1,720,989 research outputs found
Sharp Strichartz estimates for the wave equation on a rough background
No full textInternational audienceIn this paper, we obtain sharp Strichartz estimates for solutions of the wave equation ◻≫ϕ=0 where ≫ is a rough Lorentzian metric on a 4 dimensional space-time \MM. This is the last step of the proof of the bounded L2 curvature conjecture proposed in [3], and solved by S. Klainerman, I. Rodnianski and the author in [8], which also relies on the sequence of papers [16][17][18][19]. Obtaining such estimates is at the core of the low regularity well-posedness theory for quasilinear wave equations. The difficulty is intimately connected to the regularity of the Eikonal equation \gg^{\a\b}\pr_\a u\pr_\b u=0 for a rough metric ≫. In order to be consistent with the final goal of proving the bounded L2 curvature conjecture, we prove Strichartz estimates for all admissible Strichartz pairs under minimal regularity assumptions on the solutions of the Eikonal equation
Parametrix for wave equations on a rough background III : space-time regularity of the phase
No full textInternational audienceThis is the third of a sequence of four papers [21], [22], [23], [24] dedicated to the construction and the control of a parametrix to the homogeneous wave equation g φ = 0, where 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 R of g is a major step of the proof of the bounded L 2 curvature conjecture proposed in [12], and solved by S. Klainerman, I. Rodnianski and the author in [17]. 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
The bounded L2 curvature conjecture
No full textInternational audienceThis is the main paper in a sequence in which we give a complete proof of the bounded L2 curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the L2-norm of the curvature and a lower bound on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the standard scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, L2 bounds on the curvature is the minimum requirement necessary to obtain lower bounds on the radius of injectivity of causal boundaries. We note also that, while the first nontrivial improvements for well posedness for quasilinear hyperbolic systems in spacetime dimensions greater than 1+1 (based on Strichartz estimates) were obtained in [Ba-Ch1] [Ba-Ch2] [Ta1] [Ta2] [Kl-R1] and optimized in [Kl-R2] [Sm-Ta], the result we present here is the first in which the full structure of the quasilinear hyperbolic system, not just its principal part, plays a crucial role. To achieve our goals we recast the Einstein vacuum equations as a quasilinear so(3,1)-valued Yang-Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of \textit{null structure} compatible with the quasilinear, covariant nature of the equations. To prove the conjecture we formulate and establish bilinear and trilinear estimates on rough backgrounds which allow us to make use of that crucial structure. These require a careful construction and control of parametrices including L2 error bounds which is carried out in [Sz1]-[Sz4], as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in \cite{Sz5}
On collapsing ring blow up solutions to the mass supercritical NLS
No full textNon UBCUnreviewedAuthor affiliation: Ecole Normal SuperieureFacult
Going Beyond Counting First Authors in Author Co-citation Analysis
Get PDFThe present study examines one of the fundamental aspects of author co-citation analysis (ACA) - the way co-citation
counts are defined. Co-citation counting provides the data on which all subsequent statistical analyses and mappings
are based, and we compare ACA results based on two different types of co-citation counting - the traditional type that
only counts the first one among a cited work's authors on the one hand and a non-traditional type that takes into
account the first 5 authors of a cited work on the other hand. Results indicate that the picture produced through this non-traditional author co-citation counting contains more coherent author groups and is therefore considerably clearer. However, this picture represents fewer specialties in the research field being studied than that produced through the traditional first-author co-citation counting when the same number of top-ranked authors is selected and analyzed. Reasons for these effects are discussed
: École d’été 2014 - Analyse asymptotique en relativité générale
No full textIn order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations.This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background
: École d’été 2014 - Analyse asymptotique en relativité générale
No full textIn order to control locally a space-time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space-like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well-posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations.This is joint work with Sergiu Klainerman and Igor Rodnianski. These lectures will start from scratch and require no specific background
: École d’été 2014 - Analyse asymptotique en relativité générale
No full textIn order to control locally a space time which satisfies the Einstein equations, what are the minimal assumptions one should make on its curvature tensor? The bounded L2 curvature conjecture roughly asserts that one should only need L2 bounds of the curvature tensor on a given space like hypersurface. This conjecture has its roots in the remarkable developments of the last twenty years centered around the issue of optimal well posedness for nonlinear wave equations. In this context, a corresponding conjecture for nonlinear wave equations cannot hold, unless the nonlinearity has a very special nonlinear structure. I will present the proof of this conjecture, which sheds light on the specific null structure of the Einstein equations. This is joint work with Sergiu Klainerman and Igor Rodnianski
- …
