Annals of Pde最新文献

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.

In 2003, Klainerman and Nicolò [14] proved the stability of Minkowski in the case of the exterior of an outgoing null cone. Relying on the method used in [14], Caciotta and Nicolò [2] proved the stability of Kerr spacetime in external regions, i.e. outside an outgoing null cone far away from the Kerr event horizon. In this paper, we give a new proof of [2]. Compared to [2], we reduce the number of derivatives needed in the proof, simplify the treatment of the last slice, and provide a unified treatment of the decay of initial data which contains in particular the initial data considered by Klainerman and Szeftel in [20]. Also, concerning the treatment of curvature estimates, similar to [25], we replace the vectorfield method used in [2, 14] by (r^p)–weighted estimates introduced by Dafermos and Rodnianski in [8].

We study the asymptotics of complete Kähler-Einstein metrics on strictly pseudoconvex domains in (mathbb {C}^n) and derive a convergence theorem for solutions to the corresponding Monge-Ampère equation. If only a portion of the boundary is analytic, the solutions satisfy Gevrey type estimates for tangential derivatives. A counterexample for the model linearized equation suggests that there is no local convergence theorem for the complex Monge-Ampère equation.

The sphere is well-known as the only generic compact shrinker for mean curvature flow (MCF). In this paper, we characterize the generic dynamics of MCFs with a spherical singularity. In terms of the level set flow formulation of MCF, we establish that generically the arrival time function of level set flow with spherical singularity has at most (C^2) regularity.

We introduce the notions of static regular of type (I) and type (II) and show that they are sufficient conditions for local well-posedness of solving asymptotically flat, static vacuum metrics with prescribed Bartnik boundary data. We then show that hypersurfaces in a very general open and dense family of hypersurfaces are static regular of type (II). As applications, we confirm Bartnik’s static vacuum extension conjecture for a large class of Bartnik boundary data, including those that can be far from Euclidean and have large ADM masses, and give many new examples of static vacuum metrics with intriguing geometry.

We study a free transmission problem driven by degenerate fully nonlinear operators. Our first result concerns the existence of a viscosity solution to the associated Dirichlet problem. By framing the equation in the context of viscosity inequalities, we prove regularity results for the constructed viscosity solution to the problem. Our findings include regularity in ( C^{1,alpha }) spaces, and an explicit characterization of (alpha ) in terms of the degeneracy rates. We argue by perturbation methods, relating our problem to a homogeneous, fully nonlinear uniformly elliptic equation.

Given a measure (rho ) on a domain (Omega subset {mathbb {R}}^m), we study spacelike graphs over (Omega ) in Minkowski space with Lorentzian mean curvature (rho ) and Dirichlet boundary condition on (partial Omega ), which solve

The graph function also represents the electric potential generated by a charge (rho ) in electrostatic Born-Infeld’s theory. Even though there exists a unique minimizer (u_rho ) of the associated action

among functions (psi ) satisfying (|Dpsi | le 1), by the lack of smoothness of the Lagrangian density for (|Dpsi | = 1) one cannot guarantee that (u_rho ) satisfies the Euler-Lagrange equation ((mathcal{B}mathcal{I})). A chief difficulty comes from the possible presence of light segments in the graph of (u_rho ). In this paper, we investigate the existence of a solution for general (rho ). In particular, we give sufficient conditions to guarantee that (u_rho ) solves ((mathcal{B}mathcal{I})) and enjoys (log )-improved energy and (W^{2,2}_textrm{loc}) estimate. Furthermore, we construct examples which suggest a sharp threshold for the regularity of (rho ) to ensure the solvability of ((mathcal{B}mathcal{I})).