rigidity estimate of Friesecke, James and Müller for the approximate differential inclusion into $SO(n)$. The proof relies on the special properties of elliptic subsets $K\subset\mathbb R^{2\times 2}$ with respect to conformal-anticonformal decomposition, which provide a quasilinear elliptic PDE satisfied by solutions of the exact differential inclusion $Du\in K$. We also give an example showing that no analogous result can hold true in $\mathbb R^{n\times n}$ for $n\geq 3$.
We consider finite-entropy solutions of scalar conservation laws $u_t +a(u)_x =0$, that is, bounded weak solutions whose entropy productions are locally finite Radon measures. Under the assumptions that the flux function $a$ is strictly convex (with possibly degenerate convexity) and $a''$ forms a doubling measure, we obtain a characterization of finite-entropy solutions in terms of an optimal regularity estimate involving a cost function first used by Golse and Perthame.
Given any strictly convex norm $\|\cdot\|$ on $\mathbb{R}^2$ that is $C^1$ in $\mathbb{R}^2\setminus\{0\}$, we study the generalized Aviles-Giga functional \[I_ε(m):=\int_Ω \left(ε\left|\nabla m\right|^2 + \frac{1}ε\left(1-\|m\|^2\right)^2\right) \, dx,\] for $Ω\subset\mathbb R^2$ and $m\colonΩ\to\mathbb R^2$ satisfying $\nabla\cdot m=0$. Using, as in the euclidean case $\|\cdot\|=|\cdot|$, the concept of entropies for the limit equation $\|m\|=1$, $\nabla\cdot m=0$, we obtain the following. First, we prove compactness in $L^p$ of sequences of bounded energy. Second, we prove rigidity of zero-energy states (limits of sequences of vanishing energy), generalizing and simplifying a result by Bochard and Pegon. Third, we obtain optimal regularity estimates for limits of sequences of bounded energy, in terms of their entropy productions. Fourth, in the case of a limit map in $BV$, we show that lower bound provided by entropy productions and upper bound provided by one-dimensional transition profiles are of the same order. The first two points are analogous to what is known in the euclidean case $\|\cdot\|=|\cdot|$, and the last two points are sensitive to the anisotropy of the norm $\|\cdot\|$.
The anisotropic Ginzburg-Landau system \[ Δu+δ\, \nabla (\mathrm{div}\: u) +δ\, \mathrm{curl}^*(\mathrm{curl}\: u)=(|u|^2-1) u, \] for $u\colon\mathbb R^2\to\mathbb R^2$ and $δ\in (-1,1)$, models the formation of vortices in liquid crystals. We prove the existence of entire solutions such that $|u(x)|\to 1$ and $u$ has a prescribed topological degree $d\leq -1$ as $|x|\to\infty$, for small values of the anisotropy parameter $|δ| < δ_0(d)$. Unlike the isotropic case $δ=0$, this cannot be reduced to a one-dimensional radial equation. We obtain these solutions by minimizing the anisotropic Ginzburg-Landau energy in an appropriate class of equivariant maps, with respect to a finite symmetry subgroup.
We prove the existence of generalized characteristics for weak, not necessarily entropic, solutions of Burgers' equation \[ \partial_t u +\partial_x \frac{u^2}{2} =0, \] whose entropy productions are signed measures. Such solutions arise in connection with large deviation principles for the hydrodynamic limit of interacting particle systems. The present work allows to remove a technical trace assumption in a recent result by the two first authors about the $L^2$ stability of entropic shocks among such non-entropic solutions. The proof relies on the Lagrangian representation of a solution's hypograph, recently constructed by the third author. In particular, we prove a decomposition formula for the entropy flux across a given hypersurface, which is valid for general multidimensional scalar conservation laws.
We study the linear stability of entire radial solutions $u(re^{iθ})=f(r)e^{iθ}$, with positive increasing profile $f(r)$, to the anisotropic Ginzburg-Landau equation \[ -Δu -δ(\partial_x+i\partial_y)^2\bar u =(1-|u|^2)u,\quad -1<δ<1, \] which arises in various liquid crystal models. In the isotropic case $δ=0$, Mironescu showed that such solution is nondegenerately stable. We prove stability of this radial solution in the range $δ\in (δ_1,0]$ for some $-1<δ_1<0$, and instability outside this range. In strong contrast with the isotropic case, stability with respect to higher Fourier modes is \emph{not} a direct consequence of stability with respect to lower Fourier modes. In particular, in the case where $δ\approx -1$, lower modes are stable and yet higher modes are unstable.
We prove $L^2$ stability estimates for entropic shocks among weak, possibly \emph{non-entropic}, solutions of scalar conservation laws $\partial_t u+\partial_x f(u)=0$ with strictly convex flux function $f$. This generalizes previous results by Leger and Vasseur, who proved $L^2$ stability among entropy solutions. Our main result, the estimate \begin{align*} \int_{\mathbb R} |u(t,\cdot)-u_0^{shock}(\cdot -x(t))|^2\,dx\leq \int_{\mathbb R}|u_0-u_0^{shock}|^2 +Cμ_+([0,t]\times\R), \end{align*} for some Lipschitz shift $x(t)$, includes an error term accounting for the positive part of the entropy production measure $μ=\partial_t(u^2/2)+\partial_x q(u)$, where $q'(u)=uf'(u)$. Stability estimates in this general non-entropic setting are of interest in connection with large deviation principles for the hydrodynamic limit of asymmetric interacting particle systems. Our proof adapts the scheme devised by Leger and Vasseur, where one constructs a shift $x(t)$ which allows to bound from above the time-derivative of the left-hand side. The main difference lies in the fact that our solution $u(t,\cdot)$ may present a non-entropic shock at $x=x(t)$ and new bounds are needed in that situation. We also generalize this stability estimate to initial data with bounded variation.
In this paper we prove sharp regularity for a differential inclusion into a set $K\subset\mathbb{R}^{2\times 2}$ that arises in connection with the Aviles-Giga functional. The set $K$ is not elliptic, and in that sense our main result goes beyond Šverák's regularity theorem on elliptic differential inclusions. It can also be reformulated as a sharp regularity result for a critical nonlinear Beltrami equation. In terms of the Aviles-Giga energy, our main result implies that zero energy states coincide (modulo a canonical transformation) with solutions of the differential inclusion into $K$. This opens new perspectives towards understanding energy concentration properties for Aviles-Giga: quantitative estimates for the stability of zero energy states can now be approached from the point of view of stability estimates for differential inclusions. All these reformulations of our results are strong improvements of a recent work by the last two authors Lorent and Peng, where the link between the differential inclusion into $K$ and the Aviles-Giga functional was first observed and used. Our proof relies moreover on new observations concerning the algebraic structure of entropies.
We consider a nematic liquid crystal occupying the three-dimensional domain in the exterior of a spherical colloid particle. The nematic is subject to Dirichlet boundary conditions that enforce orthogonal attachment of nematic molecules to the surface of the particle. Our main interest is to understand the behavior of energy-critical configurations of the Landau-de Gennes $Q$-tensor model in the limit of vanishing correlation length. We demonstrate existence of configurations with a single Saturn-ring defect approaching the equator of the particle and no other line or point defects. We show this by analyzing asymptotics of energy minimizers under two symmetry constraints: rotational equivariance around the vertical axis and reflection across the horizontal plane. Energy blow-up at the ring defect is a significant obstacle to constructing well-behaved comparison maps needed to eliminate the possibility of point defects. The boundary estimates we develop to address this issue are new and should be applicable to a wider class of problems.
We establish small energy Hölder bounds for minimizers $u_\varepsilon$ of \[E_\varepsilon (u):=\int_ΩW(\nabla u)+ \frac{1}{\varepsilon^2} \int_Ωf(u),\] where $W$ is a positive definite quadratic form and the potential $f$ constrains $u$ to be close to a given manifold $\mathcal N$. This implies that, up to subsequence, $u_\varepsilon$ converges locally uniformly to an $\mathcal N$-valued $W$-harmonic map, away from its singular set. We treat general energies, covering in particular the 3D Landau-de Gennes model for liquid crystals, with three distinct elastic constants. Similar results are known in the isotropic case $W(\nabla u)=\vert \nabla u\vert^2$ and rely on three ingredients: a monotonicity formula for the scale-invariant energy on small balls, a uniform pointwise bound, and a Bochner equation for the energy density. In the level of generality we consider, all of these ingredients are absent. In particular, the lack of monotonicity formula is an important reason why optimal estimates on the singular set of $W$-harmonic maps constitute an open problem. Our novel argument relies on showing appropriate decay for the energy on small balls, separately at scales smaller and larger than $\varepsilon$: the former is obtained from the regularity of solutions to elliptic systems while the latter is inherited from the regularity of $W$-harmonic maps. This also allows us to handle physically relevant boundary conditions for which, even in the isotropic case, uniform convergence up to the boundary was open.
For $\varepsilon>0$, let $u_\varepsilon:Ω\to \mathbb R^2$ be a solution of the Ginzburg-Landau system $$-Δu_\varepsilon=\frac 1{\varepsilon^2} u_\varepsilon (1-|u_\varepsilon|^2)$$ in a Lipschitz bounded domain $Ω$. In an energy regime that excludes interior vortices, we prove that $1-|u_\varepsilon|$ is uniformly estimated by a positive power of $\varepsilon$ $globally$ in $Ω$ provided that the energy of $u_\varepsilon$ at the boundary $\partial Ω$ does not grow faster than $\varepsilon^{-α}$ with $α\in (0,1)$.
In this paper we study the Eikonal equation in a bounded planar domain. We prove the equivalence among optimal Besov regularity, the finiteness of every entropy production and the validity of a kinetic formulation.
We consider a nematic liquid crystal occupying the exterior region in R^3 outside of a spherical particle, with radial strong anchoring. Within the context of the Landau-de Gennes theory, we study minimizers subject to a strong external field, modelled by an additional term which favors nematic alignment parallel to the field. When the external field is high enough we obtain a scaling law for the energy. The energy scale corresponds to minimizers concentrating their energy in a boundary layer around the particle, with quadrupolar symmetry. This suggests the presence of a Saturn ring defect around the particle, rather than a dipolar director field typical of a point defect.
The study of singular perturbations of the Dirichlet energy is at the core of the phenomenological-description paradigm in soft condensed matter. Being able to pass to the limit plays a crucial role in the understanding of the geometric-driven profile of ground states. In this work we study, under very general assumptions, the convergence of minimizers towards harmonic maps. We show that the convergence is locally uniform up to the boundary, away from the lower dimensional singular set. Our results generalize related findings, most notably in the theory of liquid-crystals, to all dimensions $n\geq 3$, and to general nonlinearities. Our proof follows a well-known scheme, relying on small energy estimate and monotonicity formula. It departs substantially from previous studies in the treatment of the small energy estimate at the boundary, since we do not rely on the specific form of the potential. In particular this extends existing results in 3-dimensional settings. In higher dimensions we also deal with additional difficulties concerning the boundary monotonicity formula.
We prove that a $BV$ map with values into the projective space $\mathbb{RP}^{d-1}$ has a $BV$ lifting with values into the unit sphere $\mathbb S^{d-1}$ that satisfies an optimal $BV$-estimate. As an application to liquid crystals, this result is also stated for $BV$ maps with values into the set of uniaxial $Q$-tensors. In order to quantify $BV$ liftings, we prove an explicit formula for an intrinsic $BV$-energy of maps with values into any compact smooth manifold.
We prove regularity estimates for entropy solutions to scalar conservation laws with a force. Based on the kinetic form of a scalar conservation law, a new decomposition of entropy solutions is introduced, by means of a decomposition in the velocity variable, adapted to the non-degeneracy properties of the flux function. This allows a finer control of the degeneracy behavior of the flux. In addition, this decomposition allows to make use of the fact that the entropy dissipation measure has locally finite singular moments. Based on these observations, improved regularity estimates for entropy solutions to (forced) scalar conservation laws are obtained.