arXiv subjects

Louis Garénaux

Publications and source records attributed to Louis Garénaux.

9 recordsLinked to original sources

arXiv2026

Orbital stability of plane waves in the Klein-Gordon equation against localized perturbations

We investigate the stability and long-term behavior of spatially periodic plane waves in the complex Klein-Gordon equation under localized perturbations. Such perturbations render the wave neither localized nor periodic, placing its stability analysis outside the scope of the classical orbital stability theory for Hamiltonian systems developed by Grillakis, Shatah, and Strauss. Inspired by Zhidkov's work on the stability of time-periodic, spatially homogeneous states in the nonlinear Schrödinger equation, we develop an alternative method that relies on an amplitude-phase decomposition and leverages conserved quantities tailored to the perturbation equation. We establish an orbital stability result of plane waves that is locally uniform in space, accommodating $L^2$-localized perturbations as well as unbounded phase modulations. Our result is sharp in the sense that it holds up to the spectral stability boundary.

math.AP↗

arXiv2026

Orbital stability of monostable waves for reaction-diffusion systems

We study stability of monostable waves for reaction-diffusion systems. When the solution is initially close to a fast wave profile in optimal topology, we prove convergence to a shifted profile. The proof relies on explicit resolvent kernels estimates, allowing to handle weakly localized perturbations. It allows phase shift construction even when the translational eigenvalue is not associated to a zero of the Evans function. We further discuss distinction between Evans and Fourier eigenmodes when the marginal group velocity are directed towards the wave interface.

math.AP↗

arXiv2025

Global existence and decay of small solutions in a viscous half Klein-Gordon equation

We establish global existence and decay of solutions of a viscous half Klein-Gordon equation with a quadratic nonlinearity considering initial data, whose Fourier transform is small in L1 cap Linfty. Our analysis relies on the observation that nonresonant dispersive effects yield a transformation of the quadratic nonlinearity into a subcritical nonlocal quartic one, which can be controlled by the linear diffusive dynamics through a standard L1 - Linfty argument. This transformation can be realized by applying the normal form method of Shatah or, equivalently, through integration by parts in time in the associated Duhamel formula.

math.AP↗

arXiv2025

Linear convective stability of a front superposition with unstable connecting state

We study convective stability of a two-front superposition in a reaction-diffusion system. Due to the instability of the connecting equilibrium, long-range semi-strong interaction is expected between the two waves. When restricting to the linear dynamic, we indeed identify that convective stability of superposed waves occurs for fewer propagation speeds than for the corresponding single waves. It reflects the interaction that monostable waves have over long distances. Our method relies on numerical range estimates, that imply time-uniform resolvent bounds.

math.AP↗

arXiv2025

Long-time behavior of small solutions in the viscoelastic Klein-Gordon equation

We investigate the long-time behavior of solutions with small initial data to the viscoelastic Klein-Gordon equation with general smooth nonlinearity. Our analysis relies on the space-time resonances method to eliminate all nonresonant quadratic and cubic terms. We identify a sign condition for the remaining critical resonant term to be of absorption type, leading to global-in-time existence and diffusive decay of solutions with small initial data. Even when this condition fails, our analysis shows existence and diffusive decay of small solutions on exponentially long time intervals.

math.AP↗

arXiv2024

Existence of monostable fronts for a KPP infinite-difference numerical scheme

We study the existence of traveling wave solutions for a numerical counterpart of the KPP equation. We obtain the existence of monostable fronts for all super-critical speeds in the regime where the spatial step size is small. The key strategy is to transfer the invertibility of certain linear operators related to the front solutions from the continuous setting to the discrete case we are interested in. We rely on resolvent bounds which are uniform with respect to the step size, a procedure which is also known as spectral convergence. The approach is also able to handle infinite range discretizations with geometrically decaying coefficients that are allowed to have both signs, which prevents the use of the comparison principle.

math.NA↗

arXiv2024

Convective stability in scalar balance laws

We fill the two main remaining gaps in the full classification of non-degenerate planar traveling waves of scalar balance laws from the point of view of spectral and nonlinear stability/instability under smooth perturbations. On one hand we investigate the impact on the classification of allowing to restrict perturbations by further localization constraints, the so-called convective stability analysis. On the other hand, in the multidimensional case we extend the existing analysis to general perturbations, relaxing the artificial restriction, assumed so far, that initial perturbations are supported away from characteristic points. A striking feature of the latter analysis is that it involves infinite-dimensional families of genuinely multidimensional traveling waves that live near the perturbed planar traveling wave.

math.AP↗

arXiv2021

Nonlinear convective stability of a critical pulled front undergoing a Turing bifurcation at its back: a case study

We investigate a specific reaction-diffusion system that admits a monostable pulled front propagating at constant critical speed. When a small parameter changes sign, the stable equilibrium behind the front destabilizes, due to essential spectrum crossing the imaginary axis, causing a Turing bifurcation. Despite both equilibrium states are unstable, the front continues to exist, and is shown to be asymptotically stable, against suitably-localized perturbations, with algebraic temporal decay rate $t^{-3/2}$. To obtain such decay, we rely on point-wise semigroup estimates, and show that the Turing pattern behind the front remains bounded in time, by use of mode-filters.

math.AP↗

arXiv2021

Spectral stability of the critical front in the extended Fisher-KPP equation

We revisit the existence and stability of the critical front in the extended Fisher-KPP equation, refining earlier results of Rottschäfer and Wayne [28] which establish stability of fronts without identifying a precise decay rate. We verify that the front is marginally spectrally stable: while the essential spectrum touches the imaginary axis at the origin, there are no unstable eigenvalues and no eigenvalue (or resonance) embedded in the essential spectrum at the origin. Together with the recent work of Avery and Scheel [3], this implies nonlinear stability of the critical front with sharp $t^{-3/2}$ decay rate, as previously obtained in the classical Fisher-KPP equation. The main challenges are to regularize the singular perturbation in the extended Fisher-KPP equation and to track eigenvalues near the essential spectrum, and we overcome these difficulties with functional analytic methods.

math.AP↗