arXiv subjects

Denis Bonheure

Publications and source records attributed to Denis Bonheure.

At least 19 recordsLinked to original sources

arXiv2025

Forced Oscillations of a Spring-Mounted Body by a Viscous Liquid: Rotational Case

We study the periodic motions of the coupled system $\mathscr S$, consisting of an incompressible Navier-Stokes fluid interacting with a structure formed by a rigid body subject to {\em undamped} elastic restoring forces and torque around its rotation axis. The motion of $\mathscr S$ is driven by the uniform flow of the liquid, far away from the body, characterized by a time-periodic velocity field, $\mathbf{V}$, of frequency $f$. We show that the corresponding set of governing equations always possesses a time-periodic weak solution of the same frequency $f$, whatever $f>0$, the magnitude of $\mathbf{V}$ and the values of physical parameters. Moreover, we show that the amplitude of linear and rotational displacement is always pointwise in time uniformly bounded by one and the same constant depending on the data, regardless of whether $f$ is or is not close to a natural frequency of the structure. Thus, our result rules out the occurrence of resonant phenomena.

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arXiv2024

Stability of equilibria and bifurcations for a fluid-solid interaction problem

We study certain significant properties of the equilibrium configurations of a rigid body subject to an undamped elastic restoring force, in the stream of a viscous liquid in an unbounded 3D domain. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $\lambda$. We show that if $\lambda$ is below a critical value, $\lambda_c$ (say), there is a unique and stable time-independent configuration, where the body is in equilibrium and the flow is steady. We also prove that, if $\lambda<\lambda_c$, no oscillatory flow may occur. Successively, we investigate possible loss of uniqueness by providing necessary and sufficient conditions for the occurrence of a steady bifurcation at some $\lambda_s\ge \lambda_c$.

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arXiv2024

Flow-induced Oscillations via Hopf Bifurcation in a Fluid-Solid Interaction Problem

We furnish necessary and sufficient conditions for the occurrence of a Hopf bifurcation in a particularly significant fluid-structure problem, where a Navier-Stokes liquid interacts with a rigid body that is subject to an undamped elastic restoring force. The motion of the coupled system is driven by a uniform flow at spatial infinity, with constant dimensionless velocity $\lambda>0$. In particular, if the relevant linearized operator meets suitable spectral properties, there exists a threshold $\lambda_o>0$ above which a bifurcating time-periodic branch stems out of the branch of steady-state solutions. The most remarkable feature of our result is that no restriction is imposed on the frequency $\omega$ of the bifurcating solution, which may thus coincide with one of the natural structural frequencies $\omega_{\sf n}$ of the body. Therefore, resonance cannot occur as a result of this bifurcation. However, when $\omega\to\omega_{\sf n}$, the amplitude of oscillations may become very large when the fluid density is negligible compared to the mass of the body. To our knowledge, our result is the first {\it rigorous} investigation of the existence of a Hopf bifurcation in a fluid-structure interaction problem.

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arXiv2024

Long-time behavior of an anisotropic rigid body interacting with a Poiseuille flow in an unbounded 2D channel

We study the long-time behavior of an elliptic rigid body which is allowed to vertically translate and rotate in a 2D unbounded channel under the action of a Poiseuille flow at large distances. The motion of the fluid is modelled by the incompressible Navier-Stokes equations, while the motion of the solid is described through Newton's laws. In addition to the solid inertia and the hydrodynamic forces, we assume the dynamics of the solid is driven by internal elastic restoring forces but without any structural damping. Through a precise description of the motion of the elliptic body whenever it comes close to the channel boundaries, we prove global-in-time existence of weak solutions. Our second main contribution is a proof of return to equilibrium in case the amplitude of the Poiseuille flow is small. \black To our knowledge, this represents the first long-time analysis of fluid-solid interaction problems with a given non-trivial final state.

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arXiv2023

Global Weak Solutions to a Time-Periodic Body-Liquid Interaction Problem

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier-Stokes fluid interacting with a rigid body of finite size, subject to an {\em undamped} linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field $\bfV$ far from the body. {We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of $\bfV$ and, rather remarkably, the frequency of $\bfV$ is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical "invading domains" technique, our approach requires several new ideas.} Indeed, due to lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincar\'e map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.

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arXiv2022

Classification of radial blow-up at the first critical exponent for the Lin-Ni-Takagi problem in the ball

We investigate the behaviour of radial solutions to the Lin-Ni-Takagi problem in the ball $B_R \subset \mathbb{R}^N$ for $N \ge 3$: \begin{equation*} \left \{ \begin{aligned} - \triangle u_p + u_p & = |u_p|^{p-2}u_p & \textrm{ in } B_R, \\ \partial_\nu u_p & = 0 & \textrm{ on } \partial B_R, \end{aligned} \right. \end{equation*} when $p $ is close to the first critical Sobolev exponent $2^* = \frac{2N}{N-2}$. We obtain a complete classification of finite energy radial smooth blowing up solutions to this problem. We describe the conditions preventing blow-up as $p \to 2^*$, we give the necessary conditions in order for blow-up to occur and we establish their sharpness by constructing examples of blowing up sequences. Our approach allows for asymptotically supercritical values of $p$. We show in particular that, if $p \geq 2^\ast$, finite-energy radial solutions are precompact in $C^2(\bar{B_R})$ provided that $N\geq 7$. Sufficient conditions are also given in smaller dimensions if $p=2^\ast$. Finally we compare and interpret our results to the bifurcation analysis of Bonheure, Grumiau and Troestler in Nonlinear Anal. 147 (2016).

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arXiv2021

A sharp gradient estimate and $W^{2,q}$ regularity for the prescribed mean curvature equation in the Lorentz-Minkowski space

We consider the prescribed mean curvature equation for entire spacelike hypersurfaces in the Lorentz-Minkowski space, namely \begin{equation*} -\operatorname{div}\left(\displaystyle\frac{\nabla u}{\sqrt{1-|\nabla u|^2}}\right)= \rho \quad \hbox{in }\mathbb{R}^N, \end{equation*} where $N\geq 3$. We first prove a new gradient estimate for classical solutions with smooth data $\rho$. As a consequence we obtain that the unique weak solution of the equation satisfying a homogeneous boundary condition at infinity is locally of class $W^{2,q}$ and strictly spacelike in $\mathbb{R}^N$, provided that $\rho\in L^q(\mathbb{R}^N) \cap L^m(\mathbb{R}^N)$ with $q>N$ and $m\in[1,\frac{2N}{N+2}]$.

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arXiv2020

Bifurcation analysis of the Hardy-Sobolev equation

In this paper, we prove existence of multiple non-radial solutions to the Hardy-Sobolev equation $$\begin{cases} -\Delta u-\displaystyle\frac \gamma{|x|^2}u=\displaystyle\frac{1}{|x|^s}|u|^{p_s-2}u & \text{ in } \mathbb{R}^N\setminus\{0\},\\ u\geq 0, & \end{cases}$$ where $N\geq 3$, $s\in[0,2)$, $p_s=\frac{2(N-s)}{N-2}$ and $\gamma\in (-\infty,\frac{(N-2)^2} 4)$. We extend results of E.N. Dancer, F. Gladiali, M. Grossi, Proc. Roy. Soc. Edinburgh Sect. A 147 (2017) where only the case $s=0$ is considered. Moreover, thanks to monotonicity properties of the solutions, we separate two branches of non-radial solutions.

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arXiv2020

Concentration phenomena for the Schr\"odinger-Poisson system in $\mathbb{R}^2$

We perform a semiclassical analysis for the planar Schr\"odinger-Poisson system \[ \cases{ -\varepsilon^{2} \Delta\psi+V(x)\psi= E(x) \psi \quad \text{in $\mathbb{R}^2$},\cr -\Delta E= |\psi|^{2} \quad \text{in $\mathbb{R}^2$}, \cr } \tag{$SP_\varepsilon$} \] where $\varepsilon$ is a positive parameter corresponding to the Planck constant and $V$ is a bounded external potential. We detect solution pairs $(u_\varepsilon, E_\varepsilon)$ of the system $(SP_\varepsilon)$ as~$\ge \rightarrow 0$.

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arXiv2020

Long-time dynamics of a hinged-free plate driven by a non-conservative force

A partially hinged, partially free rectangular plate is considered, with the aim to address the possible unstable end behaviors of a suspension bridge subject to wind. This leads to a nonlinear plate evolution equation with a nonlocal stretching active in the span-wise direction. The wind-flow in the chord-wise direction is modeled through a piston-theoretic approximation, which provides both weak (frictional) dissipation and non-conservative forces. The long-time behavior of solutions is analyzed from various points of view. Compact global attractors, as well as fractal exponential attractors, are constructed using the recent quasi-stability theory. The non-conservative nature of the dynamics requires the direct construction of a uniformly absorbing ball, and this relies on the superlinearity of the stretching. For some parameter ranges, the non-triviality of the attractor is shown through the spectral analysis of the stationary linearized (non self-adjoint) equation and the existence of multiple unimodal solutions is shown. Several stability results, obtained through energy estimates under various smallness conditions and/or assumptions on the equilibrium set, are also provided. Finally, the existence of a finite set of determining modes for the dynamics is demonstrated, justifying the usual modal truncation in engineering for the study of the qualitative behavior of suspension bridge dynamics.

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arXiv2020

Equilibrium configuration of a rectangular obstacle immersed in a channel flow

Fluid flows around an obstacle generate vortices which, in turn, generate lift forces on the obstacle. Therefore, even in a perfectly symmetric framework equilibrium positions may be asymmetric. We show that this is not the case for a Poiseuille flow in an unbounded 2D channel, at least for small Reynolds number and flow rate. We consider both the cases of vertically moving obstacles and obstacles rotating around a fixed pin.

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arXiv2020

Nodal Solutions for sublinear-type problems with Dirichlet boundary conditions

We consider nonlinear second order elliptic problems of the type \[ -\Delta u=f(u) \text{ in } \Omega, \qquad u=0 \text{ on } \partial \Omega, \] where $\Omega$ is an open $C^{1,1}$-domain in $\mathbb{R}^N$, $N\geq 2$, under some general assumptions on the nonlinearity that include the case of a sublinear pure power $f(s)=|s|^{p-1}s$ with $0 1$ and $\lambda>\lambda_2(\Omega)$ (the second Dirichlet eigenvalue of the Laplacian). We prove the existence of a least energy nodal (i.e. sign changing) solution, and of a nodal solution of mountain-pass type. We then give explicit examples of domains where the associated levels do not coincide. For the case where $\Omega$ is a ball or annulus and $f$ is of class $C^1$, we prove instead that the levels coincide, and that least energy nodal solutions are nonradial but axially symmetric functions. Finally, we provide stronger results for the Allen-Cahn type nonlinearities in case $\Omega$ is either a ball or a square. In particular we give a complete description of the solution set for $\lambda\sim \lambda_2(\Omega)$, computing the Morse index of the solutions.

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arXiv2019

A Paneitz-Branson type equation with Neumann boundary conditions

We consider the best constant in a critical Sobolev inequality of second order. We show non-rigidity for the optimizers above a certain threshold, namely we prove that the best constant is achieved by a non-constant solution of the associated fourth-order elliptic problem under Neumann boundary conditions. Our arguments rely on asymptotic estimates of the Rayleigh quotient. We also show rigidity below another threshold.

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arXiv2019

Inequalities involving Aharonov-Bohm magnetic potentials in dimensions 2 and 3

This paper is devoted to a collection of results on nonlinear interpolation inequalities associated with Schr{\"o}dinger operators involving Aharonov-Bohm magnetic potentials, and to some consequences. As symmetry plays an important role for establishing optimality results, we shall consider various cases corresponding to a circle, a two-dimensional sphere or a two-dimensional torus, and also the Euclidean spaces of dimensions two and three. Most of the results are new and we put the emphasis on the methods, as very little is known on symmetry, rigidity and optimality in presence of a magnetic field. The most spectacular applications are new magnetic Hardy inequalities in dimensions 2 and 3.

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arXiv2019

Symmetry results in two-dimensional inequalities for Aharonov-Bohm magnetic fields

This paper is devoted to the symmetry and symmetry breaking properties of a two-dimensional magnetic Schr{\"o}dinger operator involving an Aharonov-Bohm magnetic vector potential. We investigate the symmetry properties of the optimal potential for the corresponding magnetic Keller-Lieb-Thir-ring inequality. We prove that this potential is radially symmetric if the intensity of the magnetic field is below an explicit threshold, while symmetry is broken above a second threshold corresponding to a higher magnetic field. The method relies on the study of the magnetic kinetic energy of the wave function and amounts to study the symmetry properties of the optimal functions in a magnetic Hardy-Sobolev interpolation inequality. We give a quantified range of symmetry by a non-perturbative method. To establish the symmetry breaking range, we exploit the coupling of the phase and of the modulus and also obtain a quantitative result.

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arXiv2018

Periodic solutions and torsional instability in a nonlinear nonlocal plate equation

A thin and narrow rectangular plate having the two short edges hinged and the two long edges free is considered. A nonlinear nonlocal evolution equation describing the deformation of the plate is introduced: well-posedness and existence of periodic solutions are proved. The natural phase space is a particular second order Sobolev space that can be orthogonally split into two subspaces containing, respectively, the longitudinal and the torsional movements of the plate. Sufficient conditions for the stability of periodic solutions and of solutions having only a longitudinal component are given. A stability analysis of the so-called prevailing mode is also performed. Some numerical experiments show that instabilities may occur. This plate can be seen as a simplified and qualitative model for the deck of a suspension bridge, which does not take into account the complex interactions between all the components of a real bridge.

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arXiv2018

Normalized solutions to the mixed dispersion nonlinear Schrödinger equation in the mass critical and supercritical regime

In this paper, we study the existence of solutions to the mixed dispersion nonlinear Schrödinger equation $$ γΔ^2 u -Δu + αu=|u|^{2 σ} u, \quad u \in H^2(\R^N), $$ under the constraint $$ \int_{\R^N}|u|^2 \, dx =c>0. $$ We assume $γ>0, N \geq 1, 4 \leq σN < \frac{4N}{(N-4)^+}$, whereas the parameter $α\in \R$ will appear as a Lagrange multiplier. Given $c \in \R^+$, we consider several questions including the existence of ground states, of positive solutions and the multiplicity of radial solutions. We also discuss the stability of the standing waves of the associated dispersive equation.

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arXiv2018

Equilibrium measures and equilibrium potentials in the Born-Infeld model

In this paper, we consider the electrostatic Born-Infeld model \begin{equation*} \tag{$\mathcal{BI}$} \left\{ \begin{array}{rcll} -\operatorname{div}\left(\displaystyle\frac{\nabla \phi}{\sqrt{1-|\nabla \phi|^2}}\right)&=& \rho & \hbox{in }\mathbb{R}^N, \\[6mm] \displaystyle\lim_{|x|\to \infty}\phi(x)&=& 0 \end{array} \right. \end{equation*} where $\rho$ is a charge distribution on the boundary of a bounded domain $\Omega\subset \mathbb{R}^N$. We are interested in its equilibrium measures, i.e. charge distributions which minimize the electrostatic energy of the corresponding potential among all possible distributions with fixed total charge. We prove existence of equilibrium measures and we show that the corresponding equilibrium potential is unique and constant in $\overline \Omega$. Furthermore, for smooth domains, we obtain the uniqueness of the equilibrium measure, we give its precise expression, and we verify that the equilibrium potential solves ($\mathcal{BI}$). Finally we characterize balls in $\mathbb{R}^N$ as the unique sets among all bounded $C^{2,\alpha}$-domains $\Omega$ for which the equilibrium distribution is a constant multiple of the surface measure on $\partial\Omega$. The same results are obtained also for Taylor approximations of the electrostatic energy.

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