Annals of Pde最新文献

Given a conformally transversally anisotropic manifold (M, g), we consider the semilinear elliptic equation

We show that an a priori unknown smooth function q can be uniquely determined from the knowledge of the Dirichlet-to-Neumann map associated to the equation. This extends the previously known results of the works Feizmohammadi and Oksanen (J Differ Equ 269(6):4683–4719, 2020), Lassas et al. (J Math Pures Appl 145:44–82, 2021). Our proof is based on over-differentiating the equation: We linearize the equation to orders higher than the order two of the nonlinearity (qu^2), and introduce non-vanishing boundary traces for the linearizations. We study interactions of two or more products of the so-called Gaussian quasimode solutions to the linearized equation. We develop an asymptotic calculus to solve Laplace equations, which have these interactions as source terms.

This is a follow-up of [5] on the general covariant modulated (GCM) procedure in perturbations of Kerr. In this paper, we construct GCM hypersurfaces, which play a central role in extending GCM admissible spacetimes in [7] where decay estimates are derived in the context of nonlinear stability of Kerr family for (|a|ll m). As in [4], the central idea of the construction of GCM hypersurfaces is to concatenate a 1–parameter family of GCM spheres of [5] by solving an ODE system. The goal of this paper is to get rid of the symmetry restrictions in the GCM procedure introduced in [4] and thus remove an essential obstruction in extending the results to a full stability proof of the Kerr family.

This is the second and last paper of a series aimed at solving the local Cauchy problem for polarized ({mathbb {U}}(1)) symmetric solutions to the Einstein vacuum equations featuring the nonlinear interaction of three small amplitude impulsive gravitational waves. Such solutions are characterized by their three singular “wave-fronts” across which the curvature tensor is allowed to admit a delta singularity. Under polarized ({mathbb {U}}(1)) symmetry, the Einstein vacuum equations reduce to the Einstein–scalar field system in ((2+1)) dimensions. In this paper, we focus on the wave estimates for the scalar field in the reduced system. The scalar field terms are the most singular ones in the problem, with the scalar field only being Lipschitz initially. We use geometric commutators to prove energy estimates which reflect that the singularities are localized, and that the scalar field obeys additional fractional-derivative regularity, as well as regularity along appropriately defined “good directions”. The main challenge is to carry out all these estimates using only the low-regularity properties of the metric. Finally, we prove an anisotropic Sobolev embedding lemma, which when combined with our energy estimates shows that the scalar field is everywhere Lipschitz, and that it obeys additional (C^{1,theta }) estimates away from the most singular region.

We construct a stable right inverse for the divergence operator in non-cylindrical domains in space-time. The domains are assumed to be Hölder regular in space and evolve continuously in time. The inverse operator is of Bogovskij type, meaning that it attains zero boundary values. We provide estimates in Sobolev spaces of positive and negative order with respect to both time and space variables. The regularity estimates on the operator depend on the assumed Hölder regularity of the domain. The results can naturally be connected to the known theory for Lipschitz domains. The most precise estimates are given in weighted spaces, where the weight depends on the distance to the boundary. This allows for the deficit to be captured precisely in the vicinity of irregularities of the boundary. As an application, we prove refined pressure estimates for weak and very weak solutions to Navier–Stokes equations in time dependent domains.

We establish uniform regularity estimates with respect to the Mach number for the three-dimensional free surface compressible Navier-Stokes system in the case of slightly well-prepared initial data in the sense that the acoustic components like the divergence of the velocity field are of size (sqrt{varepsilon }), (varepsilon ) being the Mach number. These estimates allow us to justify the convergence towards the free surface incompressible Navier-Stokes system in the low Mach number limit. One of the main difficulties is the control of the regularity of the surface in presence of boundary layers with fast oscillations.

This paper establishes a mathematical proof of the blue-shift instability at the sub-extremal Kerr Cauchy horizon for the linearised vacuum Einstein equations. More precisely, we exhibit conditions on the (s=+2) Teukolsky field, consisting of suitable integrated upper and lower bounds on the decay along the event horizon, that ensure that the Teukolsky field, with respect to a frame that is regular at the Cauchy horizon, becomes singular. The conditions are in particular satisfied by solutions of the Teukolsky equation arising from generic and compactly supported initial data by the recent work [51] of Ma and Zhang for slowly rotating Kerr.

We consider the self-dual Chern–Simons–Schrödinger equation (CSS) under equivariance symmetry. Among others, (CSS) has a static solution Q and the pseudoconformal symmetry. We study the quantitative description of pseudoconformal blow-up solutions u such that

When the equivariance index (mge 1), we construct a set of initial data (under a codimension one condition) yielding pseudoconformal blow-up solutions. Moreover, when (mge 3), we establish the codimension one property and Lipschitz regularity of the initial data set, which we call the blow-up manifold. This is a forward construction of blow-up solutions, as opposed to authors’ previous work [25], which is a backward construction of blow-up solutions with prescribed asymptotic profiles. In view of the instability result of [25], the codimension one condition established in this paper is expected to be optimal. We perform the modulation analysis with a robust energy method developed by Merle, Raphaël, Rodnianski, and others. One of our crucial inputs is a remarkable conjugation identity, which (with self-duality) enables the method of supersymmetric conjugates as like Schrödinger maps and wave maps. It suggests how we proceed to higher order derivatives while keeping the Hamiltonian form and construct adapted function spaces with their coercivity relations. More interestingly, it shows a deep connection with the Schrödinger maps at the linearized level and allows us to find a repulsivity structure for higher order derivatives.

In this paper, we initiate the study of global stability for anisotropic systems of quasilinear wave equations. Equations of this kind arise naturally in the study of crystal optics, and they exhibit birefringence. We introduce a physical space strategy based on bilinear energy estimates that allows us to prove decay for the nonlinear problem. This uses decay for the homogeneous wave equation as a black box. The proof also requires us to interface this strategy with the vector field method and take advantage of the scaling vector field. A careful analysis of the spacetime geometry of the interaction between waves is necessary in the proof.

In 1990, based on numerical and formal asymptotic analysis, Ori and Piran predicted the existence of selfsimilar spacetimes, called relativistic Larson-Penston solutions, that can be suitably flattened to obtain examples of spacetimes that dynamically form naked singularities from smooth initial data, and solve the radially symmetric Einstein-Euler system. Despite its importance, a rigorous proof of the existence of such spacetimes has remained elusive, in part due to the complications associated with the analysis across the so-called sonic hypersurface. We provide a rigorous mathematical proof. Our strategy is based on a delicate study of nonlinear invariances associated with the underlying non-autonomous dynamical system to which the problem reduces after a selfsimilar reduction. Key technical ingredients are a monotonicity lemma tailored to the problem, an ad hoc shooting method developed to construct a solution connecting the sonic hypersurface to the so-called Friedmann solution, and a nonlinear argument to construct the maximal analytic extension of the solution. Finally, we reformulate the problem in double-null gauge to flatten the selfsimilar profile and thus obtain an asymptotically flat spacetime with an isolated naked singularity.

In this paper we prove rigidity for blowup solutions to the focusing, mass-critical nonlinear Schrödinger equation in dimensions (2 le d le 15) with mass equal to the mass of the soliton. We prove that the only such solutions are the solitons and the pseudoconformal transformation of the solitons. We show that this implies a Liouville result for the nonlinear Schrödinger equation.