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Tristan Roy

Publications and source records attributed to Tristan Roy.

At least 19 recordsLinked to original sources
arXiv2025

Classification of radial solutions of energy-critical wave systems

This work concerns a general system of energy-critical wave equations in the Minkowski space of dimension $1+3$. The wave equations are coupled by the nonlinearities, which are homogeneous of degree 5. We prove that any radial solution of the system can be written asymptotically as a sum of rescaled stationary solutions plus a radiation term, along any sequence of times for which the solution is bounded in the energy space. With an additional structural assumption on the nonlinearity, we prove a continuous in time resolution result for radial solutions. The proof of the sequential resolution uses the channel of energy method, as in the scalar case treated by Duyckaerts, Kenig and Merle (Cambridge Journal of Mathematics 2013 and arXiv 1204.0031). The proof of the continous in time resolution is based on new compactness and localization arguments.

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arXiv2023

The blow-up rate for a loglog non-scaling invariant semilinear wave equation

We consider blow-up solutions of a semilinear wave equation with a loglog perturbation of the power nonlinearity in the subconformal case, and show that the blow-up rate is given by the solution of the associated ODE which has the same blow-up time. In fact, our result shows an upper bound and a lower bound of the blow-up rate, both proportional to the blow-up solution of the associated ODE. The main difficulty comes from the fact that the PDE is not scaling invariant.

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arXiv2022

Scattering above energy norm of a focusing size-dependent log energy-supercritical Schrodinger equation with radial data below ground state

Given $n \in \{ 3,4,5 \}$ and $k > 1$ (resp. $\frac{4}{3} > k > 1$) if $n \in \{ 3,4 \}$ (resp. $n=5$), we prove scattering of the radial $\tilde{H}^{k}:= \dot{H}^{k}(\mathbb{R}^{n}) \cap \dot{H}^{1}(\mathbb{R}^{n})$ solutions of a focusing size-dependent log energy-supercritical Schrodinger equation for critical energies below that of the ground states, and for critical potential energies below that of the ground states.

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arXiv2022

Local well-posedness and parabolic smoothing of solutions of fully nonlinear third-order equations on the torus

We study the initial value problem of fully nonlinear third-order equations on the torus. Under some conditions on the nonlinearity and the data, we prove that the equation behaves like a parabolic one: there exists a unique local solution in one direction of time that is infinitely smooth and the problem in not well-posed in the other direction. Under other conditions on the nonlinearity and the data, we prove that the equation behaves like a dispersive one: there exists a unique local solution in both directions of time.

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arXiv2021

Global existence of solutions of a loglog energy-supercritical Klein-Gordon equation

We prove global existence of solutions of a loglog energy-supercritical Klein-Gordon equation for n=3,4,5. Assuming that blow-up occurs at a time of maximal existence, we perform an analysis close to this time in order to find a finite bound of a Strichartz-type norm, which eventually leads to a contraction with the blow-up assumption.

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arXiv2020

On Jensen-type inequalities for nonsmooth radial scattering solutions of a loglog energy-supercritical Schrodinger equation

Given $n \in \{ 3,4 \}$ (resp. $n=5$) and $k > 1$ (resp. $\frac{4}{3} > k > 1$), we prove scattering of the radial $\tilde{H}^{k}:= \dot{H}^{k}(\mathbb{R}^{n}) \cap \dot{H}^{1}(\mathbb{R}^{n})$ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}}u log^γ (\log( 10 + |u|^{2} ))$ for $0 < γ< γ_{n}$. In order to control the barely supercritical nonlinearity for nonsmooth solutions, i.e solutions with data in $\tilde{H}^{k}$, $k \leq \frac{n}{2}$, we prove some Jensen-type inequalities.

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arXiv2018

Scattering above energy norm of solutions of a loglog energy-supercritical Schrodinger equation with radial data

We prove scattering of $\tilde{H}^{k} $ solutions of the loglog energy-supercritical Schrodinger equation $i \partial_{t} u + \triangle u = |u|^{\frac{4}{n-2}} u \log^{c} {(\log{(10+|u|^{2})})}$, $0 < c < c_{n}$, $n={3,4}$, with radial data $u(0):=u_{0} \in \tilde{H}^{k} $, $k>n/2$. This is achieved, roughly speaking, by extending Bourgain's argument (see also Grillakis) and Tao's argument in high dimensions.

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arXiv2017

A weak form of the soliton resolution conjecture for high-dimensional fourth-order Schrodinger equations

We prove a weak form of the soliton resolution conjecture of bounded solutions of high-dimensional fourth-order Schrodinger equations. The result relies upon two properties to be proved: the asymptotic frequency localization and the asymptotic spatial localization. In order to prove the asymptotic frequency localization we use the fact that the high frequency pieces of the free solution have better dispersive properties than the lower ones. In order to prove the asymptotic spatial localization, we use the symmetries of the phase of the fundamental solution.

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arXiv2016

On Control Of Sobolev Norms For Some Semilinear Wave Equations With Localized Data

We establish new bounds of the Sobolev norms of solutions of semilinear wave equations for data lying in the Hs, s<1, closure of compactly supported data inside a ball of radius R, with R a fixed and positive number. In order to do that we perform an analysis in the neighborhood of the cone, using an almost Shatah-Struwe estimate, an almost conservation law and some estimates for localized functions: this allows to prove a decay estimate and establish a low frequency estimate of the position of the solution. Then, in order to establish a high frequency estimate of the position and an estimate of the velocity, we use this decay estimate and another almost conservation law.

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arXiv2016

Introduction to scattering for radial 3D NLKG below energy norm

We prove scattering for the radial nonlinear Klein-Gordon equation $ \partial_{tt} u - Δu + u = -|u|^{p-1} u $ with $5 > p >3$ and data $ (u_{0}, u_{1}) \in H^{s} \times H^{s-1} $, $ 1 > s > 1- \frac{(5-p)(p-3)}{2(p-1)(p-2)} $ if $ 4 \geq p > 3 $ and $ 1 > s > 1 - \frac{(5-p)^{2}}{2(p-1)(6-p)}$ if $ 5> p \geq 4$. First we prove Strichartz-type estimates in $ L_{t}^{q} L_{x}^{r} $ spaces. Then by using these decays we establish some local bounds. By combining these results with a Morawetz-type estimate and a radial Sobolev inequality we control the variation of an almost conserved quantity on arbitrarily large intervals. Once we have showed that this quantity is controlled, we prove that some of these local bounds can be upgraded to global bounds. This is enough to establish scattering. All the estimates involved require a delicate analysis due to the nature of the nonlinearity and the lack of scaling.

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arXiv2016

On the interpolation with the potential bound for global solutions of the defocusing cubic wave equation on T2

We prove that the solutions of the defocusing cubic wave equation on T2 exist globally in time in Hs(T2) for s > 2/5 by contradiction. Assuming that one of the maximal times of existence is finite, we prove that the Sobolev norm of each of these solutions is bounded in an open neighborhood of it by estimating the growth of a mollified energy through the interpolation with the potential bound for the low frequency part.

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arXiv2015

Global dynamics above the ground state for the energy-critical Schrodinger equation with radial data

Consider the focusing energy critical Schrodinger equation in three space dimensions with radial initial data in the energy space. We describe the global dynamics of all the solutions of which the energy is at most slightly larger than that of the ground states, according to whether it stays in a neighborhood of them, blows up in finite time or scatters. In analogy with the paper by Schlag and the first author on the subcritical equation, the proof uses an analysis of the hyperbolic dynamics near them and the variational structure far from them. The key step that allows to classify the solutions is the one-pass lemma. The main difference from the subcritical case is that one has to introduce a scaling parameter in order to describe the dynamics near them. One has to take into account this parameter in the analysis around the ground states by introducing some orthogonality conditions. One also has to take it into account in the proof of the one-pass lemma by comparing the contribution in the variational region and in the hyperbolic region.

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arXiv2015

Blow-up of the critical Sobolev norm for nonscattering radial solutions of supercritical wave equations on $\mathbb{R}^{3}$

We consider the wave equation in space dimension $3$, with an energy-supercritical nonlinearity which can be either focusing or defocusing. For any radial solution of the equation, with positive maximal time of existence $T$, we prove that one of the following holds: (i) the norm of the solution in the critical Sobolev space goes to infinity as $t$ goes to $T$, or (ii) $T$ is infinite and the solution scatters to a linear solution forward in time. We use a variant of the channel of energy method, relying on a generalized $L^p$-energy which is almost conserved by the flow of the radial linear wave equation.

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arXiv2011

Bilinear local smoothing estimate for Airy equation

In this short note, we prove a refinement of bilinear local smoothing estimate to Airy solutions, when the frequency support of two wave are separated. As an application we prove a smoothing property of a bilinear form.

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arXiv2010

Global well-posedness of the Maxwell-Klein-Gordon equation below the energy norm

We show that the Maxwell-Klein-Gordon equations in three dimensions are globally well-posed in $H^s_x$ in the Coulomb gauge for all $s > \sqrt{3}/2 \approx 0.866$. This extends previous work of Klainerman-Machedon \cite{kl-mac:mkg} on finite energy data $s \geq 1$, and Eardley-Moncrief \cite{eardley} for still smoother data. We use the method of almost conservation laws, sometimes called the "I-method", to construct an almost conserved quantity based on the Hamiltonian, but at the regularity of $H^s_x$ rather than $H^1_x$. One then uses Strichartz, null form, and commutator estimates to control the development of this quantity. The main technical difficulty (compared with other applications of the method of almost conservation laws) is at low frequencies, because of the poor control on the $L^2_x$ norm. In an appendix, we demonstrate the equations' relative lack of smoothing - a property that presents serious difficulties for studying rough solutions using other known methods.

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