Annals of Pde最新文献

This paper provides the first rigorous construction of the self-similar algebraic spiral vortex sheet solutions to the 2-D incompressible Euler equations. These solutions are believed to represent the typical roll-up pattern of vortex sheets following the formation of curvature singularities due to the Kelvin-Helmholtz instability. Furthermore, they constitute plausible candidates for demonstrating non-uniqueness within the class of Delort’s weak solutions. The most challenging part of this paper is handling the Cauchy integral for the algebraic spiral curve, which falls outside the classical theory of singular integral operators.

We study, fully microlocally, the propagation of massive waves on the octagonal compactification

of asymptotically Minkowski spacetime, which allows a detailed analysis both at timelike and spacelike infinity (as previously investigated using Parenti–Shubin–Melrose’s sc-calculus) and, more novelly, at null infinity, denoted (mathscr {I}). The analysis is closely related to Hintz–Vasy’s recent analysis of massless wave propagation at null infinity using the “e,b-calculus” on (mathbb {O}). We prove several elementary corollaries regarding the Klein–Gordon IVP. Our main technical tool is a fully symbolic pseudodifferential calculus, (Psi _{textrm{de,sc}}(mathbb {O})), the “de,sc-calculus” on (mathbb {O}). The ‘de’ refers to the structure (“double edge”) of the calculus at null infinity, and the ‘sc’ refers to the structure (“scattering”) at the other boundary faces. We relate this structure to the hyperbolic coordinates used in other studies of the Klein–Gordon equation. Unlike hyperbolic coordinates, the de,sc- boundary fibration structure is Poincaré invariant.

We are concerned with the low regularity of self-similar solutions of two-dimensional Riemann problems for the isentropic Euler system. We establish a general framework for the analysis of the local regularity of such solutions for a class of two-dimensional Riemann problems for the isentropic Euler system, which includes the regular shock reflection problem, the Prandtl reflection problem, the Lighthill diffraction problem, and the four-shock Riemann problem. We prove that the velocity is not in (H^1) in the subsonic domain for the self-similar solutions of these problems in general. This indicates that the self-similar solutions of the Riemann problems with shocks for the isentropic Euler system are of much more complicated structure than those for the Euler system for potential flow; in particular, the velocity is not necessarily continuous in the subsonic domain. The proof is based on a regularization of the isentropic Euler system to derive the transport equation for the vorticity, a renormalization argument extended to the case of domains with boundary, and DiPerna-Lions-type commutator estimates.

A hollow vortex is a region of constant pressure suspended inside a perfect fluid and around which there is a nonzero circulation; it can therefore be interpreted as a spinning bubble of air in water. This paper gives a general method for desingularizing non-degenerate steady point vortex configurations into collections of steady hollow vortices. Our machinery simultaneously treats the translating, rotating, and stationary regimes. Through global bifurcation theory, we further obtain maximal curves of solutions that continue until the onset of a singularity. As specific examples, we give the first existence theory for co-rotating hollow vortex pairs and stationary hollow vortex tripoles, as well as a new construction of Pocklington’s classical co-translating hollow vortex pairs. All of these families extend into the non-perturbative regime, and we obtain a rather complete characterization of the limiting behavior along the global bifurcation curve.

We study energy decay rates for the damped wave equation with unbounded damping, without the geometric control condition. Our main decay result is sharp polynomial energy decay for polynomially controlled singular damping on the torus. We also prove that for normally L^p-damping on compact manifolds, the Schrödinger observability gives p-dependent polynomial decay, and finite time extinction cannot occur. We show that polynomially controlled singular damping on the circle gives exponential decay.

In this paper, we provide a new proof of nonlinear stability of the slowly-rotating Kerr-de Sitter family of black holes as a family of solutions to the Einstein vacuum equations with cosmological constant (Lambda > 0), originally established by Hintz and Vasy in their seminal work (Hintz and Vasy, Acta Mathematica 220(1):1–206, 2018). Using the linear theory developed in the companion paper (Fang in Linear stability of the slowly-rotating Kerr-de Sitter family, 2022, https://doi.org/10.1007/s40818-025-00226-y), we prove the nonlinear stability of slowly-rotating Kerr-de Sitter using a bootstrap argument, avoiding the need for a Nash-Moser argument, and requiring the initial data to be small only in the (H^6) norm.

We prove local well-posedness for the Muskat problem on the half-plane, which models motion of an interface between two fluids of distinct densities (e.g., oil and water) in a porous medium (e.g., an aquifer) that sits atop an impermeable layer (e.g., bedrock). Our result allows for the interface to touch the bottom, and hence applies to the important scenario of the heavier fluid invading a region occupied by the lighter fluid along the impermeable layer. We use this result in the companion paper Zlatoš [The 2D Muskat problem II: Stable regime small data singularity on the half-plane, preprint], to prove existence of finite time stable regime singularities in this model, including for arbitrarily small initial data. We do not require the interface and its derivatives to vanish at (pminfty) or be periodic, and even allow it to be (O(|x|^{1-})), which is an optimal bound on the power of growth. We also extend our result to the Muskat problem on the whole plane and on horizontal strips.

In this paper, we prove the linear stability of weakly charged and slowly rotating Kerr-Newman black holes under coupled gravitational and electromagnetic perturbations. We show that the solutions to the linearized Einstein-Maxwell equations decay at an inverse polynomial rate to a linearized Kerr-Newman solution plus a pure gauge term. This work builds on the framework developed in Häfner (Invent Math 223(3):1227–1406, 2021) for the study of the Einstein vacuum equations. We work in the generalized wave map and Lorenz gauge. The proof involves the analysis of the resolvent of the Fourier transformed linearized Einstein-Maxwell operator on asymptotically flat spaces, which relies on recent advances in microlocal analysis and non-elliptic Fredholm theory developed in Vasy (Invent Math 194(2):381–513, 2013). The most delicate part of the proof is the description of the resolvent at low frequencies.

We consider randomly forced resistive magnetic relaxation equations (MRE) with resistivity (kappa > 0) and a force proportional to (sqrt{kappa}) on the flat (d)-torus (mathbb{T}^{d}) for (dgeq 2). We show the path-wise global well-posedness of the system and the existence of the invariant measures, and construct a random magnetohydrostatic (MHS) equilibrium (B(x)) in (H^{1}(mathbb{T}^{d})) with law (mathcal{D}(B)=mu_0) as a non-resistive limit (kappato 0) of statistically stationary solutions (B_{kappa}(x,t)). For (d=2), the measure (mu_0) does not concentrate on any compact subset in (H^{1}(mathbb{T}^{2})) with finite Hausdorff dimension. In particular, all realizations of the random MHS equilibrium (B(x)) are almost surely not finite Fourier mode solutions.

In this article, we prove the global well-posedness in the critical Sobolev space (H_{rad}^2left(mathbb{R}^2right) times H_{rad}^1 left(mathbb{R}^2right)) for the radial time-like extremal hypersurface equation in (left(1+3right))- dimensional Minkowski space-time. This is achieved by deriving a new div-curl type lemma and combined it with energy and “momentum” balance law to get some space-time estimates of the nonlinearity.