Annals of Pde最新文献

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})).

We prove global in time well-posedness for perturbations of the 2D stochastic Navier–Stokes equations

driven by additive space-time white noise ( xi ), with perturbation ( zeta ) in the Hölder–Besov space (mathcal {C}^{-2 + 3kappa } ), periodic boundary conditions and initial condition ( u_{0} in mathcal {C}^{-1 + kappa } ) for any ( kappa >0 ). The proof relies on an energy estimate which in turn builds on a dynamic high-low frequency decomposition and tools from paracontrolled calculus. Our argument uses that the solution to the linear equation is a ( log )–correlated field, yielding a double exponential growth bound on the solution. Notably, our method does not rely on any explicit knowledge of the invariant measure to the SPDE, hence the perturbation ( zeta ) is not restricted to the Cameron–Martin space of the noise, and the initial condition may be anticipative. Finally, we introduce a notion of weak solution that leads to well-posedness for all initial data ( u_{0}) in ( L^{2} ), the critical space of initial conditions.

We prove asymptotic stability of the Poisson homogeneous equilibrium among solutions of the Vlasov–Poisson system in the Euclidean space (mathbb {R}^3). More precisely, we show that small, smooth, and localized perturbations of the Poisson equilibrium lead to global solutions of the Vlasov–Poisson system, which scatter to linear solutions at a polynomial rate as (trightarrow infty ). The Euclidean problem we consider here differs significantly from the classical work on Landau damping in the periodic setting, in several ways. Most importantly, the linearized problem cannot satisfy a “Penrose condition”. As a result, our system contains resonances (small divisors) and the electric field is a superposition of an electrostatic component and a larger oscillatory component, both with polynomially decaying rates.

In this paper, we address for the 2D Euler equations the existence of rigid time periodic solutions close to stationary radial vortices of type (f_0(|x|)textbf{1}_{{{,mathrm{mathbb {D}},}}}(x)), with ({{,mathrm{mathbb {D}},}}) the unit disc and (f_0) being a strictly monotonic profile with constant sign. We distinguish two scenarios according to the sign of the profile: defocusing and focusing. In the first regime, we have scarcity of the bifurcating curves associated with lower symmetry. However in the focusing case we get a countable family of bifurcating solutions associated with large symmetry. The approach developed in this work is new and flexible, and the explicit expression of the radial profile is no longer required as in [41] with the quadratic shape. The alternative for that is a refined study of the associated spectral problem based on Sturm-Liouville differential equation with a variable potential that changes the sign depending on the shape of the profile and the location of the time period. Deep hidden structure on positive definiteness of some intermediate integral operators are also discovered and used in a crucial way. Notice that a special study will be performed for the linear problem associated with the first mode founded on Prüfer transformation and Kneser’s Theorem on the non-oscillation phenomenon.

We prove a local existence theorem for the free boundary problem for a relativistic fluid in a fixed spacetime. Our proof involves an a priori estimate which only requires control of derivatives tangential to the boundary, which holds also in the Newtonian compressible case.