Sergiu Klainerman

On almost global existence for nonrelativistic wave equations in 3D

Almost global solutions are constructed to three-dimensional, quadratically nonlinear wave equations. The proof relies on generalized energy estimates and a new decay estimate. The method applies to equations that are only classically invariant, such as the nonlinear system of hyperelasticity.

The Evolution Problem in General Relativity

Preface * Introduction * Analytic methods in the initial value problem * Definitions and results * Estimates for the connection coefficients * Estimates for the curvature tensor * The error estimates * The initial hypersurface and the last slice * Conclusions * Bibliography * Index

Remark on the optimal regularity for equations of wave maps type

The goal of this paper is to review the estimates proved in [3] and extend them to all dimensions, in particular to the harder case of space dimension 2. As in [3], the main application we have in view is to equations of Wave Maps type, namely systems of equations of the form φ + ΓIJK(φ)Q0(φ J , φ) = 0. (1) Here, = −∂ t +∆ denotes the standard D’Alembertian in R , and Q0 is the null form Q0(φ, ψ) = ∂αφ · ∂ ψ = −∂tφ∂tψ + n ∑

BILINEAR ESTIMATES AND APPLICATIONS TO NONLINEAR WAVE EQUATIONS

We undertake a systematic review of results proved in [26, 27, 30-32] concerning local well-posedness of the Cauchy problem for certain systems of nonlinear wave equations, with minimal regularity assumptions on the initial data. Moreover we give a considerably simplified and unified treatment of these results and provide also complete proofs for large data. The paper is also intended as an introduction to and survey of current research in the very active area of nonlinear wave equations. The key ingredients throughout the survey are the use of the null structure of the equations we consider and, intimately tied to it, bilinear estimates.

On the Uniqueness of Solutions to the Gross-Pitaevskii Hierarchy

The purpose of this note is to give a new proof of uniqueness of the Gross-Pitaevskii hierarchy, first established in [1], in a different space, based on space-time estimates similar in spirit to those of [2].

Decay and regularity for the schrödinger equation

Consider the Schrödinger equation {fx25-1}.The following estimates are proved: (A) IfV≡0 then for any 0≤α<1/2, {fx25-2}, and for α=1/2,s>1/2, {fx25-3} (B) If |V(x)|≤C(1+|x|2)−1−δ, δ>0, then (if 0 is neither an eigenvalue nor a resonance of −Δ+V), {fx25-4}.

Bilinear space-time estimates for homogeneous wave equations

Abstract In this paper, we pursue a systematic treatment of the regularity theory for products and bilinear forms of solutions of the homogeneous wave equation. We discuss necessary and sufficient conditions for the validity of bilinear estimates, based on L 2 norms in space and time, of derivatives of products of solutions. Also, we give necessary conditions and formulate some conjectures for similar estimates based on L q t L x r norms.

Finite energy solutions of the Yang-Mills equations in R3+1

Yang-Mills equations in R3+1 is well-posed in the energy norm. This means that for an appropriate gauge condition, we construct local, unique solutions in a time interval which depends only on the size of the energy norm of the data. Since the energy norm is left invariant by the Yang-Mills flow the local solution is automatically extended to the entire space-time. Thus our results, which settle a problem stated in [Str], imply the well-known, fundamental, regularity result of Eardley and Moncrief [E-M]. That result, proved in the temporal gauge, requires a higher degree of smoothness for the data. Our main result, proved also in the temporal gauge, allows us to extend the concept of solutions to arbitrary finite energy initial data. The solutions are automatically unique in the class of solutions obtained by our procedure. Moreover, the global regularity proof given by Eardley and Moncrief depends in an essential way on the specific properties of the fundamental solution of the wave equation in the flat Minkowski space-time R3+1, namely the strong Huygens Principle. Indeed due mainly to this fact their proof does not seem to extend to general curved space-times. We have reasons to hope that the very different approach we take here will resolve this difficulty. The basic ingredients of our method are: 1. The introduction of appropriate local Coulomb gauges adapted to the causal structure of the equations. 2. An appropriate method of localizing the new space-time estimates for

On the optimal local regularity for the Yang-Mills equations in ℝ⁴⁺¹

Here Fαβ = ∂aAβ − ∂bAb + [ Aα , Aβ ] represents the curvature of a connection 1form, or gauge field, A = Aαdx with values in the Lie algebra of a classical Lie group of matrices such as SU(N) or SO(N). The equations (1.1) are invariant, up to a conjugation, under the gauge transformations Aα −→ OAαO−1 − ∂αOO−1, with O elements of the corresponding group. They are obtained by considering the critical points corresponding to the Lagrangian 14 〈Fαβ , Fαβ〉 with 〈 , 〉 the positive definite Killing form of the Lie algebra. The equations have a finite number of conservation laws, among them the total energy,

On the uniqueness of smooth, stationary black holes in vacuum

A fundamental conjecture in general relativity asserts that the domain of outer communication of a regular, stationary, four dimensional, vacuum black hole solution is isometrically diffeomorphic to the domain of outer communication of a Kerr black hole. So far the conjecture has been resolved, by combining results of Hawking [17], Carter [4] and Robinson [28], under the additional hypothesis of non-degenerate horizons and real analyticity of the space-time. We develop a new strategy to bypass analyticity based on a tensorial characterization of the Kerr solutions, due to Mars [24], and new geometric Carleman estimates. We prove, under a technical assumption (an identity relating the Ernst potential and the Killing scalar) on the bifurcate sphere of the event horizon, that the domain of outer communication of a smooth, regular, stationary Einstein vacuum spacetime of dimension 4 is locally isometric to the domain of outer communication of a Kerr spacetime.