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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.

The authors consider the energy super critical semilinear heat equation \\partial _{t}u=\\Delta u+u^{p}, x\\in \\mathbb{R}^3, p>5. The authors first revisit the construction of radially symmetric self similar solutions performed through an ode approach and propose a bifurcation type argument which allows for a sharp control of the spectrum of the corresponding linearized operator in suitable weighted spaces. They then show how the sole knowledge of this spectral gap in weighted spaces implies the finite codimensional nonradial stability of these solutions for smooth well localized initial data using energy bounds. The whole scheme draws a route map for the derivation of the existence and stability of self-similar blow up in nonradial energy super critical settings.

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for | a | ≪ m . The paper builds on the strategy laid out in [ 6 ] in the context of the nonlinear stability of Schwarzschild for axially symmetric polarized perturbations. In fact the central idea of [ 6 ] was the introduction and construction of generally covariant modulated (GCM) spheres on which specific geometric quantities take Schwarzschildian values. This was made possible by taking into account the full general covariance of the Einstein vacuum equations. The goal of this, and its companion paper [ 7 ], is to get rid of the symmetry restriction in the construction of GCM spheres in [ 6 ] and thus remove an essential obstruction in extending the result to a full stability proof of the Kerr family.

This is a follow-up of our paper (Klainerman and Szeftel in Construction of GCM spheres in perturbations of Kerr, Accepted for publication in Annals of PDE) on the construction of general covariant modulated (GCM) spheres in perturbations of Kerr, which we expect to play a central role in establishing their nonlinear stability. We reformulate the main results of that paper using a canonical definition of ℓ = 1 modes on a 2-sphere embedded in a 1 + 3 vacuum manifold. This is based on a new, effective, version of the classical uniformization theorem which allows us to define such modes and prove their stability for spheres with comparable metrics. The reformulation allows us to prove a second, intrinsic, existence theorem for GCM spheres, expressed purely in terms of geometric quantities defined on it. A natural definition of angular momentum for such GCM spheres is also introduced, which we expect to play a key role in determining the final angular momentum for general perturbations of Kerr.

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.

We consider the 2-dimensional focusing mass critical NLS with an inhomogeneous nonlinearity: i∂tu+Δu+k(x)|u|2u=0i\\partial _tu+\\Delta u+k(x)|u|^{2}u=0. From a standard argument, there exists a threshold Mk>0M_k>0 such that H1H^1 solutions with ‖u‖L2>Mk\\|u\\|_{L^2}>M_k are global in time while a finite time blow-up singularity formation may occur for ‖u‖L2>Mk\\|u\\|_{L^2}>M_k. In this paper, we consider the dynamics at threshold ‖u0‖L2=Mk\\|u_0\\|_{L^2}=M_k and give a necessary and sufficient condition on kk 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 nondegenerate point, hence extending the pioneering work by Merle who treated the pseudoconformal invariant case k≡1k\\equiv 1.

We design and analyze a Schwarz waveform relaxation algorithm for domain decomposition of advection-diffusion-reaction problems with strong heterogeneities. The interfaces are curved, and we use optimized Ventcell transmission conditions. We analyze the semidiscretization in time with discontinuous Galerkin as well. We also show two-dimensional numerical results using generalized mortar finite elements in space.

We consider the two dimensional$L^2$critical nonlinear Schrödinger equation$i\\partial_tu+\\Delta u+uu^2=0$ . In their pioneering 1997 work, Bourgain and Wang have constructed smooth solutions which blow up in finite time$T<+\\infty$with the pseudo conformal speed$$||\\nabla u(t)||_{L^2}\\sim {1\\over T-t},$$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 work by F. Merle and P. Rapha\"el. We moreover exhibit some continuation properties of the scattering solution after blow up time and recover the chaotic phase behavior first exhibited in F. Merle's 1992 ıt Comm. Pure. Appl. Math. paper in the critical mass case.

We introduce a nonoverlapping variant of the Schwarz waveform relaxation algorithm for semilinear wave propagation in one dimension. Using the theory of absorbing boundary conditions, we derive a new nonlinear algorithm. We show that the algorithm is well-posed and we prove its convergence by energy estimates and a Galerkin method. We then introduce an explicit scheme. We prove the convergence of the discrete algorithm with suitable assumptions on the nonlinearity. We finally illustrate our analysis with numerical experiments.