Annals of Pde最新文献

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.

Let ({{mathcal {H}}}) denote the future outgoing null hypersurface emanating from a spacelike 2-sphere S in a vacuum spacetime (({{mathcal {M}}},textbf{g})). In this paper we study the so-called canonical foliation on ({{mathcal {H}}}) introduced in [13, 22] and show that the corresponding geometry is controlled locally only in terms of the initial geometry on S and the (L^2) curvature flux through ({{mathcal {H}}}). In particular, we show that the ingoing and outgoing null expansions ({textrm{tr}}chi ) and ({textrm{tr}}{{{underline{chi }}}}) are both locally uniformly bounded. The proof of our estimates relies on a generalisation of the methods of [15,16,17] and [1, 2, 26, 32] where the geodesic foliation on null hypersurfaces ({{mathcal {H}}}) is studied. The results of this paper, while of independent interest, are essential for the proof of the spacelike-characteristic bounded (L^2) curvature theorem [12].

In this paper we study the spacelike-characteristic Cauchy problem for the Einstein vacuum equations. Given initial data on a maximal spacelike hypersurface (Sigma simeq overline{B_1} subset {{mathbb {R}}}^3) and the outgoing null hypersurface ({{mathcal {H}}}) emanating from ({partial }Sigma ), we prove a priori estimates for the resulting future development in terms of low-regularity bounds on the initial data at the level of curvature in (L^2). The proof uses the bounded (L^2) curvature theorem [22], the extension procedure for the constraint equations [12], Cheeger-Gromov theory in low regularity [13], the canonical foliation on null hypersurfaces in low regularity [15] and global elliptic estimates for spacelike maximal hypersurfaces.

We show novel types of uniqueness and rigidity results for Schrödinger equations in either the nonlinear case or in the presence of a complex-valued potential. As our main result we obtain that the trivial solution (u=0) is the only solution for which the assumptions (u(t=0)vert _{D}=0, u(t=T)vert _{D}=0) hold, where (Dsubset mathbb {R}^d) are certain subsets of codimension one. In particular, D is discrete for dimension (d=1). Our main theorem can be seen as a nonlinear analogue of discrete Fourier uniqueness pairs such as the celebrated Radchenko–Viazovska formula in [21], and the uniqueness result of the second author and M. Sousa for powers of integers [22]. As an additional application, we deduce rigidity results for solutions to some semilinear elliptic equations from their zeros.

This paper is concerned with the relativistic Boltzmann equation without angular cutoff. We establish the global-in-time existence, uniqueness and asymptotic stability for solutions nearby the relativistic Maxwellian. We work in the case of a spatially periodic box. We assume the generic hard-interaction and soft-interaction conditions on the collision kernel that were derived by Dudyński and Ekiel-Je(dot{text {z}})ewska (Comm. Math. Phys. 115(4):607–629, 1985) in [32], and our assumptions include the case of Israel particles (J. Math. Phys. 4:1163–1181, 1963) in [56]. In this physical situation, the angular function in the collision kernel is not locally integrable, and the collision operator behaves like a fractional diffusion operator. The coercivity estimates that are needed rely crucially on the sharp asymptotics for the frequency multiplier that has not been previously established. We further derive the relativistic analogue of the Carleman dual representation for the Boltzmann collision operator. This resolves the open question of perturbative global existence and uniqueness without the Grad’s angular cut-off assumption.