Benoit Pausader
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Benoit Pausader has a total of 21 co-authors. Here are the top co-authors by number of shared publications:
- Alexandru D. Ionescu
Princeton University, Princeton, USA
7 shared publications - Huy Q. Nguyen
Department of Mathematics, Brown University, Providence, USA
3 shared publications - Fabio Pusateri
Princeton University
2 shared publications - Yu Deng
Courant Institute of Mathematical Sciences, New York University, New York, NY, USA
2 shared publications Alexandru D Ionescu
Princeton University , NJ 08544, USA
1 shared publication- Alexandru D. Ionescu
University of Wisconsin–Madison
1 shared publication
Latest Publications
- Open Access
- 30 July 2021
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (12) , 8865-8889
On the Asymptotic Behavior of Solutions to the Vlasov–Poisson System
We prove small data modified scattering for the Vlasov–Poisson system in dimension $d=3$, using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamics related to the scattering mass.- 30 July 2021
- Published by Oxford University Press (OUP) in International Mathematics Research Notices
- Vol. 2022 (12) , 8865-8889
Open AccessOn the Asymptotic Behavior of Solutions to the Vlasov–Poisson System
We prove small data modified scattering for the Vlasov–Poisson system in dimension $d=3$, using a method inspired from dispersive analysis. In particular, we identify a simple asymptotic dynamics related to the scattering mass. - Open Access
The profile decomposition for the hyperbolic Schrödinger equation
In this note, we prove the profile decomposition for hyperbolic Schr\"odinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot H}^{\frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature.Open AccessThe profile decomposition for the hyperbolic Schrödinger equation
In this note, we prove the profile decomposition for hyperbolic Schr\"odinger (or mixed signature) equations on $\mathbb{R}^2$ in two cases, one mass-supercritical and one mass-critical. First, as a warm up, we show that the profile decomposition works for the ${\dot H}^{\frac12}$ critical problem, which gives a simple generalization of for instance one of the results in Fanelli-Visciglia (2013). Then, we give the derivation of the profile decomposition in the mass-critical case by proving an improved Strichartz estimate. We will use a very similar approach to that laid out in the notes of Killip-Visan (2008), but we are forced to do a double Whitney decomposition to accommodate an extra scaling symmetry that arises in the problem with mixed signature. - Open Access
- 17 March 2020
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 237 (1) , 101
Publisher Correction to: A Paradifferential Approach for Well-Posedness of the Muskat Problem
- 17 March 2020
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 237 (1) , 101
Open AccessPublisher Correction to: A Paradifferential Approach for Well-Posedness of the Muskat Problem
- 21 February 2020
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 237 (1) , 35-100
A Paradifferential Approach for Well-Posedness of the Muskat Problem
- 21 February 2020
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 237 (1) , 35-100
A Paradifferential Approach for Well-Posedness of the Muskat Problem
- Open Access
A paradifferential approach for well-posedness of the Muskat problem
We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh-Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet-Neumann operator in rough domains.Open AccessA paradifferential approach for well-posedness of the Muskat problem
We study the Muskat problem for one fluid or two fluids, with or without viscosity jump, with or without rigid boundaries, and in arbitrary space dimension $d$ of the interface. The Muskat problem is scaling invariant in the Sobolev space $H^{s_c}(\mathbb{R}^d)$ where $s_c=1+\frac{d}{2}$. Employing a paradifferential approach, we prove local well-posedness for large data in any subcritical Sobolev spaces $H^s(\mathbb{R}^d)$, $s>s_c$. Moreover, the rigid boundaries are only required to be Lipschitz and can have arbitrarily large variation. The Rayleigh-Taylor stability condition is assumed for the case of two fluids with viscosity jump but is proved to be automatically satisfied for the case of one fluid. The starting point of this work is a reformulation solely in terms of the Drichlet-Neumann operator. The key elements of proofs are new paralinearization and contraction results for the Drichlet-Neumann operator in rough domains. - 23 September 2019
- Published by American Mathematical Society (AMS) in Quarterly of Applied Mathematics
- Vol. 78 (2) , 305-332
Derivation of the ion equation
We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion m e / M i ≪ 1 m_e/M_i\ll 1 is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on R 3 \mathbb {R}^3 , the validity of such an expansion is established even with “ill-prepared” Cauchy data, by including an additional initial layer correction.
- 23 September 2019
- Published by American Mathematical Society (AMS) in Quarterly of Applied Mathematics
- Vol. 78 (2) , 305-332
Derivation of the ion equation
We consider the classical Euler-Poisson system for electrons and ions, interacting through an electrostatic field. The mass ratio of an electron and an ion m e / M i ≪ 1 m_e/M_i\ll 1 is small and we establish an asymptotic expansion of solutions, where the main term is obtained from a solution to a self-consistent equation involving only the ion variables. Moreover, on R 3 \mathbb {R}^3 , the validity of such an expansion is established even with “ill-prepared” Cauchy data, by including an additional initial layer correction.
- 19 September 2019
- Published by American Mathematical Society (AMS) in Quarterly of Applied Mathematics
- Vol. 78 (2) , 277-303
Global regularity of solutions of the Einstein-Klein-Gordon system: A review
In this article we consider the Einstein field equations of General Relativity for self-gravitating massive scalar fields (the Einstein-Klein-Gordon system). Our goal is to review the main results and ideas in our work [The Einstein-Klein-Gordon coupled system: Global stability of the Minkowski solution, preprint (2019)] on the global regularity, modified scattering, and asymptotic analysis of solutions of this system with initial data in a small neighborhood of the Minkowski space-time.
- 19 September 2019
- Published by American Mathematical Society (AMS) in Quarterly of Applied Mathematics
- Vol. 78 (2) , 277-303
Global regularity of solutions of the Einstein-Klein-Gordon system: A review
In this article we consider the Einstein field equations of General Relativity for self-gravitating massive scalar fields (the Einstein-Klein-Gordon system). Our goal is to review the main results and ideas in our work [The Einstein-Klein-Gordon coupled system: Global stability of the Minkowski solution, preprint (2019)] on the global regularity, modified scattering, and asymptotic analysis of solutions of this system with initial data in a small neighborhood of the Minkowski space-time.
- Open Access
Global solutions of the gravity-capillary water-wave system in three dimensions
In this paper we prove global regularity for the full water waves system in 3 dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models. To construct global solutions we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than $|t|^{-5/6}$) and the presence of a large, codimension 1, set of quadratic time-resonances. To deal with such a situation we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the nonlinearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions we use Harmonic Analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.
Open AccessGlobal solutions of the gravity-capillary water-wave system in three dimensions
In this paper we prove global regularity for the full water waves system in 3 dimensions for small data, under the influence of both gravity and surface tension. This problem presents essential difficulties which were absent in all of the earlier global regularity results for other water wave models. To construct global solutions we use a combination of energy estimates and matching dispersive estimates. There is a significant new difficulty in proving energy estimates in our problem, namely the combination of slow pointwise decay of solutions (no better than $|t|^{-5/6}$) and the presence of a large, codimension 1, set of quadratic time-resonances. To deal with such a situation we propose here a new mechanism, which exploits a non-degeneracy property of the time-resonant hypersurfaces and some special structure of the quadratic part of the nonlinearity, connected to the conserved energy of the system. The dispersive estimates rely on analysis of the Duhamel formula in the Fourier space. The main contributions come from the set of space-time resonances, which is a large set of dimension 1. To control the corresponding bilinear interactions we use Harmonic Analysis techniques, such as orthogonality arguments in the Fourier space and atomic decompositions of functions. Most importantly, we construct and use a refined norm which is well adapted to the geometry of the problem.
- 13 April 2017
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 225 (2) , 771-871
The Euler–Maxwell System for Electrons: Global Solutions in 2D
- 13 April 2017
- Published by Springer Nature in Archive for Rational Mechanics and Analysis
- Vol. 225 (2) , 771-871
The Euler–Maxwell System for Electrons: Global Solutions in 2D
- Open Access
- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 37 (1) , 281-293
Discrete Schrödinger equation and ill-posedness for the Euler equation
We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by $U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$ can be well approximated by the linear Schrödinger equation, at least for a short period of time.- 1 January 2017
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 37 (1) , 281-293
Open AccessDiscrete Schrödinger equation and ill-posedness for the Euler equation
We consider the 2D Euler equation with periodic boundary conditions in a family of Banach spaces based on the Fourier coefficients, and show that it is ill-posed in the sense that 'norm inflation' occurs. The proof is based on the observation that the evolution of certain perturbations of the 'Kolmogorov flow' given in velocity by $U(x,y) = \left( {\begin{array}{*{20}{c}}{\cos \;y}\\0\end{array}} \right)$ can be well approximated by the linear Schrödinger equation, at least for a short period of time.