Annals of Pde最新文献

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.

In this paper, we establish the uniqueness and nonlinear stability of concentrated symmetric traveling vortex patch-pairs for the 2D Euler equation. We also prove the uniqueness of concentrated rotating polygons as well. The proofs are achieved by a combination of the local Pohozaev identity, a detailed description of asymptotic behaviors of the solutions and some symmetry properties obtained by the method of moving planes.

In this paper, we give the first rigorous justification of the Benjamin-Ono equation:

as an internal water wave model on the physical time scale. Here, ({varepsilon }) is a small parameter measuring the weak nonlinearity of the waves, (mu ) is the shallowness parameter, and (gamma in (0,1)) is the ratio between the densities of the two fluids. To be precise, we first prove the existence of a solution to the internal water wave equations for a two-layer fluid with surface tension, where one layer is of shallow depth and the other is of infinite depth. The existence time is of order ({mathcal {O}}(frac{1}{{varepsilon }})) for a small amount of surface tension such that ({varepsilon }^2 le textrm{bo}^{-1} ) where (textrm{bo}) is the Bond number. Then, we show that these solutions are close, on the same time scale, to the solutions of the BO equation with a precision of order ({mathcal {O}}(mu + textrm{bo}^{-1})). In addition, we provide the justification of new equations with improved dispersive properties, the Benjamin equation, and the Intermediate Long Wave (ILW) equation in the deep-water limit.

The long-time well-posedness of the two-layer fluid problem was first studied by Lannes [Arch. Ration. Mech. Anal., 208(2):481-567, 2013] in the case where both fluids have finite depth. Here, we adapt this work to the case where one of the fluid domains is of finite depth, and the other one is of infinite depth. The novelties of the proof are related to the geometry of the problem, where the difference in domains alters the functional setting for the Dirichlet-Neumann operators involved. In particular, we study the various compositions of these operators that require a refined symbolic analysis of the Dirichlet-Neumann operator on infinite depth and derive new pseudo-differential estimates that might be of independent interest.

The 2-D Peskin problem describes a 1-D closed elastic string immersed and moving in a 2-D Stokes flow that is induced by its own elastic force. The geometric shape of the string and its internal stretching configuration evolve in a coupled way, and they combined govern the dynamics of the system. In this paper, we show that certain geometric quantities of the moving string satisfy extremum principles and decay estimates. As a result, we can prove that the 2-D Peskin problem admits a unique global solution when the initial data satisfies a medium-size geometric condition on the string shape, while no assumption on the size of stretching is needed.