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Nicolas Burq

Publications and source records attributed to Nicolas Burq.

At least 19 recordsLinked to original sources
arXiv2026

Gauge transforms, random averaging operator ansatz and improved probabilistic well-posedness for the radial NLS on the $3d$ ball

We construct probabilistic strong solutions to the cubic Schr\"odinger equation on the three-dimensional ball with radial initial data, which is a significant improvement of a result by Bourgain--Bulut. These solutions lie in the supercritical regime with respect to the probabilistic scaling introduced by Deng--Nahmod--Yue. We achieve this result through gauge transformations that do not modify the equation, combined with a refined modulation analysis using random averaging operators.

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arXiv2025

A measure-$L^\infty$ div-curl lemma

In this note we give a very short proof of the div-curl lemma in the limit conjugate case $\mathcal M-L^\infty$, where $\mathcal{M}$ is the set of Radon measures on $\mathbb{R}^d$. The proof follows the classical approach by defining here the product in the sense of distributions via a non unique microlocal Hodge's decomposition. The result is valid for many other spaces than $\mathcal M-L^\infty$, including the classical div-curl lemma spaces $L^p-L^{p'}$ for $1<p<\infty$, and spaces of non conjugated regularity.

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arXiv2025

Resolvent bounds imply observability from measurable time sets for Schr\"odinger equations

We prove that on a compact Riemannian manifold, resolvent bounds for the Laplace--Beltrami operator imply observability, and thus controllability, for the Schr\"odinger propagator from time sets of positive Lebesgue measure. Applications include almost all cases where observability and controllability hold from time intervals, particularly when the geometric control condition is satisfied or when the manifold is a compact surface of negative curvature.

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arXiv2025

Observability of Schr\"odinger propagators on tori in rough settings

On tori of arbitrary dimensions, Schr\"odinger propagators with bounded potentials are conjectured to be observable from space-time domains of positive Lebesgue measure. We reduce this conjecture to certain integrability bounds for free Schr\"odinger waves, thereby proving the conjecture on the one-dimensional torus and producing new examples of observation domains. These bounds are far weaker than Bourgain's conjectured periodic Strichartz estimates, yet remain highly nontrivial.

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arXiv2025

Trace and Observability Inequalities for Laplace Eigenfunctions on the Torus

We investigate trace and observability inequalities for Laplace eigenfunctions on the d-dimensional torus, with respect to arbitrary Borel measures $\mu$. Specifically, we characterize the measures $\mu$ for which the inequalities $$ \int |u|^2 d \mu \lesssim \int |u|^2 d x \quad \text{(trace)}, \qquad \int |u|^2 d \mu \gtrsim \int |u|^2 d x \quad \text{(observability)}$$ hold uniformly for all eigenfunctions $u$ of the Laplacian. Sufficient conditions are derived based on the integrability and regularity of $\mu$, while necessary conditions are formulated in terms of the dimension of the support of the measure. These results generalize classical theorems of Zygmund and Bourgain--Rudnick to higher dimensions. Applications include results in the spirit of Cantor--Lebesgue theorems, constraints on quantum limits, and control theory for the Schr\"odinger equation. Our approach combines several tools: the cluster structure of lattice points on spheres; decoupling estimates; and the construction of eigenfunctions exhibiting strong concentration or vanishing behavior, tailored respectively to the trace and observability inequalities.

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arXiv2024

Strichartz Estimates for a Class of Baouendi-Grushin Operators

We prove Strichartz estimates for a class of Baouendi--Grushin operators acting either on the Euclidean space or a product of the type $\mathbb{R}^{d_1} \times M$, where $(M,g)$ is a smooth compact manifold with no boundary. We then give an application of these Strichartz estimates to the Cauchy theory for the associated Schr\"odinger equations.

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arXiv2024

Nonlinear interpolation and the flow map for quasilinear equations

We prove an interpolation theorem for nonlinear functionals defined on scales of Banach spaces that generalize Besov spaces. It applies to functionals defined only locally, requiring only some weak Lipschitz conditions, extending those introduced by Lions and Peetre. Our analysis is self-contained and independent of any previous results about interpolation theory. It depends solely on the concepts of Friedrichs' mollifiers, seen through the formalism introduced by Hamilton, combined with the frequency envelopes introduced by Tao and used recently by two of the authors and others to study the Cauchy problem for various quasilinear evolutions in partial differential equations. Inspired by this latter work, our main application states that, for an abstract flow map of a quasilinear problem, both the continuity of the flow as a function of time and the continuity of the data to solution map follow automatically from the estimates that are usually proven when establishing the existence of solutions: propagation of regularity via tame a priori estimates for higher regularities and contraction for weaker norms.

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arXiv2024

Delocalized eigenvectors of transitive graphs and beyond

We prove delocalization of eigenvectors of vertex-transitive graphs via elementary estimates of the spectral projector. We recover in this way known results which were formerly proved using representation theory. Similar techniques show that for general symmetric matrices, most approximate eigenvectors spectrally localized in a given window containing sufficiently many eigenvalues are delocalized in $L^q$ norms. Building upon this observation, we prove a delocalization result for approximate eigenvectors of large graphs containing few short loops, under an assumption on the resolvent which is verified in some standard cases, for instance random lifts of a fixed base graph.

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arXiv2024

Measure and continuous vector field at a boundary I: propagation equations and wave observability

The celebrated geometric control condition of Bardos, Lebeau, and Rauch is necessary and sufficient for wave observability [1,7] and exact controllability. It requires that any point in phase-space be transported by the generalized geodesic flow to the region of observation in some finite time. The initial smoothness (\Cinf) required on the coefficients of the metric to prove that exact control and geometric control are essentially equivalent was subsequently relaxed to \Con2-metrics/coefficients and \Con3-domains [2] which is close to the optimal smoothness required to preserve a generalized geodesic flow. In this article, we investigate a natural generalization of the geometric control condition that makes sense for \Con1-metrics and we prove that wave observability holds under this condition. Moreover, we establish that the observability property is stable under rougher (Lipschitz) perturbation of the metric. We also provide a geometric necessary condition for wave observability to hold. Transport equations that describe the propagation of semi-classical measures are at the heart of the arguments. They are natural extensions to geometries with boundaries of usual transport equations. This article is mainly dedicated to the proof of such propagation equations in this very rough context.

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arXiv2024

Measure and continuous vector field at a boundary II: geodesics and support propagation

Nonnegative measures that are solutions to a transport equation with continuous coefficients have been widely studied. Because of the low regularity of the associated vector field, there is no natural flow since nonuniqueness of integral curves is the general rule. It has been known since the works by L. Ambrosio [2] and L. Ambrosio and G. Crippa [3, 4] that such measures can be described as a superposition of $\delta$-measures supported on integral curves. In this article, motivated by some observability questions for the wave equation, we are interested in such transport equations in the case of domains with boundary. Associated with a wave equation with C^1-coefficients are bicharacteristics that are integral curves of a continuous Hamiltonian vector field. We first study in details their behaviour in the presence of a boundary and define their natural generalisation that follows the laws of geometric optics. Then, we introduce a natural class of transport equations with a source term on the boundary, and we prove that any nonnegative measure satisfying such an equation has a union of maximal generalized bicharacteristics for support. This result is a weak form of the superposition principle in the presence of a boundary. With its companion article [7], this study completes the proof of wave observability generalizing the celebrated result of Bardos, Lebeau, and Rauch [5] in a low regularity framework where coefficients of the wave equation (and associated metric) are C^1 and the boundary and the manifold are C^2.

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arXiv2024

Almost surely smoothed scattering for cubic NLS

We consider cubic NLS in dimensions 2, 3, 4 and we prove that almost surely solutions with randomized initial data at low regularity scatter. Moreover, we establish some smoothing properties of the associated scattering operator and precise the rate of convergence.

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arXiv2024

Probabilistic well-posedeness for the nonlinear Schr\"odinger equation on the $2d$ sphere I: positive regularities

We establish the probabilistic well-posedness of the nonlinear Schr\"odinger equation on the $2d$ sphere $\mathbb{S}^{2}$. The initial data are distributed according to Gaussian measures with typical regularity $H^{s}(\mathbb{S}^{2})$, for $s>0$. This level of regularity goes significantly beyond existing deterministic results, in a regime where the flow map cannot be extended uniformly continuously.

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arXiv2024

The Second Picard iteration of NLS on the $2d$ sphere does not regularize Gaussian random initial data

We consider the Wick ordered cubic Schr\"odinger equation (NLS) posed on the two-dimensional sphere, with initial data distributed according to a Gaussian measure. We show that the second Picard iteration does not improve the regularity of the initial data in the scale of the classical Sobolev spaces. This is in sharp contrast with the Wick ordered NLS on the two-dimensional tori, a model for which we know from the work of Bourgain that the second Picard iteration gains one half derivative. Our proof relies on identifying a singular part of the nonlinearity. We show that this singular part is responsible for a concentration phenomenon on a large circle (i.e. a stable closed geodesic), which prevents any regularization in the second Picard iteration.

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arXiv2023

Domains of dependence for subelliptic wave equations and unique continuation for fractional powers of H\"ormander's operators

We prove the sharp domain of dependence property for solutions to subelliptic wave equations for sums of squares of vector fields satisfying H\"ormander bracket condition. We deduce a unique continuation property for the square root of subelliptic Laplace operators under an additional analyticity condition. Then, with a different, more involved method, we prove the same result of unique continuation for more general $s$-powers ($0<s<1$).

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arXiv2023

Measure propagation along a $\mathscr{C}^0$-vector field and wave controllability on a rough compact manifold

The celebrated Rauch-Taylor/Bardos-Lebeau-Rauch geometric control condition is central in the study of the observability of the wave equation linking this property to high-frequency propagation along geodesics that are therays of geometric optics. This connection is best understood through the propagation properties of microlocal defect measures that appear as solutions to the wave equation concentrate. For a sufficiently smooth metric this propagation occurs along the bicharacteristic flow. If one considers a merely $\mathscr{C}^1$-metric this bicharacteristic flow may however not exist. The Hamiltonian vector field is only continuous; bicharacteristics do exist (as integral curves of this continuous vector field) but uniqueness is lost. Here, on a compact manifold without boundary, we consider this low regularity setting, revisit the geometric control condition, and address the question of support propagation for a measure solution to an ODE with continuous coefficients. This leads to a sufficient condition for the observability and equivalently the exact controllability of the wave equation. Moreover, we investigate the stabililty of the observability property and the sensitivity of the control process under a perturbation of the metric of regularity as low as Lipschitz.

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arXiv2023

Strichartz estimates for Maxwell equations on domains with perfectly conducting boundary conditions

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.

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arXiv2023

Bilinear Strichartz estimates and almost sure global solutions for the nonlinear Schr{\"o}dinger equation

The purpose of this article is to construct global solutions, in a probabilistic sense, for the nonlinear Schr{\"o}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{\"o}dinger equation without harmonic potential. Secondly, we prove a Kato type smoothing estimate for the linear Schr{\"o}dinger equation with harmonic potential. This allows us to consider the Schr{\"o}dinger equation with a nonlinearity of odd degree in a supercritical regime, in any dimension $d\geq 2$.

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