The paper provides a description of the wave map problem with a specific focus on the breakthrough work of T. Tao which showed that a wave map, a dynamic lorentzian analog of a harmonic map, from Minkowski space into a sphere with smooth initial data and a small critical Sobolev norm exists globally in time and remains smooth. When the dimension of the base Minkowski space is $(2+1)$, the critical norm coincides with energy, the only manifestly conserved quantity in this (lagrangian) theory. As a consequence,...

We review recent results concerning the study of rough solutions to the initial value problem for the Einstein vacuum equations expressed relative to wave coordinates. We develop new analytic methods based on Strichartz type inequalities which results in a gain of half a derivative relative to the classical result. Our methods blend paradifferential techniques with a geometric approach to the derivation of decay estimates. The latter allows us to take full advantage of the specific structure of...

This note summarizes the results obtained in []. We exhibit stable finite time blow up regimes for the energy critical co-rotational Wave Map with the ${\mathbb{S}}^{2}$ target in all homotopy classes and for the equivariant critical $SO\left(4\right)$ Yang-Mills problem. We derive sharp asymptotics on the dynamics at blow up time and prove quantization of the energy focused at the singularity.

In this article, we study small perturbations of the family of Friedmann-Lemaître-Robertson-Walker cosmological background solutions to the coupled Euler-Einstein system with a positive cosmological constant in $1+3$ spacetime dimensions. The background solutions model an initially uniform quiet fluid of positive energy density evolving in a spacetime undergoing exponentially accelerated expansion. Our nonlinear analysis shows that under the equation of state $p={c}^{2}\rho ,0<{c}^{2}<1/3$, the background metric + fluid solutions...

We provide ${L}^{1}$ estimates for a transport equation which contains singular integral operators. The form of the equation was motivated by the study of Kirchhoff–Sobolev parametrices
in a Lorentzian space-time satisfying the Einstein equations. While our main application is for a specific problem in General Relativity we believe that the phenomenon which our result illustrates
is of a more general interest.

We report on recent progress obtained on the construction and control of a parametrix to the homogeneous wave equation ${\square}_{\mathbf{g}}\phi =0$, where $\gg $ 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 $\mathbf{R}$ of $\gg $ is a major step towards the proof of the bounded ${L}^{2}$ curvature conjecture.

This paper reports on the recent proof of the bounded ${L}^{2}$ curvature conjecture. More precisely we show that the time of existence of a classical solution to the Einstein-vacuum equations depends only on the ${L}^{2}$-norm of the curvature and a lower bound of the volume radius of the corresponding initial data set.

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