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We consider the energy supercritical defocusing nonlinear Schrödinger equation i ∂ t u + Δ u - u | u | p - 1 = 0 in dimension d ≥ 5 . In a suitable range of energy supercritical parameters ( d ,  p ), we prove the existence of C ∞ well localized spherically symmetric initial data such that the corresponding unique strong solution blows up in finite time. Unlike other known blow up mechanisms, the singularity formation does not occur by concentration of a soliton or through a self similar solution, which are unknown in the defocusing case, but via a front mechanism . Blow up is achieved by compression for the associated hydrodynamical flow which in turn produces a highly oscillatory singularity. The front blow up profile is chosen among the countable family of C ∞ spherically symmetric self similar solutions to the compressible Euler equation whose existence and properties in a suitable range of parameters are established in the companion paper (Merle et al. in Preprint (2019)) under a non degeneracy condition which is checked numerically.

We consider the energy critical Schrödinger map problem with the 2-sphere target for equivariant initial data of homotopy index k =1. We show the existence of a codimension one set of smooth well localized initial data arbitrarily close to the ground state harmonic map in the energy critical norm, which generates finite time blowup solutions. We give a sharp description of the corresponding singularity formation which occurs by concentration of a universal bubble of energy.

A well-known open problem in general relativity, dating back to 1972, has been to prove Price's law for an appropriate model of gravitational collapse. This law postulates inverse-power decay rates for the gravitational radiation flux through the event horizon and null infinity with respect to appropriately normalized advanced and retarded time coordinates. It is intimately related both to astrophysical observations of black holes and to the fate of observers who dare cross the event horizon. In this paper, we prove a well-defined (upper bound) formulation of Price's law for the collapse of a self-gravitating scalar field with spherically symmetric initial data. We also allow the presence of an additional gravitationally coupled Maxwell field. Our results are obtained by a new mathematical technique for understanding the long-time behavior of large data solutions to the resulting coupled non-linear hyperbolic system of p.d.e.'s in 2 independent variables. The technique is based on the interaction of the conformal geometry, the celebrated red-shift effect, and local energy conservation; we feel it may be relevant for the problem of non-linear stability of the Kerr solution. When combined with previous work of the first author concerning the internal structure of charged black holes, which had assumed the validity of Price's law, our results can be applied to the strong cosmic censorship conjecture for the Einstein-Maxwell-real scalar field system with complete spacelike asymptotically flat spherically symmetric initial data. Under Christodoulou's C^sup 0^-formulation, the conjecture is proven to be false. [PUBLICATION ABSTRACT]

We linearize the Einstein-scalar field equations, expressed relative to constant mean curvature (CMC)-transported spatial coordinates gauge, around members of the well-known family of Kasner solutions on (0, ∞) × 𝕋³. The Kasner solutions model a spatially uniform scalar field evolving in a (typically) spatially anisotropic spacetime that expands towards the future and that has a \"Big Bang\" singularity at {t = 0}. We place initial data for the linearized system along {t = 1} ≃ 𝕋³ and study the linear solution's behavior in the collapsing direction t ↓ 0. Our first main result is the proof of an approximate L² monotonicity identity for the linear solutions. Using it, we prove a linear stability result that holds when the background Kasner solution is sufficiently close to the Friedmann-Lemaître-Robertson-Walker (FLRW) solution. In particular, we show that as t ↓ 0, various time-rescaled components of the linear solution converge to regular functions defined along {t = 0}. In addition, we motivate the preferred direction of the approximate monotonicity by showing that the CMC-transported spatial coordinates gauge can be viewed as a limiting version of a family of parabolic gauges for the lapse variable; an approximate monotonicity identity and corresponding linear stability results also hold in the parabolic gauges, but the corresponding parabolic PDEs are locally well posed only in the direction t ↓ 0. Finally, based on the linear stability results, we outline a proof of the following result, whose complete proof will appear elsewhere: the FLRW solution is globally nonlinearly stable in the collapsing direction t ↓ 0 under small perturbations of its data at {t = 1}.

We give a new proof of the global stability of Minkowski space originally established in the vacuum case by Christodoulou and Klainerman. The new approach, which relies on the classical harmonic gauge, shows that the Einstein-vacuum and the Einstein-scalar field equations with asymptotically flat initial data satisfying a global smallness condition produce global (causally geodesically complete) solutions asymptotically convergent to the Minkowski space-time.

In the mathematical physics literature, there are heuristic arguments, going back three decades, suggesting that for an open set of initially smooth solutions to the Einstein-vacuum equations in high dimensions, stable, approximately monotonic curvature singularities can dynamically form along a spacelike hypersurface. In this article, we study the Cauchy problem and give a rigorous proof of this phenomenon in sufficiently high dimensions, thereby providing the first constructive proof of stable curvature-blowup (without symmetry assumptions) along a spacelike hypersurface as an effect of pure gravity. Our proof applies to an open subset of regular initial data satisfying the assumptions of Hawking’s celebrated “singularity” theorem, which shows that the solution is geodesically incomplete but does not reveal the nature of the incompleteness. Specifically, our main result is a proof of the dynamic stability of the Kasner curvature singularity for a subset of Kasner solutions whose metrics exhibit only moderately (as opposed to severely) spatially anisotropic behavior. Of independent interest is our method of proof, which is more robust than earlier approaches in that (i) it does not rely on approximate monotonicity identities and (ii) it accommodates the possibility that the solution develops very singular high-order spatial derivatives, whose blowup-rates are allowed to be, within the scope of our bootstrap argument, much worse than those of the base-level quantities driving the fundamental blowup. For these reasons, our approach could be used to obtain similar blowup-results for various Einstein-matter systems in any number of spatial dimensions for solutions corresponding to an open set of moderately spatially anisotropic initial data, thus going beyond the nearly spatially isotropic regime treated in earlier works.

This paper concludes the series begun in [M. Dafermos and I. Rodnianski, Decay for solutions of the wave equation on Kerr exterior spacetimes I–II: the cases |a| ≪ M or axisymmetry, arXiv:1010.5132], providing the complete proof of definitive boundedness and decay results for the scalar wave equation on Kerr backgrounds in the general subextremal |a| < M case without symmetry assumptions. The essential ideas of the proof (together with explicit constructions of the most difficult multiplier currents) have been announced in our survey [M. Dafermos and I. Rodnianski, The black hole stability problem for linear scalar perturbations, in Proceedings of the 12th Marcel Grossmann Meeting on General Relativity, T. Damour et al. (ed.), World Scientific, Singapore, 2011, pp. 132–189, arXiv:1010.5137]. Our proof appeals also to the quantitative mode-stability proven in [Y. Shlapentokh-Rothman, Quantitative Mode Stability for the Wave Equation on the Kerr Spacetime, arXiv:1302.6902, to appear, Ann. Henri Poincaré], together with a streamlined continuity argument in the parameter a, appearing here for the first time. While serving as Part III of a series, this paper repeats all necessary notation so that it can be read independently of previous work.

We consider Kerr spacetimes with parameters a and M such that | a |≪ M , Kerr-Newman spacetimes with parameters | Q |≪ M , | a |≪ M , and more generally, stationary axisymmetric black hole exterior spacetimes which are sufficiently close to a Schwarzschild metric with parameter M >0 and whose Killing fields span the null generator of the event horizon. We show uniform boundedness on the exterior for solutions to the wave equation □ g ψ =0. The most fundamental statement is at the level of energy: We show that given a suitable foliation Σ τ , then there exists a constant C depending only on the parameter M and the choice of the foliation such that for all solutions ψ , a suitable energy flux through Σ τ is bounded by C times the initial energy flux through Σ 0 . This energy flux is positive definite and does not degenerate at the horizon, i.e. it agrees with the energy as measured by a local observer. It is shown that a similar boundedness statement holds for all higher order energies, again without degeneration at the horizon. This leads in particular to the pointwise uniform boundedness of ψ , in terms of a higher order initial energy on Σ 0 . Note that in view of the very general assumptions, the separability properties of the wave equation or geodesic flow on the Kerr background are not used. In fact, the physical mechanism for boundedness uncovered in this paper is independent of the dispersive properties of waves in the high-frequency geometric optics regime.

This is the first in a series of papers in which we initiate the study of very rough solutions to the initial value problem for the Einstein-vacuum equations expressed relative to wave coordinates. By very rough we mean solutions which cannot be constructed by the classical techniques of energy estimates and Sobolev inequalities. Following [Kl-Ro] we develop new analytic methods based on Strichartz-type inequalities which result 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 the Einstein equations.

This is the main paper in a sequence in which we give a complete 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 on the volume radius of the corresponding initial data set. We note that though the result is not optimal with respect to the scaling of the Einstein equations, it is nevertheless critical with respect to its causal geometry. Indeed, L 2 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 Bahouri and Chemin (Am J Math 121:1337–1777, 1999 ; IMRN 21:1141–1178, 1999 ), Tataru (Am J Math 122:349–376, 2000 ; JAMS 15(2):419–442, 2002 ), Klainerman and Rodnianski (Duke Math J 117(1):1–124, 2003 ) and optimized in Klainerman and Rodnianski (Ann Math 161:1143–1193, 2005 ), Smith and Tataru (Ann Math 162:291–366, 2005 ), 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 s o ( 3 , 1 ) -valued Yang–Mills theory and introduce a Coulomb type gauge condition in which the equations exhibit a specific new type of 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 L 2 error bounds which is carried out in Szeftel (Parametrix for wave equations on a rough background I: regularity of the phase at initial time, arXiv:1204.1768 , 2012 ; Parametrix for wave equations on a rough background II: construction of the parametrix and control at initial time, arXiv:1204.1769 , 2012 ; Parametrix for wave equations on a rough background III: space-time regularity of the phase, arXiv:1204.1770 , 2012 ; Parametrix for wave equations on a rough background IV: control of the error term, arXiv:1204.1771 , 2012 ), as well as a proof of sharp Strichartz estimates for the wave equation on a rough background which is carried out in Szeftel (Sharp Strichartz estimates for the wave equation on a rough background, arXiv:1301.0112 , 2013 ). It is at this level that our problem is critical. Indeed, any known notion of a parametrix relies in an essential way on the eikonal equation, and our space-time possesses, barely, the minimal regularity needed to make sense of its solutions.