Annals of Pde最新文献

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface (Sigma simeq overline{B_1} subset {{mathbb {R}}}^3) and the outgoing null hypersurface ({{mathcal {H}}}) emanating from ({partial }Sigma ), we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in (L^2). The proof uses the bounded (L^2) curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution (u=0) is the only solution for which the assumptions (u(t=0)vert _{D}=0, u(t=T)vert _{D}=0) hold, where (Dsubset mathbb {R}^d) are certain subsets of codimension one. In particular, D is discrete for dimension (d=1). Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko–Viazovska formula in [21], and the uniqueness result of the second author and M. Sousa for powers of integers [22]. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros.

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je(dot{text {z}})ewska (Comm. Math. Phys. 115(4):607–629, 1985) in [32], and our assumptions include the case of Israel particles (J. Math. Phys. 4:1163–1181, 1963) in [56]. In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad’s angular cut-off assumption.

In this paper, we initiate the study of the global stability of nonlinear wave equations with initial data that are not required to be localized around a single point. More precisely, we allow small initial data localized around any finite collection of points which can be arbitrarily far from one another. Existing techniques do not directly apply to this setting because they require norms with radial weights away from some center to be small. The smallness we require on the data is measured in a norm which does not depend on the scale of the configuration of the data. Our method of proof relies on a close analysis of the geometry of the interaction between waves originating from different sources. We prove estimates on the bilinear forms encoding the interaction, which allow us to show improved bounds for the energy of the solution. We finally apply a variant of the vector field method involving modified Klainerman–Sobolev estimates to prove global stability. As a corollary of our proof, we are able to show global existence for a class of data whose (H^1) norm is arbitrarily large.

This the first in a series of papers whose ultimate goal is to establish the full nonlinear stability of the Kerr family for (|a|ll 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 (ell =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.

The question about behavior of gaps between zeros of polynomials under differentiation is classical and goes back to Marcel Riesz. Recently, Stefan Steinerberger [42] formally derived a nonlocal nonlinear partial differential equation which models dynamics of roots of polynomials under differentiation. In this paper, we connect rigorously solutions of Steinerberger’s PDE and evolution of roots under differentiation for a class of trigonometric polynomials. Namely, we prove that the distribution of the zeros of the derivatives of a polynomial and the corresponding solutions of the PDE remain close for all times. The global in time control follows from the analysis of the propagation of errors equation, which turns out to be a nonlinear fractional heat equation with the main term similar to the modulated discretized fractional Laplacian ((-Delta )^{1/2}).

In this article, we prove various illposedness results for the Cauchy problem for the incompressible Hall- and electron-magnetohydrodynamic (MHD) equations without resistivity. These PDEs are fluid descriptions of plasmas, where the effect of collisions is neglected (no resistivity), while the motion of the electrons relative to the ions (Hall current term) is taken into account. The Hall current term endows the magnetic field equation with a quasilinear dispersive character, which is key to our mechanism for illposedness. Perhaps the most striking conclusion of this article is that the Cauchy problems for the Hall-MHD (either viscous or inviscid) and the electron-MHD equations, under one translational symmetry, are ill-posed near the trivial solution in any sufficiently high regularity Sobolev space (H^{s}) and even in any Gevrey spaces. This result holds despite obvious wellposedness of the linearized equations near the trivial solution, as well as conservation of the nonlinear energy, by which the (L^{2}) norm (energy) of the solution stays constant in time. The core illposedness (or instability) mechanism is degeneration of certain high frequency wave packet solutions to the linearization around a class of linearly degenerate stationary solutions of these equations, which are essentially dispersive equations with degenerate principal symbols. The method developed in this work is sharp and robust, in that we also prove nonlinear (H^{s})-illposedness (for s arbitrarily high) in the presence of fractional dissipation of any order less than 1, matching the previously known wellposedness results. The results in this article are complemented by a companion work, where we provide geometric conditions on the initial magnetic field that ensure wellposedness(!) of the Cauchy problems for the incompressible Hall and electron-MHD equations. In particular, in stark contrast to the results here, it is shown in the companion work that the nonlinear Cauchy problems are well-posed near any nonzero constant magnetic field.

We consider the cubic and quintic nonlinear Schrödinger equations (NLS) under the ({mathbb {R}}^{d}) and ({mathbb {T}}^{d}) energy-supercritical setting. Via a newly developed unified scheme, we prove the unconditional uniqueness for solutions to NLS at critical regularity for all dimensions. Thus, together with [19, 20], the unconditional uniqueness problems for (H^{1})-critical and (H^{1})-supercritical cubic and quintic NLS are completely and uniformly resolved at critical regularity for these domains. One application of our theorem is to prove that defocusing blowup solutions of the type in [59] are the only possible (C([0,T);{dot{H}}^{s_{c}})) solutions if exist in these domains.

The spherically symmetric null hypersurfaces in a Schwarzschild spacetime are smooth away from the singularities and foliate the spacetime. We prove the existence of more general foliations by null hypersurfaces without the spherical symmetry condition. In fact we also relax the spherical symmetry of the ambient spacetime and prove a more general result: in a perturbed Schwarzschild spacetime (not necessary being vacuum), nearly round null hypersurfaces can be extended regularly to the past null infinity, thus there exist many foliations by regular null hypersurfaces in the exterior region of a perturbed Schwarzschild black hole. A significant point of the result is that the ambient spacetime metric is not required to be differentiable in all directions.