arXiv subjects
Publications and source records attributed to Nicolas Burq.
For the wave and the Schrödinger equations we show how observability can be deduced from the observability of solutions localized in frequency according to a dyadic scale.
We consider Maxwell equations on a smooth domain with perfectly conducting boundary conditions in isotropic media in two and three dimensions. In the charge-free case we recover Strichartz estimates due to Blair--Smith--Sogge for wave equations on domains up to endpoints. For the proof we suitably extend Maxwell equations over the boundary, which introduces coefficients on the full space with codimension-$1$ Lipschitz singularity. This system can be diagonalized to half-wave equations amenable to the results of Blair--Smith--Sogge. In two dimensions, we improve the local well-posedness of the Maxwell system with Kerr nonlinearity via Strichartz estimates.
The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{ö}dinger equation posed on $\mathbb{R}^d$, in a supercritical regime. Firstly, we establish Bourgain type bilinear estimates for the harmonic oscillator which yields a gain of half a derivative in space for the local theory with randomised initial conditions, for the cubic equation in $\mathbb{R}^3$. Then, thanks to the lens transform, we are able to obtain global in time solutions for the nonlinear Schr{ö}dinger equation without harmonic potential. Secondly, we prove a Kato type smoothing estimate for the linear Schr{ö}dinger equation with harmonic potential. This allows us to consider the Schr{ö}dinger equation with a nonlinearity of odd degree in a supercritical regime, in any dimension $d\geq 2$.
It is well known that both the heat equation with Dirichlet or Neumann boundary conditions are null controlable as soon as the control acts in a non trivial domain (i.e. a set of positive measure, see [10, 11, 12, 1, 6]. In this article, we show that for any couple of initial data (u0, v0) we can achieve the null control for both equations (Dirichlet and Neumann boundary conditions respectively) simultaneously with the same control function for both equations.
In this paper we consider a compact Riemannian manifold (M, g) of class C 1 $\cap$ W 2,$\infty$ and the damped wave or Schrödinger equations on M , under the action of a damping function a = a(x). We establish the following fact: if the measure of the set {x $\in$ M ; a(x) = 0} is strictly positive, then the decay in time of the associated energy is at least logarithmic.
The purpose of this article is to investigate the structure of singular measures from a microlocal perspective. Motivated by the result of De Philippis-Rindler [10] and the notions of wave cone of Murat-Tartar [19,20,26,27] and of polarisation set of Denker [9] we introduce a notion of $L^1$-regularity wave front set for scalar and vector distributions. Our main result is a proper microlocal characterisation of the support of the singular part of tempered Radon measures and of their polar functions at these points. The proof is based on De Philippis-Rindler's approach reinforced by microlocal analysis techniques and some extra geometric measure theory arguments. We deduce a sharp $L^1$ elliptic regularity result which appears to be new even for scalar measures and which enlightens the interest of the techniques from geometric measure theory to the study of harmonic analysis questions. For instance we prove that $ Ψ^0 L^1\cap \mathcal M_{loc}\subseteq L^1_{loc},$ and in particular we obtain $L^1$ elliptic regularity results as $Δu\in L^1_{loc}, D^2 u \in \mathcal M_{loc} \Longrightarrow D^2 u\in L^1_{loc}.$ We also deduce several consequences including extensions of the results in [10] giving constraints on the polar function at singular points for measures constrained by a PDE, and of Alberti's rank one theorem. Finally, we also illustrate the interest of this microlocal approach with a result of propagation of singularities for constrained measures.
In this note we investigate propagation of smallness properties for solutions to heat equations. We consider spectral projector estimates for the Laplace operator with Dirichlet or Neumann boundary conditions on a Riemanian manifold with or without boundary. We show that using the new approach for the propagation of smallness from Logunov-Malinnikova [7, 6, 8] allows to extend the spectral projector type estimates from Jerison-Lebeau [3] from localisation on open set to localisation on arbitrary sets of non zero Lebesgue measure; we can actually go beyond and consider sets of non vanishing d -- $δ$ ($δ$ > 0 small enough) Hausdorf measure. We show that these new spectral projector estimates allow to extend the Logunov-Malinnikova's propagation of smallness results to solutions to heat equations. Finally we apply these results to the null controlability of heat equations with controls localised on sets of positive Lebesgue measure. A main novelty here with respect to previous results is that we can drop the constant coefficient assumptions (see [1, 2]) of the Laplace operator (or analyticity assumption, see [4]) and deal with Lipschitz coefficients. Another important novelty is that we get the first (non one dimensional) exact controlability results with controls supported on zero measure sets.
We study the energy decay rate of the Kelvin-Voigt damped wave equation with piecewise smooth damping on the multi-dimensional domain. Under suitable geometric assumptions on the support of the damping, we obtain the optimal polynomial decay rate which turns out to be different from the one-dimensional case studied in \cite{LR05}. This optimal decay rate is saturated by high energy quasi-modes localised on geometric optics rays which hit the interface along non orthogonal neither tangential directions. The proof uses semi-classical analysis of boundary value problems.
The question of unique continuation of harmonic functions in a domain $Ω$ $\subset$ R d with boundary $\partial$$Ω$, satisfying Dirichlet boundary conditions and with normal derivatives vanishing on a subset $ω$ of the boundary is a classical problem. When $ω$ contains an open subset of the boundary it is a consequence of Carleman estimates (uniqueness for second order elliptic operators across an hypersurface). The case where $ω$ is a set of positive (d -- 1) dimensional measure has attracted a lot of attention, see e.g. [10, 3, 15], where qualitative results have been obtained in various situations. The main open questions (about uniqueness) concern now Lipschitz domains and variable coefficients. Here, using results by Logunov and Malinnikova [13, 14], we consider the simpler case of W 2,$\infty$ domains but prove quantitative uniqueness both for Dirichlet and Neumann boundary conditions. As an application, we deduce quantitative estimates for the Dirichlet and Neumann Laplace eigenfunctions on a W 2,$\infty$ domain with boundary.
The purpose of this article is to show that the spectral projector estimates for Laplace operators can be deduced from Logunov-Malinnikova's Propagation of smallness estimates for harmonic functions [11, 10, 9]. The main point is to pass from the local estimates obtained in [4] (on a compact manifold) to global estimates. We also state classical consequences in terms of observability and control for heat equations, which are direct consequences of these spectral projector estimates..
In this paper we consider the Cauchy problem for gravity water waves, in a domain with a flat bottom and in arbitrary space dimension. We prove that if the data are of size $\varepsilon$ in a space of analytic functions which have a holomorphic extension in a strip of size $σ$, then the solution exists up to a time of size $C/\varepsilon$ in a space of analytic functions having at time $t$ a holomorphic extension in a strip of size $σ- C'\varepsilon t$.
We study in this article decay rates for Kelvin-Voigt damped wave equations under a geometric control condition. We prove that when the damping coefficient is sufficiently smooth ($C^1$ vanishing nicely) we show that exponential decay follows from geometric control conditions (see~\cite{BuCh, Te12} for similar results under stronger assumptions on the damping function).
We consider two dimensional Grushin Schrödinger equation posed on a finite cylinder $Ω=(-1,1)_x\times \T_y$ with Dirichlet boundary condition. We obtain the sharp observability by any horizontal strip, with the optimal time $T_*>0$ depending on the size of the strip. Consequently, we prove the exact controllability for Grushin Schrödinger equation. By exploiting the concentration of eigenfunctions of harmonic oscillator at $x=0$, we also show that the observability fails for any $T\leq T_*$.
We consider the one-dimensional nonlinear Schrödinger equation with a nonlinearity of degree $p>1$. We exhibit measures on the space of initial data for which we describe the non trivial evolution by the linear Schrödinger flow and we show that their nonlinear evolution is absolutely continuous with respect to this linear evolution. We deduce from this precise description the global well-posedness of the equation for $p>1$ and scattering for $p>3$. To the best of our knowledge, it is the first occurence where the description of quasi-invariant measures allows to get quantitative asymptotics (here scattering properties) for the nonlinear evolution.
The purpose of this short note is to show how it is possible to combine existing results in the literature to get the unique continuation from sets of positive measure for time dependent parabolic equations with Lipschitz principal part and bounded lower order terms, result which was known in the case of analytic coefficients in [3].
We show in this article how perturbative approaches~from our work with Hitrik (see also the work by Anantharaman-Macia) and the {\em black box} strategy from~ our work with Zworski allow to obtain decay rates for Kelvin-Voigt damped wave equations from quite standard resolvent estimates : Carleman estimates or geometric control estimates for Helmoltz equationCarleman or other resolvent estimates for the Helmoltz equation. Though in this context of Kelvin Voigt damping, such approach is unlikely to allow for the optimal results when additional geometric assumptions are considered (see \cite{BuCh, Bu19}), it turns out that using this method, we can obtain the usual logarithmic decay which is optimal in general cases. We also present some applications of this approach giving decay rates in some particular geometries (tori).
For the damped wave equation on a compact manifold with {\em continuous} dampings, the geometric control condition is necessary and sufficient for {uniform} stabilisation. In this article, on the two dimensional torus, in the special case where $a(x) = \sum\_{j=1}^N a\_j 1\_{x\in R\_j}$ ($R\_j$ are polygons), we give a very simple necessary and sufficient geometric condition for uniform stabilisation. We also propose a natural generalization of the geometric control condition which makes sense for $L^\infty$ dampings. We show that this condition is always necessary for uniform stabilisation (for any compact (smooth) manifold and any $L^\infty$ damping), and we prove that it is sufficient in our particular case on $\mathbb{T}^2$ (and for our particular dampings).
We introduce a novel data randomisation for the free wave equation which leads to the same range of Strichartz estimates as for radial data, albeit in a non-radial context. We then use these estimates to establish global well-posedness for a wave maps type nonlinear wave equation for certain supercritical data, provided the data are suitably small and randomised.