[Paper Review] Parametrix for wave equations on a rough background II: construction and control at initial time

Jérémie Szeftel|arXiv (Cornell University)|Apr 8, 2012

Advanced Mathematical Physics Problems18 references19 citations

TL;DR

This paper constructs and controls a parametrix for the wave equation □_gφ = 0 on a rough Lorentzian background metric g satisfying the Einstein vacuum equations, under only L² bounds on the curvature tensor R. Using a plane wave parametrix and careful analysis of error terms involving vector fields and geometric quantities, it establishes uniform L² control of the parametrix and its error at initial time, a critical step toward proving the bounded L² curvature conjecture for general relativity.

ABSTRACT

This is the second of a sequence of four papers \cite{param1}, \cite{param2}, \cite{param3}, \cite{param4} dedicated to the construction and the control of a parametrix to the homogeneous wave equation $\square_{\bf g} ϕ=0$, where ${\bf g}$ is a rough metric satisfying the Einstein vacuum equations. Controlling such a parametrix as well as its error term when one only assumes $L^2$ bounds on the curvature tensor ${\bf R}$ of ${\bf g}$ is a major step of the proof of the bounded $L^2$ curvature conjecture proposed in \cite{Kl:2000}, and solved by S. Klainerman, I. Rodnianski and the author in \cite{boundedl2}. On a more general level, this sequence of papers deals with the control of the eikonal equation on a rough background, and with the derivation of $L^2$ bounds for Fourier integral operators on manifolds with rough phases and symbols, and as such is also of independent interest.

Motivation & Objective

• To construct a parametrix for the wave equation □_gφ = 0 on a rough Lorentzian background metric g satisfying the Einstein vacuum equations.
• To control the parametrix and its error term using only L² bounds on the curvature tensor R of g, without higher regularity assumptions.
• To provide the foundational analysis at initial time necessary for the full proof of the bounded L² curvature conjecture.
• To develop tools for handling Fourier integral operators with rough phases and symbols on manifolds with limited regularity.
• To establish uniform L² control of geometric quantities and derivatives of the parametrix at t = 0, essential for subsequent energy estimates.

Proposed method

• Uses a plane wave parametrix representation: Sf(t,x) = ∫_{S²}∫₀^∞ e^{iλu(t,x,ω)} f(λω) λ² dλ dω.
• Constructs the parametrix via a geometric phase function u(t,x,ω) solving the eikonal equation on a rough background.
• Analyzes the error term in the wave equation by decomposing it into components involving vector fields N, N′ and geometric quantities like g(N,N′).
• Applies structure equations for null vector fields N and N′ to control derivatives of the metric and connection coefficients.
• Uses a decomposition of the divergence term into components involving A₁ to A₅, with careful control of singularities via (1 - g(N,N′)²)⁻¹ and (λ - λ′(a/a′)g(N,N′))⁻¹ factors.
• Establishes control of key terms like ∇_N(g(N,N′)) / (1 - g(N,N′))¹ᐟ² and ∇_{N−g(N,N′)N′}(g(N,N′)) / (1 - g(N,N′))³ᐟ² via prior estimates in Appendix A.

Experimental results

Research questions

• RQ1 How can a parametrix be constructed for the wave equation □_gφ = 0 when the metric g is only known to have L² curvature?
• RQ2 What is the precise structure of the error term in the parametrix approximation under minimal regularity assumptions?
• RQ3 How can the divergence terms arising from the parametrix error be controlled uniformly at initial time?
• RQ4 What role do geometric quantities like g(N,N′), ∇_N(g(N,N′)), and second derivatives of the metric play in the error analysis?
• RQ5 Can uniform L² bounds on the parametrix and its error be established using only L² bounds on curvature, without higher Sobolev regularity?

Key findings

• The parametrix construction is valid under only L² bounds on the curvature tensor R, achieving the minimal regularity required for the bounded L² curvature conjecture.
• The error term in the parametrix approximation is controlled in L² at initial time, with uniform bounds depending only on the L² norm of R and initial data size.
• The analysis establishes control of critical geometric terms such as ∇_N(g(N,N′)) / (1 - g(N,N′))¹ᐟ² and ∇_{N−g(N,N′)N′}(g(N,N′)) / (1 - g(N,N′))³ᐟ².
• The divergence terms in the error decomposition are shown to take a form compatible with subsequent energy estimates, via explicit control of A₁ to A₅ terms.
• The parametrix and its error are uniformly bounded in L² on Σ₀ under the assumption that ‖R‖_{L²(Σ₀)} ≤ ε and r_vol(Σ₀,1) ≥ 1/2.
• This control at initial time is a necessary ingredient in the full proof of the bounded L² curvature conjecture, as established in [12].