Annals of Pde最新文献

In this paper we prove integrated energy and pointwise decay estimates for solutions of the vacuum linearized Einstein equation on the domain of outer communication of the Kerr black hole spacetime. The estimates are valid for the full subextreme range of Kerr black holes, provided integrated energy estimates for the Teukolsky equation hold. For slowly rotating Kerr backgrounds, such estimates are known to hold, due to the work of one of the authors. The results in this paper thus provide the first stability results for linearized gravity on the Kerr background, in the slowly rotating case, and reduce the linearized stability problem for the full subextreme range to proving integrated energy estimates for the Teukolsky equation. This constitutes an essential step towards a proof of the black hole stability conjecture, i.e. the statement that the Kerr family is dynamically stable, one of the central open problems in general relativity.

We show a stability result for the Schwarzschild singularity (inside the black hole region) for the Einstein vacuum equations (EVE). The result is proven in the class of polarized axial symmetry, under perturbations of the Schwarzschild data induced on a hypersurface ({r=epsilon }), (epsilon<<2M). Our result is only partly a stability result, in that we show that while a (space-like) singularity persists under perturbations as above, the behavior of the metric approaching the singularity is much more involved than for the Schwarzschild solution. Indeed, we find that the solution displays asymptocially-velocity-term-dominated dynamics and approaches a different Kasner solution at each point of the singularity. These Kasner-type asymptotics are very far from isotropic, since (as in Schwarzschild) there are two contracting directions and one expanding one. Our proof relies on energy methods and on a new approach to the EVE in axial symmetry, which we believe has wider applicability: In this symmetry class and under a suitable geodesic gauge, the EVE can be studied as a free wave coupled to (nonlinear) ODEs, which couple the geometry of the projected, 2+1 space-time to the free wave. The fact that the nonlinear part of the Einstein equations is described by ODEs lies at the heart of how one can overcome a certain linear instability exhibited by the singularity.

The question of whether the hyper-dissipative (HD) Navier-Stokes (NS) system can exhibit spontaneous formation of singularities in the super-critical regime–the hyperviscous effects being represented by a fractional power of the Laplacian, say (beta ), confined to interval (bigl (1, frac{5}{4}bigr ))–has been a major open problem in the mathematical fluid dynamics since the foundational work of J.L. Lions in 1960s. In this work, an evidence of criticality of the Laplacian is presented, more precisely, a class of plausible blow-up scenarios is ruled out as soon as (beta ) is greater than one. While the framework is based on the ‘scale of sparseness’ of the super-level sets of the positive and negative parts of the components of the higher-order derivatives of the velocity previously introduced by the authors, a major novelty in the current work is classification of the HD flows near a potential spatiotemporal singularity in two main categories, ‘homogeneous’ (the case consistent with a near-steady behavior) and ‘non-homogenous’ (the case consistent with the formation and decay of turbulence). The main theorem states that in the non-homogeneous case any (beta ) greater than one prevents a singularity. In order to illustrate the impact of this result in a methodology-free setting, a two-parameter family of dynamically rescaled blow-up profiles is considered, and it is shown that as soon as (beta ) is greater than one, a new region in the parameter space is ruled out. More importantly, the region is a neighborhood (in the parameter space) of the self-similar profile, i.e., the approximately self-similar blow-up, a prime suspect in possible singularity formation, is ruled out for all HD NS models.

We prove the nonlinear inviscid damping for a class of monotone shear flows with non-constant background density for the two-dimensional ideal inhomogeneous fluids in (mathbb {T}times [0,1]) when the initial perturbation is in Gevrey-(frac{1}{s}) ((frac{1}{2}<s<1)) class with compact support.

We provide a geometric optics description in spaces of low regularity, (L^2) and (H^1), of the transport of oscillations in solutions to linear and some semilinear second-order hyperbolic boundary problems along rays that graze the boundary of a convex obstacle to arbitrarily high finite or infinite order. The fundamental motivating example is the case where the spacetime manifold is (M=(mathbb {R}^nsetminus mathcal {O})times mathbb {R}_t), where (mathcal {O}subset mathbb {R}^n) is an open convex obstacle with (C^infty ) boundary, and the governing hyperbolic operator is the wave operator (Box :=Delta -partial _t^2). The main theorem says that high frequency exact solutions are well approximated in spaces of low regularity by approximate solutions constructed from fairly explicit solutions to relatively simple profile equations. The theorem has two main assumptions. The first is that the grazing set, that is, the set of points on the spacetime boundary at which incoming characteristics meet the boundary tangentially, is a codimension two, (C^1) submanifold of spacetime. The second is that the reflected flow map, which sends points on the spacetime boundary forward in time to points on reflected and grazing rays, is injective and has appropriate regularity properties near the grazing set. Both assumptions are in general hard to verify, but we show that they are satisfied for the diffraction of incoming plane waves by a large class of strictly convex obstacles in all dimensions, involving grazing points of arbitrarily high finite or infinite order.

We prove fine higher regularity results of Calderón-Zygmund-type for equations involving nonlocal operators modelled on the fractional p-Laplacian with possibly discontinuous coefficients of VMO-type. We accomplish this by establishing precise pointwise bounds in terms of certain fractional sharp maximal functions. This approach is new already in the linear setting and enables us to deduce sharp regularity results also in borderline cases.

We study the properties of spacelike singularities in spherically symmetric spacetimes obeying the Einstein equations, in the presence of matter. Building upon previous work of An–Zhang [4], we consider matter described by a scalar field, both in the presence of an electromagnetic field and without. We prove that, if a spacelike singularity obeying several reasonable assumptions is formed, then the Hawking mass, the Kretschmann scalar, and the matter fields have inverse polynomial blow-up rates near the singularity that may be described precisely. Furthermore, one may view the resulting spacetime in the context of the BKL heuristics regarding spacelike singularities in relativistic cosmology. In particular, near any point p on the singular boundary in our spherically symmetric spacetime, we obtain a leading order BKL-type expansion, including a description of Kasner exponents associated to p. This confirms heuristics of Buonanno–Damour–Veneziano [14]. As a result, we provide a rigorous description of a detailed, quantitative correspondence between Kasner-like singularities most often associated to the cosmological setting, and the singularities observed in (spherically symmetric) gravitational collapse. Moreover, we outline a program concerning the study of the stability and instability of spacelike singularities in the latter picture, both outside of spherical symmetry and within (where the electromagnetic field acts as a proxy for angular momentum).

A Stokes wave is a traveling free-surface periodic water wave that is constant in the direction transverse to the direction of propagation. In 1981 McLean discovered via numerical methods that Stokes waves at infinite depth are unstable with respect to transverse perturbations of the initial data. Even for a Stokes wave that has very small amplitude (varepsilon ), we prove rigorously that transverse perturbations, after linearization, will lead to exponential growth in time. To observe this instability, extensive calculations are required all the way up to order (O(varepsilon ^3)). All previous rigorous results of this type were merely two-dimensional, in the sense that they only treated long-wave perturbations in the longitudinal direction. This is the first rigorous proof of three-dimensional instabilities of Stokes waves.

We study the interior of black holes in the presence of charged scalar hair of small amplitude (epsilon ) on the event horizon and show their terminal boundary is a crushing Kasner-like singularity. These spacetimes are spherically symmetric, spatially homogeneous and they differ significantly from the hairy black holes with uncharged matter previously studied in [M. Van de Moortel, Violent nonlinear collapse inside charged hairy black holes, Arch. Rational. Mech. Anal., 248, 89, 2024] in that the electric field is dynamical and subject to the backreaction of charged matter. We prove this charged backreaction causes drastically different dynamics compared to the uncharged case that ultimately impact the formation of the spacelike singularity, exhibiting novel phenomena such as

• Collapsed oscillations: oscillatory growth of the scalar hair, nonlinearly induced by the collapse

• A fluctuating collapse: The final Kasner exponents’ dependency in (epsilon ) is via an expression of the form

  (|sin left( omega _0 cdot epsilon ^{-2}+ O(log (epsilon ^{-1}))right) |).

• A Kasner bounce: a transition from an unstable Kasner metric to a different stable Kasner metric

The Kasner bounce occurring in our spacetime is reminiscent of the celebrated BKL scenario in cosmology.

We additionally propose a construction indicating the relevance of the above phenomena – including Kasner bounces – to spacelike singularities inside more general (asymptotically flat) black holes, beyond the hairy case.

While our result applies to all values of (Lambda in mathbb {R}), in the (Lambda <0) case, our spacetime corresponds to the interior region of a charged asymptotically Anti-de-Sitter stationary black hole, also known as a holographic superconductor in high-energy physics, and whose exterior region was rigorously constructed in the recent mathematical work [W. Zheng, Asymptotically Anti-de Sitter Spherically Symmetric Hairy Black Holes, arXiv.2410.04758].

For every (alpha < nicefrac 13), we construct an explicit divergence-free vector field ({textbf {b}}(t,x)) which is periodic in space and time and belongs to (C^0_t C^{alpha }_x cap C^{alpha }_t C^0_x) such that the corresponding scalar advection-diffusion equation

exhibits anomalous dissipation of scalar variance for arbitrary (H^1) initial data:

The vector field is deterministic and has a fractal structure, with periodic shear flows alternating in time between different directions serving as the base fractal. These shear flows are repeatedly inserted at infinitely many scales in suitable Lagrangian coordinates. Using an argument based on ideas from quantitative homogenization, the corresponding advection-diffusion equation with small (kappa ) is progressively renormalized, one scale at a time, starting from the (very small) length scale determined by the molecular diffusivity up to the macroscopic (unit) scale. At each renormalization step, the effective diffusivity is enhanced by the influence of advection on that scale. By iterating this procedure across many scales, the effective diffusivity on the macroscopic scale is shown to be of order one.