Rough Solutions of Vacuum Einstein Equations in Cmc Gauge: Energy Estimates

We provide a fully tensorial approach to bound the Bel-Robinson energy of -order under the assumption ∫ t0+T t0 ( ‖k‖L∞(Σt) + ∑ λ≥0 ‖Pλ∆ 1 2Ric‖L∞(Σt) ) dt <∞, which is our first step to establish local well-posedness for the solution of Einstein vacuum equation in CMC gauge with H2+ data . 1. Geometric set-up and Main result 1.1. Heat flow. Let 4 = g∇i∇j denote the Laplace-Betrami operator on Σ. For any tensor field F we consider the heat flow ∂τU(τ)F −4U(τ)F = 0, U(0)F = F. This defines a family of linear operators {U(τ) : τ ≥ 0} which are self-adjoint and form a semigroup, that is, U(τ)U(τ ′) = U(τ + τ ′) and 〈U(τ)F,G〉 = 〈F,U(τ)G〉 for any tensor fields F,G and any τ, τ ′ ≥ 0. Moreover, U(τ) commutes with 4 for all τ ≥ 0. The following properties of heat flow have been proved in [3]. Proposition 1.1. For any tensor field F defined on Σ there hold ‖∇U(τ)F‖2 ≤ ‖∇F‖2, (1.1) ‖∇U(τ)F‖2 ≤ τ‖F‖2, (1.2) ‖U(τ)∇F‖2 ≤ τ‖F‖2, (1.3) ‖4U(τ)F‖2 ≤ τ‖F‖2. (1.4) Moreover, for every 2 ≤ p ≤ ∞ there holds (1.5) ‖U(τ)F‖p ≤ ‖F‖p. As in [3], for any a ∈ C we may introduce the two families of operators Λ = (I −4) and D = (−4) in the way that when <(a) < 0 we define ΛF = 1 Γ(−a/2) ∫ ∞ 0 τ−a/2−1e−τU(τ)Fdτ, DF = 1 Γ(−a/2) ∫ ∞ 0 τ−a/2−1U(τ)Fdτ while when <(a) < 0 we define Λ = Λa−2l(I −4), D = Da−2l(−4)l, where l is a positive integer such that <(a) < 2l. We remark that ΛF with a ∈ C and D with <(a) > 0 are well-defined for all tensor fields F . However, DF with <(a) < 0 makes sense only for F orthogonal to the kernel of 4. Thus, for the expression DF with <(a) < 0, it always means that D acts on the orthogonal 1