A Generalized Representation Formula for Geometric Extensions The - PowerPoint PPT Presentation

Generalized Representation Formula Arick Shao Preliminaries Minkowski Spacetime A Generalized Representation Formula for Geometric Extensions The Kirchhoff-Sobolev Parametrix Tensor Wave Equations on Curved The Main Result Reasons to Generalize Spacetimes A New Derivation The Main Formula - Preliminary Version The Precise Formulation Arick Shao The Basic Setting The Required Quantities The Main Formula - More Precise Version University of Toronto Derivation of the Main Formula Overview March 8, 2012 Main Steps Completion of the Proof

Generalized The Model Equation Representation Formula Arick Shao Preliminaries Minkowski Spacetime ◮ Consider first the Minkowski spacetime R 1 + 3 . Geometric Extensions The Kirchhoff-Sobolev Parametrix ◮ Consider the (scalar) wave equation, The Main Result Reasons to Generalize A New Derivation � φ = − ∂ 2 φ, ψ ∈ C ∞ ( R 1 + 3 ) , t φ + ∆ φ = ψ , The Main Formula - Preliminary Version The Precise with initial data Formulation The Basic Setting The Required Quantities φ | t = 0 = α 0 ∈ C ∞ ( R 3 ) , ∂ t φ | t = 0 = α 1 ∈ C ∞ ( R 3 ) . The Main Formula - More Precise Version Derivation of the Main Formula ◮ One has an explicit solution for φ – Kirchhoff’s formula – Overview in terms of ψ , α 0 , and α 1 . Main Steps Completion of the Proof

Generalized The Model Formula Representation Formula Arick Shao ◮ Write φ = φ 1 + φ 2 , where Preliminaries Minkowski Spacetime ◮ φ 1 satisfies � φ = ψ , with zero initial data. Geometric Extensions The Kirchhoff-Sobolev ◮ φ 2 satisfies � φ ≡ 0, with initial data α 0 , α 1 . Parametrix The Main Result ◮ Then, we have the representation formula Reasons to Generalize A New Derivation The Main Formula - Preliminary Version 1 � φ 2 ( t , x ) = [ α 0 ( y ) + ( y − x ) · ∇ α 0 ( y )] d σ y The Precise 4 π t 2 Formulation ∂ B ( x , t ) The Basic Setting � 1 The Required Quantities + α 1 ( y ) d σ y , The Main Formula - More Precise Version 4 π t ∂ B ( x , t ) Derivation of the � t Main Formula φ 1 ( t , x ) = 1 � ψ ( y , t − r ) d σ y dr . Overview 4 π r Main Steps ∂ B ( x , r ) 0 Completion of the Proof ◮ B ( x , r ) is the ball in R 3 about x of radius r .

Generalized Curved Spacetimes Representation Formula Arick Shao ◮ Main Question: Can we extend this representation to Preliminaries geometric settings, i.e., to curved spacetimes? Minkowski Spacetime Geometric Extensions ◮ Curved spacetime: any general ( 1 + 3 ) -dimensional The Kirchhoff-Sobolev Parametrix Lorentzian manifold ( M , g ) . The Main Result ◮ Equation: Covariant tensorial wave equation, Reasons to Generalize A New Derivation The Main Formula - � g Φ = g αβ D 2 αβ Φ = Ψ , Preliminary Version The Precise Formulation with appropriate “initial conditions”. The Basic Setting ◮ Goal: representation formula The Required Quantities The Main Formula - More Precise Version Φ | p = F (Ψ) + error (Φ) + initial data. Derivation of the Main Formula Overview ◮ Some classical applications: Main Steps Completion of the Proof 1. (Y. Choqu´ et-Bruhat) Local well-posedness of the Einstein-vacuum equations. 2. (Chru´ sciel-Shatah) Global existence of the Yang-Mills equations in curved spacetimes.

Generalized Infinite-Order Formulas Representation Formula Arick Shao Preliminaries Minkowski Spacetime ◮ Infinite-order, or “Hadamard-type”, representation Geometric Extensions The Kirchhoff-Sobolev Parametrix formulas are more explicit and precise. The Main Result ◮ Require infinitely many derivatives of metric g . Reasons to Generalize A New Derivation ◮ Formula is only local: require geodesic convexity . The Main Formula - Preliminary Version ◮ Wave equations in curved spacetimes no longer satisfy The Precise Formulation the strong Huygens principle . The Basic Setting The Required Quantities ◮ Representation formula at point p depends on entire The Main Formula - More Precise Version causal, rather than null, past (or future) of p . Derivation of the Main Formula ◮ These severe restrictions for infinite-order formulas Overview Main Steps often make them undesirable for nonlinear PDEs. Completion of the Proof

Generalized First-Order Formulas Representation Formula Arick Shao Preliminaries Minkowski Spacetime ◮ In contrast, one can also derive first-order, or Geometric Extensions The Kirchhoff-Sobolev “Kirchhoff-Sobolev-type”, representation formulas. Parametrix The Main Result ◮ Again, formula is only local. Reasons to Generalize ◮ Require only limited number of derivatives of g . A New Derivation The Main Formula - ◮ Formula not explicit – contains recursive error terms: Preliminary Version The Precise Formulation Φ | p = F (Ψ) + error (Φ) + initial data. The Basic Setting The Required Quantities The Main Formula - More ◮ Representation formula can be supported on only the Precise Version Derivation of the null past (or future) of p . Main Formula Overview ◮ Require smoothness of null, rather than causal, cone. Main Steps Completion of the Proof ◮ The price to be paid is the recursive error terms.

Generalized A Recent Result Representation Formula Arick Shao ◮ “Kirchhoff-Sobolev Parametrix” [KSP] Preliminaries Minkowski Spacetime (Klainerman-Rodnianski, 2007): first-order Geometric Extensions The Kirchhoff-Sobolev representation formula on curved spacetimes. Parametrix The Main Result ◮ Valid within null radius of injectivity . Reasons to Generalize ◮ Supported entirely on past null cone. A New Derivation The Main Formula - ◮ Handles covariant tensorial wave equations, using only Preliminary Version The Precise fully covariant (coordinate-independent) techniques. Formulation ◮ Extendible to wave equations on vector bundles. The Basic Setting The Required Quantities The Main Formula - More ◮ Rough statement of KSP: Precise Version Derivation of the Main Formula � 4 π · g (Φ | p , J p ) = [ g ( A , Ψ) + Err ( A , Φ)] + i . v . . Overview Main Steps N − ( p ) Completion of the Proof ◮ A corresponds to r − 1 in Minkowski space.

Generalized Applications of KSP Representation Formula Arick Shao Preliminaries Minkowski Spacetime Geometric Extensions ◮ Applications of this formula: The Kirchhoff-Sobolev Parametrix 1. Gauge-invariant proof of global existence of Yang-Mills. The Main Result Reasons to Generalize ◮ The classical result (Eardley-Moncrief, 1982) relies on A New Derivation The Main Formula - Cronstr¨ om gauge. Preliminary Version The Precise 2. Breakdown criterion for Einstein-vacuum equations Formulation The Basic Setting (Klainerman-Rodnianski, 2010). The Required Quantities The Main Formula - More ◮ Needed pointwise bound for Riemann curvature R of Precise Version ( M , g ) , which satisfies tensor wave equation Derivation of the Main Formula Overview � g R = R · R . Main Steps Completion of the Proof

Generalized Extending KSP Representation Formula Arick Shao Preliminaries Minkowski Spacetime ◮ The main result of this presentation is a generalization Geometric Extensions The Kirchhoff-Sobolev Parametrix of KSP , which we call [GKSP]. The Main Result ◮ Q. Why generalize KSP? Reasons to Generalize A New Derivation The Main Formula - 1. Want to handle systems of tensor wave equations with Preliminary Version (covariant) first-order terms : The Precise Formulation The Basic Setting n The Required Quantities c · D Φ c = Ψ m , � � g Φ m + 1 ≤ m ≤ n . P m The Main Formula - More Precise Version c = 1 Derivation of the Main Formula 2. Removal of extraneous assumptions needed in KSP . Overview Main Steps 3. Explicit formula for initial value terms. Completion of the Proof

Generalized Handling First-Order Terms Representation Formula Arick Shao Preliminaries Minkowski Spacetime Geometric Extensions ◮ Analogous breakdown criterion for Einstein-Maxwell The Kirchhoff-Sobolev Parametrix equations (S., 2010) The Main Result ◮ Curvature R and electromagnetic tensor F satisfy Reasons to Generalize A New Derivation The Main Formula - = F · D 2 F + ( R + DF ) 2 + l . o . , Preliminary Version � g R ∼ The Precise = F · DR + ( R + DF ) 2 + l . o . . Formulation � g DF ∼ The Basic Setting The Required Quantities The Main Formula - More ◮ Right hand side has first-order terms. Precise Version Derivation of the ◮ In KSP , these become part of the inhomogeneity Ψ , but Main Formula this does not yield the necessary estimates. Overview Main Steps ◮ For GKSP , we must treat these terms differently. Completion of the Proof

Generalized Removing Assumptions Representation Formula Arick Shao Preliminaries Minkowski Spacetime Geometric Extensions ◮ Assumptions for KSP: The Kirchhoff-Sobolev Parametrix 1. Smoothness/regularity of all past null cones in a The Main Result Reasons to Generalize neighborhood of the base point p . A New Derivation The Main Formula - 2. Local hyperbolicity – spacelike “initial” hypersurface Preliminary Version passed by null cone exactly once. The Precise Formulation ◮ Less assumptions for GKSP: The Basic Setting The Required Quantities The Main Formula - More 1. Smoothness/regularity of past null cone from p . Precise Version Derivation of the ◮ (1) for KSP weakened to only null regularity at p . Main Formula Overview ◮ (2) for KSP is not needed at all. Main Steps Completion of the Proof