This paper is the second part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first paper of this sequence [4], we established well posedness of the characteristic problem with general initial data.In this second paper, we generalize the results of Dafermos [6] on the stability of the radius function at the Cauchy horizon by including a cosmological constant. This requires a considerable deviation from the strategy followed in [6], focusing on the level sets of the radius function instead of the red-shift and blue-shift regions. We also present new results on the global structure of the solution when the free data is not identically zero in a neighborhood of the origin.In the third and final paper [5], we will consider the issue of mass inflation and extendibility of solutions beyond the Cauchy horizon.

This paper is the third part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development as a "suitably regular" Lorentzian manifold.In the first part [7] of this series we established the well posedness of the characteristic problem, whereas in the second part [8] we studied the stability of the radius function at the Cauchy horizon.In this third and final paper we show that, depending on the decay rate of the initial data, mass inflation may or may not occur. When the mass is controlled, it is possible to obtain continuous extensions of the metric across the Cauchy horizon with square integrable Christoffel symbols. Under slightly stronger conditions, we can bound the gradient of the scalar field. This allows the construction of (non-isometric) extensions of the maximal development which are classical solutions of the Einstein equations. Our results provide evidence against the validity of the strong cosmic censorship conjecture when Λ > 0.

This paper is the first part of a trilogy dedicated to the following problem: given spherically symmetric characteristic initial data for the EinsteinMaxwell-scalar field system with a cosmological constant Λ, with the data on the outgoing initial null hypersurface given by a subextremal ReissnerNordström black hole event horizon, study the future extendibility of the corresponding maximal globally hyperbolic development (MGHD) as a "suitably regular" Lorentzian manifold.In this first part we establish well posedness of the Einstein equations for characteristic data satisfying the minimal regularity conditions leading to classical solutions. We also identify the appropriate notion of maximal solution, from which the construction of the corresponding MGHD follows, and determine breakdown criteria. This is the unavoidable starting point of the analysis; our main results will depend on the detailed understanding of these fundamentals.In the second part of this series [12] we study the stability of the radius function at the Cauchy horizon. In the third and final paper [13] we show that, depending on the decay rate of the initial data, mass inflation may or may not occur; in fact, it is even possible to have (non-isometric) extensions of the spacetime across the Cauchy horizon as classical solutions of the Einstein equations.

ABSTRACT. We consider the nonlinear Schrödinger equation under a partial quadratic confinement. We show that the global dispersion corresponding to the direction(s) with no potential is enough to prove global in time Strichartz estimates, from which we infer the existence of wave operators thanks to suitable vector-fields. Conversely, given an initial Cauchy datum, the solution is global in time and asymptotically free, provided that confinement affects one spatial direction only. This stems from anisotropic Morawetz estimates, involving a marginal of the position density.

In this paper we study the spherically symmetric characteristic initial data problem for the Einstein-Maxwell-scalar field system with a positive cosmological constant in the interior of a black hole, assuming an exponential Price law along the event horizon. More precisely, we construct open sets of characteristic data which, on the outgoing initial null hypersurface (taken to be the event horizon), converges exponentially to a reference Reissner-Nördstrom black hole at infinity.We prove the stability of the radius function at the Cauchy horizon, and show that, depending on the decay rate of the initial data, mass inflation may or may not occur. In the latter case, we find that the solution can be extended across the Cauchy horizon with continuous metric and Christoffel symbols in L 2 loc , thus violating the Christodoulou-Chruściel version of strong cosmic censorship. Contents2010 Mathematics Subject Classification. Primary 83C05; Secondary 35Q76, 83C22, 83C57, 83C75. References 491. Introduction 1.1. Strong cosmic censorship and spherical symmetry. Determinism of a physical system, modeled mathematically by evolution equations, is embodied in the questions of existence and uniqueness of solutions for given initial data. The initial value problem (or Cauchy problem) is therefore the appropriate setting for studying these models.Well known examples of equations where the Cauchy problem is quintessential are those of Newtonian mechanics, the Euler and Navier-Stokes systems in hydrodynamics and Maxwell's equations of electromagnetism. Historically, the geometric nature and mathematical complexity of the Einstein equations made it difficult to recognize that they also fit into this framework. It was not until the seminal work of Y. , and her later work with R. Geroch [5], that the central role of the Cauchy problem in general relativity was established. These results relied crucially on recognizing the hyperbolic character of the Einstein equations. Uniqueness of the solutions, as for any hyperbolic PDE, then follows from a domain of dependence property. The essence of [5] consists precisely in showing that given initial data there exists a maximal globally hyperbolic development (MGHD) for the corresponding Cauchy problem, that is, a maximal spacetime where this domain of dependence property holds.For the Einstein equations, global uniqueness fails, and therefore determinism breaks down, if extensions of MGHDs to strictly larger spacetimes can be found. The statement that generically, for suitable Cauchy initial data, * the corresponding MGHD cannot be extended is known as the strong cosmic censorship conjecture (SCCC) [7,9,27].A crucial point in the precise formulation of this conjecture is deciding what exactly is meant by an extension. Various proposals have been advanced, differing on the degree of regularity that is demanded for the larger spacetime. The strongest formulation would correspond to the impossibility of extending the MGHD with a continuous Lorentzian metric. This happens for instance in the Schwa...

We establish local well-posedness results for the Initial Value Problem associated to the Schrödinger-Debye system in dimensions N = 2, 3 for data in H s × H ℓ , with s and ℓ satisfying max{0, s − 1} ≤ ℓ ≤ min{2s, s + 1}. In particular, these include the energy space H 1 × L 2 . Our results improve the previous ones obtained in [2], [3] and [9]. Moreover, in the critical case (N = 2) and for initial data in H 1 × L 2 , we prove that solutions exist for all times, thus providing a negative answer to the open problem mentioned in [11] concerning the formation of singularities for these solutions.

In this article we study the generalized dispersion version of the Kadomtsev-Petviashvili II equation, on $\T \times \R$ and $\T \times \R^2$. We start by proving bilinear Strichartz type estimates, dependent only on the dimension of the domain but not on the dispersion. Their analogues in terms of Bourgain spaces are then used as the main tool for the proof of bilinear estimates of the nonlinear terms of the equation and consequently of local well-posedness for the Cauchy problem.Comment: 32 page

We consider the large time behavior of solutions to defocusing nonlinear Schrödinger equation in the presence of a time dependent external potential. The main assumption on the potential is that it grows at most quadratically in space, uniformly with respect to the time variable. We show a general exponential control of first order derivatives and momenta, which yields a double exponential bound for higher Sobolev norms and momenta. On the other hand, we show that if the potential is an isotropic harmonic potential with a time dependent frequency which decays sufficiently fast, then Sobolev norms are bounded, and momenta grow at most polynomially in time, because the potential becomes negligible for large time: there is scattering, even though the potential is unbounded in space for fixed time.