Joackim Berniér
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View article: Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity
Almost global existence for some nonlinear Schrödinger equations on $\mathbb{T}^d$ in low regularity Open
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View article: On almost periodic solutions to NLS without external parameters
On almost periodic solutions to NLS without external parameters Open
In this note, we present a result established in [BGR24] where we prove that nonlinear Schrödinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, ple…
View article: Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori
Long time stability for cubic nonlinear Schrödinger equations on non-rectangular flat tori Open
We consider nonlinear Schrödinger equations on flat tori satisfying a simple and explicit Diophantine non-degeneracy condition. Provided that the nonlinearity contains a cubic term, we prove the almost global existence and stability of mos…
View article: On almost periodic solutions to NLS without external parameters
On almost periodic solutions to NLS without external parameters Open
In this note, we present a result established in [BGR24] where we prove that nonlinear Schrodinger equations on the circle, without external parameters, admit plenty of infinite dimensional non resonant invariant tori, or equivalently, ple…
View article: On alternating-conjugate splitting methods
On alternating-conjugate splitting methods Open
The new class of alternating-conjugate splitting methods is presented and analyzed. They are obtained by concatenating a given composition involving complex coefficients with the same composition but with the complex conjugate coefficients…
View article: Almost global existence for Hamiltonian PDEs on compact manifolds
Almost global existence for Hamiltonian PDEs on compact manifolds Open
We prove an abstract result of almost global existence of small solutions to semi-linear Hamiltonian partial differential equations satisfying very weak non resonance conditions and basic multilinear estimates. Thanks to works by Delort--S…
View article: Infinite dimensional invariant tori for nonlinear Schr\"odinger equations
Infinite dimensional invariant tori for nonlinear Schr\"odinger equations Open
We prove that nonlinear Schr\"odinger equations on the circle, without external parameters, admits plenty of almost periodic solutions. Indeed, we prove that arbitrarily close to most of the finite dimensional KAM tori constructed by Kuksi…
View article: Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity
Almost conservation of the harmonic actions for fully discretized nonlinear Klein--Gordon equations at low regularity Open
Close to the origin, the nonlinear Klein--Gordon equations on the circle are nearly integrable Hamiltonian systems which have infinitely many almost conserved quantities called harmonic actions or super-actions. We prove that, at low regul…
View article: Exponential stability of solutions to the Schrödinger–Poisson equation
Exponential stability of solutions to the Schrödinger–Poisson equation Open
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View article: Exponential stability of solutions to the Schr{ö}dinger-Poisson equation
Exponential stability of solutions to the Schr{ö}dinger-Poisson equation Open
We prove an exponential stability result for the small solutions of the Schr{ö}dinger-Poisson equation on the circle without exterior parameters in Gevrey class. More precisely we prove that for most of the initial data of Gevrey-norm smal…
View article: Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$
Dynamics of quintic nonlinear Schr{ö}dinger equations in $H^{2/5+}(\mathbb{T})$ Open
In this paper, we succeed in integrating Strichartz estimates (encoding the dispersive effects of the equations) in Birkhoff normal form techniques. As a consequence, we deduce a result on the long time behavior of quintic NLS solutions on…
View article: Symmetric-conjugate splitting methods for linear unitary problems
Symmetric-conjugate splitting methods for linear unitary problems Open
We analyze the preservation properties of a family of reversible splitting methods when they are applied to the numerical time integration of linear differential equations defined in the unitary group. The schemes involve complex coefficie…
View article: Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects
Polar decomposition of semigroups generated by non-selfadjoint quadratic differential operators and regularizing effects Open
We characterize geometrically the regularizing effects of the semigroups\ngenerated by accretive non-selfadjoint quadratic differential operators. As a\nbyproduct, we establish the subelliptic estimates enjoyed by these operators,\nbeing e…
View article: Dynamics of nonlinear Klein–Gordon equations in low regularity on $\mathbb{S}^2$
Dynamics of nonlinear Klein–Gordon equations in low regularity on $\mathbb{S}^2$ Open
We describe the long-time behavior of small nonsmooth solutions to the nonlinear Klein–Gordon equations on the sphere \mathbb{S}^2 . More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved f…
View article: Birkhoff normal forms for Hamiltonian PDEs in their energy space
Birkhoff normal forms for Hamiltonian PDEs in their energy space Open
We study the long time behavior of small solutions of semi-linear dispersive Hamiltonian partial differential equations on confined domains. Provided that the system enjoys a new non-resonance condition and a sufficiently strong energy est…
View article: Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity
Almost global existence for some nonlinear Schr{ö}dinger equations on $\mathbb{T}^d$ in low regularity Open
We are interested in the long time behavior of solutions of the nonlinear Schr{ö}dinger equation on the $d$-dimensional torus in low regularity, i.e. for small initial data in the Sobolev space $H^{s_0}(\mathbb T^d)$ with $s_0>d/2$. We pro…
View article: Gains of integrability and local smoothing effects for quadratic\n evolution equations
Gains of integrability and local smoothing effects for quadratic\n evolution equations Open
We characterize geometrically the semigroups generated by non-selfadjoint\nquadratic differential operators $(e^{-tq^w})_{t\\geq 0}$ enjoying local\nsmoothing effects and providing gains of integrability. More precisely, we\nprove that the…
View article: Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2
Dynamics of nonlinear Klein-Gordon equations in low regularity on S^2 Open
We describe the long time behavior of small non-smooth solutions to the nonlinear Klein-Gordon equations on the sphere S^2. More precisely, we prove that the low harmonic energies (also called super-actions) are almost preserved for times …
View article: A note on some microlocal estimates used to prove the convergence of splitting methods relying on pseudo-spectral discretizations
A note on some microlocal estimates used to prove the convergence of splitting methods relying on pseudo-spectral discretizations Open
In [BCC20], we used some classical microlocal estimates to prove the convergence of our splitting methods (for example page A671). In this note, through Corollary 2 and Remark 1, we provide a detailed proof of these estimates. All the proo…
View article: Splitting Methods for Rotations: Application to Vlasov Equations
Splitting Methods for Rotations: Application to Vlasov Equations Open
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LONG TIME BEHAVIOR OF THE SOLUTIONS OF NLW ON THE -DIMENSIONAL TORUS Open
We consider the nonlinear wave equation (NLW) on the $d$ -dimensional torus $\mathbb{T}^{d}$ with a smooth nonlinearity of order at least 2 at the origin. We prove that, for almost any mass, small and smooth solutions of high Sobolev indic…
View article: Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators
Exact splitting methods for semigroups generated by inhomogeneous quadratic differential operators Open
We introduce some general tools to design exact splitting methods to compute numerically semigroups generated by inhomogeneous quadratic differential operators. More precisely, we factorize these semigroups as products of semigroups that c…
View article: Exact splitting methods for kinetic and Schr{ö}dinger equations
Exact splitting methods for kinetic and Schr{ö}dinger equations Open
In [8], some exact splittings are proposed for inhomogeneous quadratic differential equations including, for example, transport equations, kinetic equations, and Schr{ö}dinger type equations with a rotation term. In this work, these exact …