Annals of Pde最新文献

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.

Global behavior of solutions is studied for the nonlinear Klein-Gordon equation with a focusing power nonlinearity and a damping term in the energy space on the Euclidean space. We give a complete classification of solutions into 5 types of global behavior for all initial data in a small neighborhood of each superposition of two ground states (2-solitons) with the opposite signs and sufficient spatial distance. The neighborhood contains, for each sign of the ground state, the manifold with codimension one in the energy space, consisting of solutions that converge to the ground state at time infinity. The two manifolds are joined at their boundary by the manifold with codimension two of solutions that are asymptotic to 2-solitons moving away from each other. The connected union of these three manifolds separates the rest of the neighborhood into the open set of global decaying solutions and that of blow-up.

We study the vanishing viscosity limit of the one-dimensional Burgers equation near nondegenerate shock formation. We develop a matched asymptotic expansion that describes small-viscosity solutions to arbitrary order up to the moment the first shock forms. The inner part of this expansion has a novel structure based on a fractional spacetime Taylor series for the inviscid solution. We obtain sharp vanishing viscosity rates in a variety of norms, including (L^infty ). Comparable prior results break down in the vicinity of shock formation. We partially fill this gap.

A fundamental question in fluid dynamics concerns the formation of discontinuous shock waves from smooth initial data. We prove that from smooth initial data, smooth solutions to the 2d Euler equations in azimuthal symmetry form a first singularity, the so-called (C^{frac{1}{3}} ) pre-shock. The solution in the vicinity of this pre-shock is shown to have a fractional series expansion with coefficients computed from the data. Using this precise description of the pre-shock, we prove that a discontinuous shock instantaneously develops after the pre-shock. This regular shock solution is shown to be unique in a class of entropy solutions with azimuthal symmetry and regularity determined by the pre-shock expansion. Simultaneous to the development of the shock front, two other characteristic surfaces of cusp-type singularities emerge from the pre-shock. These surfaces have been termed weak discontinuities by Landau & Lifschitz [12, Chapter IX, §96], who conjectured some type of singular behavior of derivatives along such surfaces. We prove that along the slowest surface, all fluid variables except the entropy have (C^{1, {frac{1}{2}} }) one-sided cusps from the shock side, and that the normal velocity is decreasing in the direction of its motion; we thus term this surface a weak rarefaction wave. Along the surface moving with the fluid velocity, density and entropy form (C^{1, {frac{1}{2}} }) one-sided cusps while the pressure and normal velocity remain (C^2); as such, we term this surface a weak contact discontinuity.

In this work, we derive the globally precise late-time asymptotics for the spin-({mathfrak {s}}) fields on a Schwarzschild background, including the scalar field (({mathfrak {s}}=0)), the Maxwell field (({mathfrak {s}}=pm 1)) and the linearized gravity (({mathfrak {s}}=pm 2)). The conjectured Price’s law in the physics literature which predicts the sharp rates of decay of the spin (s=pm {mathfrak {s}}) components towards the future null infinity as well as in a compact region is shown. Further, we confirm the heuristic claim by Barack and Ori that the spin (+1, +2) components have an extra power of decay at the event horizon than the conjectured Price’s law. The asymptotics are derived via a unified, detailed analysis of the Teukolsky master equation that is satisfied by all these components.

Inspired by the numerical evidence of a potential 3D Euler singularity [54, 55], we prove finite time singularity from smooth initial data for the HL model introduced by Hou-Luo in [54, 55] for the 3D Euler equations with boundary. Our finite time blowup solution for the HL model and the singular solution considered in [54, 55] share some essential features, including similar blowup exponents, symmetry properties of the solution, and the sign of the solution. We use a dynamical rescaling formulation and the strategy proposed in our recent work in [11] to establish the nonlinear stability of an approximate self-similar profile. The nonlinear stability enables us to prove that the solution of the HL model with smooth initial data and finite energy will develop a focusing asymptotically self-similar singularity in finite time. Moreover the self-similar profile is unique within a small energy ball and the (C^gamma ) norm of the density (theta ) with (gamma approx 1/3) is uniformly bounded up to the singularity time.