arXiv subjects

Igor Rodnianski

Publications and source records attributed to Igor Rodnianski.

62 records · Page 4Linked to original sources

arXiv2004

Global existence for the Einstein vacuum equations in wave coordinates

We prove global stability of Minkowski space for the Einstein vacuum equations in harmonic (wave) coordinate gauge for the set of restricted data coinciding with Schwartzschild solution in the neighborhood of space-like infinity. The result contradicts previous beliefs that wave coordinates are "unstable in the large" and provides an alternative approach to the stability problem originally solved (for unrestricted data, in a different gauge and with a precise description of the asymptotic behavior at null infinity) by D. Christodoulou and S. Klainerman. Using the wave coordinate gauge we recast the Einstein equations as a system of quasilinear wave equations and, in absence of the classical null condition, establish a small data global existence result. In our previous work we introduced the notion of a eak nul condition and showed that the Einstein equations in wave coordinates satisfy this condition. The result of this paper relies on this observation and combines it with the vector field method based on the symmetries of the standard Minkowski space. In a forthcoming paper we will address the question of stability of Minkowski space for the Einstein vacuum equations in wave coordinates for all "small" asymptotically flat data and the case of the Einstein equations coupled to a scalar field.

math.AP↗

arXiv2004

Global regularity for the Maxwell-Klein-Gordon equation with small critical Sobolev norm in high dimensions

We show that in dimensions $n \geq 6$ that one has global regularity for the Maxwell-Klein-Gordon equations in the Coulomb gauge provided that the critical Sobolev norm $\dot H^{n/2-1} \times \dot H^{n/2-2}$ of the initial data is sufficiently small. These results are analogous to those recently obtained for the high-dimensional wave map equation but unlike the wave map equation, the Coulomb gauge non-linearity cannot be iterated away directly. We shall use a different approach, proving Strichartz estimates for the covariant wave equation. This in turn will be achieved by use of Littlewood-Paley multipliers, and a global parametrix for the covariant wave equation constructed using a truncated, microlocalized Cronstrom gauge.

math.AP↗

arXiv2004

A note on boundary value problems for black hole evolutions

In recent work of Allen at. al., heuristic and numerical arguments were put forth to suggest that boundary value problems for black hole evolution, where an appropriate Sommerfeld radiation condition is imposed, would fail to produce Price law tails. The interest in this issue lies in its possible implications for numerical relativity, where black hole evolution is typically studied in terms of such boundary formulations. In this note, it is shown rigorously that indeed, Price law tails do not arise in this case, i.e. that Sommerfeld (and more general) radiation conditions lead to decay faster than any polynomial power. Our setting is the collapse of a spherically symmetric self-gravitating scalar field. We allow an additional gravitationally coupled Maxwell field. The proof also applies to the easier problem of a spherically symmetric solution of the wave equation on a Schwarzschild or Reissner-Nordstrom background. The method relies on previous work of the authors.

gr-qc↗

arXiv2003

Sharp trace theorems for null hypersurfaces on Einstein metrics with finite curvature flux

The main objective of the paper is to prove a geometric version of sharp trace and product estimates on null hypersurfaces with finite curvature flux. These estimates play a crucial role to control the geometry of such null hypersurfaces. The paper is based on an invariant version of the classical Littlewood -Paley theory, in a noncommutative setting, defined via heat flow on surfaces.

math.AP↗

arXiv2003

A geometric approach to the Littlewood-Paley theory

We develop a geometric invariant Littlewood-Paley theory for arbitrary tensors on a compact 2 dimensional manifold. We show that all the important features of the classical LP theory survive with estimates which depend only on very limited regularity assumptions on the metric. We give invariant descriptions of Sobolev and Besov spaces and prove some sharp product inequalities. This theory has being developed in connection to the work of the authors on the geometry of null hypersurfaces with a finite curvature flux condition, see \cite{KR1}, \cite{KR3}. We are confident however that it can be applied, and extended, to many different situations.

math.AP↗

arXiv2003

Causal Geometry of Einstein-Vacuum Spacetimes with Finite Curvature Flux

One of the central difficulties of settling the $L^2$-bounded curvature conjecture for the Einstein -Vacuum equations is to be able to control the causal structure of spacetimes with such limited regularity. In this paper we show how to circumvent this difficulty by showing that the geometry of null hypersurfaces of Enstein-Vacuum spacetimes can be controlled in terms of initial data and the total curvature flux through the hypersurface.

math.AP↗

arXiv2001

A physical space approach to wave equation bilinear estimates

Bilinear estimates for the wave equation in Minkowski space are normally proven using the Fourier transform and Plancherel's theorem. However, such methods are difficult to carry over to non-flat situations (such as wave equations with rough metrics, or with connections with non-zero curvature). In this note we give some techniques to prove these estimates which rely more on physical space methods such as vector fields, tube localization, splitting into coarse and fine scales, and induction on scales (in the spirit of recent papers of Wolff).

math.AP↗

arXiv1997

Does a quantum particle know the time?

We study the spatial regularity of the fundamental solution E(t,x) of the Schrödinger equation on the circle in a scale of Besov spaces. Although the fundamental solution is not smooth, we reveal a fine change of regularity of E(t,x) at different times t. For rational t, E(t,x) is a weighted sum of delta-functions, and, therefore, exhibits the same regularity as at t=0. For irrational t, the regularity of E(t,x) is better and depends on how well t is approximated by rationals. For badly approximated t (e.g., when t is a quadratic irrational, or, more generally, when t has bounded quotients in its continued fraction expansion), E(t,x) is a "1/2-derivative" more regular than E(0,x). For a generic irrational t, E(t,x) is almost "1/2-derivative" more regular. However, the better t is approximated by rationals, the lower is the regularity of E(t,x). We describe different thin classes of irrationals which prescribe their particular regularity to the fundamental solution. These classes are singled out and characterized by the behavior of the continued fraction expansions of their members.

quant-ph↗