Annals of Pde最新文献

In this work, we develop a wavelet-inspired, (L^3)-based convex integration framework for constructing weak solutions to the three-dimensional incompressible Euler equations. The main innovations include a new multi-scale building block, which we call an intermittent Mikado bundle; a wavelet-inspired inductive set-up which includes assumptions on spatial and temporal support, in addition to (L^p) and pointwise estimates for Eulerian and Lagrangian derivatives; and sharp decoupling lemmas, inverse divergence estimates, and space-frequency localization technology which is well-adapted to functions satisfying (L^p) estimates for p other than 1, 2, or (infty ). We develop these tools in the context of the Euler-Reynolds system, enabling us to give both a new proof of the intermittent Onsager theorem from Novack and Vicol (Invent Math 233(1):223–323, 2023) in this paper, and a proof of the (L^3)-based strong Onsager conjecture in the companion paper Giri et al. (The (L^3)-based strong Onsager theorem, arxiv).

In this paper, we consider the stability of the generalized Lagrangian mean curvature flow of graph case in the cotangent bundle, which is first defined by Smoczyk-Tsui-Wang (Smoczyk et al. J für die reine und angewandte Mathematik 750: 97–121, 2019). By new estimates of derivatives along the flow, we weaken the initial condition and remove the positive curvature condition in Smoczyk et al. (J für die reine und angewandte Mathematik 750: 97–121, 2019). More precisely, we prove that if the graph induced by a closed 1-form is a special Lagrangian submanifold in the cotangent bundle of a Riemannian manifold, then the generalized Lagrangian mean curvature flow is stable near it.

In this paper, we investigate the dynamics of solutions of the Muskat equation with initial interface consisting of multiple corners allowing for linear growth at infinity. Specifically, we prove that if the initial data contains a finite set of small corners then we can find a precise description of the solution showing how these corners desingularize and move at the same time. At the analytical level, we are solving a small data critical problem which requires renormalization. This is accomplished using a nonlinear change of variables which serves as a logarithmic correction and accurately describes the motion of the corners during the evolution.

We consider the global evolution problem for Einstein’s field equations in the near-Minkowski regime and study the long-time dynamics of a massive scalar field evolving under its own gravitational field. We establish the existence of a globally hyperbolic Cauchy development associated with any initial data set that is sufficiently close to a data set in Minkowski spacetime. In addition to applying to massive fields, our theory allows us to cover metrics with slow decay in space. The strategy of proof, proposed here and referred to as the Euclidean-Hyperboloidal Foliation Method, applies, more generally, to nonlinear systems of coupled wave and Klein-Gordon equations. It is based on a spacetime foliation defined by merging together asymptotically Euclidean hypersurfaces (covering spacelike infinity) and asymptotically hyperboloidal hypersurfaces (covering timelike infinity). A transition domain (reaching null infinity) limited by two asymptotic light cones is introduced in order to realize this merging. On the one hand, we exhibit a boost-rotation hierarchy property (as we call it) which is associated with Minkowski’s Killing fields and is enjoyed by commutators of curved wave operators and, on the other hand, we exhibit a metric hierarchy property (as we call it) enjoyed by components of Einstein’s field equations in frames associated with our Euclidean-hyperboloidal foliation. The core of the argument is, on the one hand, the derivation of novel integral and pointwise estimates which lead us to almost sharp decay properties (at timelike, null, and spacelike infinity) and, on the other hand, the control of the (quasi-linear and semi-linear) coupling between the geometric and matter parts of the Einstein equations.

This paper investigates the global dynamics of the apparent horizon. We present an approach to establish its existence and its long-term behaviors. Our apparent horizon is constructed by solving the marginally outer trapped surface (MOTS) along each incoming null hypersurface. Based on the nonlinear hyperbolic estimates established in [21] by Klainerman-Szeftel under polarized axial symmetry, we prove that the corresponding apparent horizon is smooth, asymptotically null and converging to the event horizon eventually. To further address the local achronality of the apparent horizon, a new concept, called the null comparison principle, is introduced in this paper. For three typical scenarios of gravitational collapse, our null comparison principle is tested and verified, which guarantees that the apparent horizon must be piecewise spacelike or piecewise null. In addition, we also validate and provide new proofs for several physical laws along the apparent horizon.

We consider the 3D Boltzmann equation with the constant collision kernel. We investigate the well/ill-posedness problem using the methods from nonlinear dispersive PDEs. We construct a family of special solutions, which are neither near equilibrium nor self-similar, to the equation, and prove that the well/ill-posedness threshold in (H^{s}) Sobolev space is exactly at regularity (s=1), despite the fact that the equation is scale invariant at ( s=frac{1}{2}).

In this paper, we study the 2D free boundary incompressible Euler equations with surface tension, where the fluid domain is periodic in (x_1), and has finite depth. We construct initial data with a flat free boundary and arbitrarily small velocity, such that the gradient of vorticity grows at least double-exponentially for all times during the lifespan of the associated solution. This work generalizes the celebrated result by Kiselev–Šverák [17] to the free boundary setting. The free boundary introduces some major challenges in the proof due to the deformation of the fluid domain and the fact that the velocity field cannot be reconstructed from the vorticity using the Biot-Savart law. We overcome these issues by deriving uniform-in-time control on the free boundary and obtaining pointwise estimates on an approximate Biot-Savart law.

In the paper by Klainerman, Rodnianski and Tao [7], they give a physical space proof to a classical result of Klainerman and Machedon [3] for the bilinear space-time estimates of null forms. In this paper, we shall give an alternative and very simple physical space proof of the same bilinear estimates by applying div-curl type lemma of Zhou [14] and Wang and Zhou [12, 13]. We have only attained the limited goal of proving the bilinear estimates for the dyadic piece of the solution. Summing up the dyadic parts leads to the bilinear estimates with a Besov loss. As far as we know, the later development of wave maps [1, 2, 8,9,10,11], and the proof of bounded curvature theorem [5, 6] rely on basic ideas of Klainerman and Machedon [3] as well as Klainerman, Rodnianski and Tao [7].

We consider the two-dimensional stationary Navier–Stokes equations on the whole plane (mathbb {R}^2). In the higher-dimensional cases (mathbb {R}^n) with (n geqslant 3), the well-posedness and ill-posedness in scaling critical spaces are well-investigated by numerous papers. However, the corresponding problem in the two-dimensional whole plane case has been known as an open problem due to inherent difficulties of two-dimensional analysis. The aim of this paper is to address this issue and solve it negatively. More precisely, we prove the ill-posedness in the scaling critical Besov spaces based on (L^p(mathbb {R}^2)) for all (1 leqslant p leqslant 2) in the sense of the discontinuity of the solution map. To overcome the difficulties, we propose a new method based on the contradictory argument that reduces the problem to the analysis of the corresponding nonstationary Navier–Stokes equations and shows the existence of nonstationary solutions with strange large time behavior, if we suppose to contrary that the stationary problem is well-posed.