Breakdown Criteria for The Main Results Nonvacuum Spacetimes - PowerPoint PPT Presentation

Breakdown Criteria Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations Breakdown Criteria for The Main Results Nonvacuum Spacetimes Nonvacuum Einstein Equations The Main Theorem The Cauchy Problem Energy Estimates Generalized EMT’s Global Energy Estimates Arick Shao Local Energy Estimates Representation Formulas University of Toronto Preliminaries Applying the Parametrix The Generalized Formula October 7, 2011 The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria The Breakdown Problem Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations The Main Results Nonvacuum Spacetimes ◮ General question: Under what conditions can an The Main Theorem The Cauchy Problem existing local solution of an evolution equation on a Energy Estimates finite interval [ 0 , T ) be further extended past T? Generalized EMT’s Global Energy Estimates ◮ Why is this useful? Local Energy Estimates Representation Formulas 1. Characterize breakdown of solutions. Preliminaries 2. Global existence problem. Applying the Parametrix The Generalized Formula The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria Nonlinear Wave Equations Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum ◮ Equations of the form Equations The Main Results � φ = ( ∂φ ) 2 , Nonvacuum Spacetimes φ | t = 0 = φ 0 , ∂ t φ | t = 0 = φ 1 . The Main Theorem The Cauchy Problem Energy Estimates ◮ Local existence for H s -spaces. Generalized EMT’s Global Energy Estimates ◮ If local solution on [ 0 , T ) satisfies Local Energy Estimates Representation Formulas � ∂φ � L ∞ < ∞ , (1) Preliminaries Applying the Parametrix The Generalized Formula then solution can be extended past T . The Geometry of ◮ Time of existence controlled by H s -norms, which can be Null Cones uniformly controlled on [ 0 , T ) using (1). Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria Incompressible 3-d Euler equations Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations ◮ u : R 1 + 3 → R 3 , p : R 1 + 3 → R . The Main Results Nonvacuum Spacetimes The Main Theorem ∂ t u + u · ∇ u + ∇ p = 0, The Cauchy Problem Energy Estimates ∇ · u = 0. Generalized EMT’s Global Energy Estimates Local Energy Estimates Vorticity: ω = ∇ × u . Representation Formulas ◮ Beale, Kato, Majda (1984): If a local solution has ω Preliminaries Applying the Parametrix bounded in L 1 t L ∞ x , then it can be extended. The Generalized Formula The Geometry of ◮ Need not bound all of ∇ u . Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria Yang-Mills Equations Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations The Main Results ◮ Eardley, Moncrief (1982): global existence in R 1 + 3 . Nonvacuum Spacetimes The Main Theorem The Cauchy Problem ◮ Continuation criterion: � F � L ∞ < ∞ Energy Estimates ◮ F - Yang-Mills “curvature”. Generalized EMT’s ◮ � F � L ∞ controlled using wave equations and Global Energy Estimates Local Energy Estimates fundamental solutions. Representation Formulas ◮ Chru´ sciel, Shatah (1997): generalized to globally Preliminaries Applying the Parametrix hyperbolic ( 1 + 3 ) -dim. Lorentz manifolds. The Generalized Formula The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria Results for Vacuum Equations Arick Shao Introduction The Breakdown Problem Some Classical Results ◮ Einstein vacuum: ( 1 + 3 ) -dim. spacetimes ( M , g ) , The Einstein Vacuum Equations The Main Results Ric g = 0. Nonvacuum Spacetimes The Main Theorem The Cauchy Problem ◮ Anderson (2001): � R g � L ∞ < ∞ ⇒ solution can be Energy Estimates Generalized EMT’s extended. Global Energy Estimates Local Energy Estimates ◮ Geometric, requires two derivatives of g . Representation Formulas ◮ Other continuation criteria: Preliminaries Applying the Parametrix The Generalized Formula � ∂ g � L ∞ < ∞ , or � ∂ g � L 1 x < ∞ . The Geometry of t L ∞ Null Cones Preliminaries The Ricci Coefficients ◮ Not geometric, depends on choice of coordinates. The Sharp Trace Theorems

Breakdown Criteria Improved Results Arick Shao Introduction The Breakdown Problem ◮ Klainerman, Rodnianski (2008): improved breakdown Some Classical Results The Einstein Vacuum criterion for vacuum: Equations The Main Results Nonvacuum Spacetimes � k � L ∞ + �∇ ( log n ) � L ∞ < ∞ The Main Theorem The Cauchy Problem Energy Estimates ◮ CMC foliation, compact time slices. Generalized EMT’s ◮ k , n - second fundamental form, lapse of time slices. Global Energy Estimates Local Energy Estimates ◮ Geometric, do not need full coordinate system. Representation ◮ k and ∇ ( log n ) at level of ∂ g , but do not cover all Formulas Preliminaries components of ∂ g . Applying the Parametrix The Generalized Formula ◮ D. Parlongue (2008): vacuum, maximal foliation, The Geometry of asymptotically flat time slices, replaced L ∞ by L 2 Null Cones t L ∞ x . Preliminaries The Ricci Coefficients ◮ Q. Wang (2010): vacuum, CMC, compact time slices, The Sharp Trace Theorems replaced L ∞ by L 1 t L ∞ x .

Breakdown Criteria General Einstein Equations Arick Shao ◮ Spacetime ( M , g , Φ ) , Φ - matter fields. Introduction The Breakdown Problem ◮ Einstein equations: Some Classical Results The Einstein Vacuum Equations R αβ − 1 The Main Results 2 Rg αβ = Q αβ . Nonvacuum Spacetimes The Main Theorem The Cauchy Problem Q - energy-momentum tensor. Energy Estimates Generalized EMT’s ◮ Einstein-scalar ( Φ = φ - scalar): Global Energy Estimates Local Energy Estimates Representation Q αβ = ∂ α φ∂ β φ − 1 Formulas 2 g αβ ∂ µ φ∂ µ φ . � g φ = 0, Preliminaries Applying the Parametrix The Generalized Formula ◮ Einstein-Maxwell ( Φ = F - 2-form): The Geometry of Null Cones Preliminaries D α F αβ = 0, D [ α F βγ ] = 0, The Ricci Coefficients The Sharp Trace Theorems Q αβ = F αµ F βµ − 1 4 g αβ F µν F µν .

Breakdown Criteria The Main Questions Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations The Main Results Nonvacuum Spacetimes The Main Theorem The Cauchy Problem ◮ Does there exist a “breakdown criterion” similar to K-R Energy Estimates Generalized EMT’s for Einstein-scalar and Einstein-Maxwell spacetimes. Global Energy Estimates Local Energy Estimates ◮ Other nonvacuum settings? Representation Formulas Preliminaries Applying the Parametrix The Generalized Formula The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria The Basic Setting Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations The Main Results ◮ Same setting as K-R, but with E-S or E-M spacetime Nonvacuum Spacetimes The Main Theorem ( M , g , Φ ) rather than E-V. The Cauchy Problem ◮ Time foliation: Energy Estimates Generalized EMT’s Global Energy Estimates � M = Local Energy Estimates Σ τ , t 0 < t 1 < 0. Representation t 0 <τ< t 1 Formulas Preliminaries ◮ Σ τ ’s are compact. Applying the Parametrix The Generalized Formula ◮ CMC foliation: tr k = τ < 0 on Σ τ . The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems

Breakdown Criteria The Main Theorem Arick Shao Introduction The Breakdown Problem Theorem Some Classical Results The Einstein Vacuum Assume an Einstein-scalar or Einstein-Maxwell spacetime Equations ( M , g , Φ ) in the setting of the previous slide. If The Main Results Nonvacuum Spacetimes The Main Theorem sup ( � k ( τ ) � L ∞ + �∇ ( log n ) ( τ ) � L ∞ ) < ∞ , (2) The Cauchy Problem Energy Estimates t 0 � τ< t 1 Generalized EMT’s Global Energy Estimates and the following bounds hold for the matter field, Local Energy Estimates Representation Formulas (E-S) sup � D φ ( τ ) � L ∞ < ∞ , (3) Preliminaries Applying the Parametrix t 0 � τ< t 1 The Generalized Formula � F ( τ ) � L ∞ < ∞ , (E-M) sup (4) The Geometry of Null Cones t 0 � τ< t 1 Preliminaries The Ricci Coefficients then ( M , g , Φ ) can be extended as a CMC foliation beyond The Sharp Trace Theorems time t 1 .

Breakdown Criteria Additional Remarks Arick Shao Introduction The Breakdown Problem Some Classical Results The Einstein Vacuum Equations The Main Results Nonvacuum Spacetimes ◮ Strategy of proof analogous to K-R. The Main Theorem The Cauchy Problem ◮ We focus on E-M setting, since E-S is easier. Energy Estimates Generalized EMT’s ◮ The theorem extends to Einstein-Klein-Gordon and Global Energy Estimates Local Energy Estimates Einstein-Yang-Mills spacetimes (nontrivial). Representation ◮ Result can likely be extended to L 2 x and L 1 Formulas t L ∞ t L ∞ x Preliminaries breakdown criteria. Applying the Parametrix The Generalized Formula The Geometry of Null Cones Preliminaries The Ricci Coefficients The Sharp Trace Theorems