arXiv subjects

Jeremie Szeftel

Publications and source records attributed to Jeremie Szeftel.

27 records · Page 2Linked to original sources

arXiv2012

Parametrix for wave equations on a rough background I: regularity of the phase at initial time

This is the first of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved jointly with S. Klainerman and I. Rodnianski in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

math.AP↗

arXiv2012

Parametrix for wave equations on a rough background II: construction and control at initial time

This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

math.AP↗

arXiv2012

Parametrix for wave equations on a rough background III: space-time regularity of the phase

This is the third of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

math.AP↗

arXiv2012

Parametrix for wave equations on a rough background IV: control of the error term

This is the last of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

math.AP↗

arXiv2012

On collapsing ring blow up solutions to the mass supercritical NLS

We consider the nonlinear Schrödinger equation $i\partial_tu+Δu+u|u|^{p-1}=0$ in dimension $N\geq 2$ and in the mass super critical and energy subcritical range $1+\frac 4N<p<\min\{\frac{N+2}{N-2},5\}.$ For initial data $u_0\in H^1$ with radial symmetry, we prove a universal upper bound on the blow up speed. We then prove that this bound is sharp and attained on a family of collapsing ring blow up solutions first formally predicted by Gavish, Fibich and Wang.

math.AP↗

arXiv2010

The instability of Bourgain-Wang solutions for the L^2 critical NLS

We consider the two dimensional $L^2$ critical nonlinear Schrödinger equation $i\pa_tu+Δu+u|u|^2=0$. In the pioneering work \cite{BW}, Bourgain and Wang have constructed smooth solutions which blow up in finite time $T<+\infty$ with the pseudo conformal speed $$\|\nabla u(t)\|_{L^2}\sim \frac{1}{T-t},$$ and which display some decoupling between the regular and the singular part of the solution at blow up time. We prove that this dynamic is unstable. More precisely, we show that any such solution with small super critical $L^2$ mass lies on the boundary of both $H^1$ open sets of global solutions that scatter forward and backwards in time, and solutions that blow up in finite time on the right in the log-log regime exhibited in \cite{MR1}, \cite{MR4}, \cite{R1}. We moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in \cite{Mcpam} in the critical mass case

math.AP↗

arXiv2010

Existence and uniqueness of minimal blow up solutions to an inhomogeneous mass critical NLS

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: $i\partial_tu+Δu+k(x)|u|^{2}u=0$. From standard argument, there exists a threshold $M_k>0$ such that $H^1$ solutions with $\|u\|_{L^2} M_k$. In this paper, we consider the dynamics at threshold $\|u_0\|_{L^2}=M_k$ and give a necessary and sufficient condition on $k$ to ensure the existence of critical mass finite time blow up elements. Moreover, we give a complete classification in the energy class of the minimal finite time blow up elements at a non degenerate point, hence extending the pioneering work by Merle who treated the pseudo conformal invariant case $k\equiv 1$.

math.AP↗

arXiv2009

Stable self similar blow up dynamics for slightly L^2 supercritical NLS equations

We consider the focusing nonlinear Schrödinger equations $i\partial_t u+Δu +u|u|^{p-1}=0$ in dimension $1\leq N\leq 5$ and for slightly $L^2$ supercritical nonlinearities $p_c<p<(1+\e)p_c$ with $p_c=1+\frac{4}{N}$ and $0<\e\ll 1$. We prove the existence and stability in the energy space $H^1$ of a self similar finite time blow up dynamics and provide a qualitative description of the singularity formation near the blow up time

math.AP↗

arXiv2005

Almost global existence for Hamiltonian semi-linear Klein-Gordon equations with small Cauchy data on Zoll manifolds

This paper is devoted to the proof of almost global existence results for Klein-Gordon equations on Zoll manifolds (e.g. spheres of arbitrary dimension) with Hamiltonian nonlinearities, when the Cauchy data are smooth and small. The proof relies on Birkhoff normal form methods and on the specific distribution of eigenvalues of the laplacian perturbed by a potential on Zoll manifolds.

math.DS↗