Geometric Methods in Hyperbolic PDEs

Geometric Methods in Hyperbolic PDEs Jared Speck jspeck@math.princeton.edu Department of Mathematics Princeton University January 24, 2011

Unifying mathematical themes Many physical phenomena are modeled by systems of hyperbolic PDEs. Keywords associated with hyperbolic PDEs: Initial value problem formulation (the evolution problem ) Finite speed of propagation Energy estimates Goal: Show how geometrically motived techniques can help us extract information about solutions.

The Einstein equations M = def 1 + 3 dimensional manifold Ric µν 1 2 Rg µν = T µν Matter equations = 0 g µν = Lorentzian metric of signature (, +, +, +) Ric µν = Ric µν (g, g, 2 g) = Ricci tensor R = (g 1 ) αβ Ric αβ = scalar curvature T µν = energy-momentum tensor of the matter Convention: X ν = def (g 1 ) να X α, etc.

Euler-Einstein equations with Λ > 0 Ric µν 1 Λ > 0 Dark Energy 2 Rg {}}{ µν + Λg µν = T µν α T }{{ αµ = 0 } Euler equations Cosmology T µν = (ρ + p)u µ u ν + pg µν g αβ u α u β = 1 u 0 > 0 (future-directed) p = cs 2 ρ = equation of state; c s = speed of sound Stability assumption: 0 < c s < 1/3

The Friedmann-Lemaître-Robertson-Walker (FLRW) family on (, ) T 3 Background solution ansatz: p = p(t), ũ µ = (1, 0, 0, 0), 3 g = dt 2 + a 2 (t) (dx i ) 2 i=1 pa 3(1+c2 s ) p = const Λ ȧ = a 3 + const, a(0) = 1 3a 3(1+c2 s ) }{{} Friedmann s equation Can show: a(t) e Ht, H = Λ/3 = Hubble s constant

Accelerated expansion of the universe S. Weinberg: Almost all of modern cosmology is based on this Robertson-Walker metric. g = dt 2 + a 2 (t) a(t) e Ht 3 (dx i ) 2 i=1 Notation : Ω(t) = def ln a(t) Ω(t) Ht Side remark: Λ = 0 = a(t) t C, C = 2 3(1+c 2 s ) ; t = 0 Big Bang

Stability S. Hawking: We now have a good idea of [the universe s] behavior at late times: the universe will continue to expand at an ever-increasing rate. Fundamental question What happens if we slightly perturb the initial conditions of the special FLRW solution g, p, ũ? Trustworthiness of FLRW solution its salient features are stable under small perturbations of the initial conditions (I. Rodnianski & J.S., 2009; J.S., 2010: Nonlinear stability holds!)

The Cauchy problem for the Einstein equations Fundamental facts: Local existence (Choquet-Bruhat, 1952) Constraint equations Elliptic Ric µν 1 2 Rg µν + Λg µν = T µν Data: 3 manifold Σ, Riemannian metric g, Evolution equations Hyperbolic wave equations second fundamental form K, matter data (Choquet-Bruhat & Geroch, 1969) Each data set launches a unique maximal solution: the Maximal Globally Hyperbolic Development of the data (MGHD)

In other words Rephrasing of fundamental question: What is the nature of the MGHD of data near that of the special FLRW solution g, p, ũ?

Continuation principle (to avoid blow-up) Lemma (heuristic) If a singularity forms at time T max, then lim T T max sup 0 t T S(t) = S(t) = suitably-defined Sobolev-type (i.e., L 2 x type) norm Moral conclusion: To avoid blow-up, show that S(t) remains finite.

Fundamental result with moral implications Theorem (Littman, 1963) m Assume that t 2 φ + i 2 φ = 0 in m 2 spatial dimensions. i=1 If and only if p = 2, we have φ(t) L p x f (t) φ(0) L p x, f = independent of φ Moral conclusion: L 2 x based estimates (energy estimates) play an indispensable role in hyperbolic PDE.

Norms ( 0 < q 1, h jk = e 2Ω g jk, P = e 3(1+c2 s )Ω p, Ω Ht, H = def Λ/3 ) S g00 +1;N = e qω t g 00 H N + e (q 1)Ω g 00 H N + e qω g 00 + 1 H N, 3 ( ) S g0 ;N = e (q 1)Ω t g 0j H N + e (q 2)Ω g 0j H N + e (q 1)Ω g 0j H N, S h ;N = j=1 3 j,k=1 ( ) e qω t h jk H N + e (q 1)Ω h jk H N + h jk H N 1, U N = e (1+q)Ω 3 u j 2, H N j=1 S P p,u ;N= e Ω u j H N + P p H N, S N = S g00 +1;N + S g0 ;N + S h ;N + U N 1 + S P p,u ;N. }{{} Measures the deviation from the FLRW solution

The FLRW stability theorem Theorem (I. Rodnianski & J.S. 2009, J.S. 2010) Assume that N 3 and 0 < c 2 s < 1/3. Then there exist constants ɛ 0 > 0 and C > 1 such that for all ɛ ɛ 0, if S N (0) C 1 ɛ, then there exists a future global causal geodesically complete solution to the Euler-Einstein system with Λ > 0. Furthermore, S N (t) ɛ ( continuation principle) holds for all t 0. (i.e., the MGHD is singularity-free in the future direction) Previous work: Anderson, Brauer/Rendall/Reula, Friedrich, Ringström, Isenberg,

Strategy of proof: derive norm inequalities Lemma (J.S. 2010) t UN 1 2 (t) U2 N 1 (t 1) + τ=t 1 S 2 P p,u ;N (t) S2 P p,u ;N (t 1) + C S 2 g 00 +1;N (t) S2 g 00 +1;N (t 1) + S 2 g 0 ;N (t) S2 g 0 ;N (t 1) + S 2 h ;N (t) S2 h ;N (t 1) + t t 1 t t 1 t t 1 <0 {}}{ 2(3c 2 s 1 + q)e (1+q)Ω U 2 N 1 + CS NU N 1 dτ, t t 1 e qhτ S 2 N dτ, 4qHS 2 g 00 +1;N + Ce qhτ S N S g00 +1;N dτ, 4qHS 2 g 0 ;N + CS h ;NS g0 ;N + Ce qhτ S N S g0 ;N dτ, He qhτ S 2 h ;N + Ce qhτ S N S h ;N dτ, def S N = U N 1 + S P p,u ;N + S g00 +1;N + S g0 ;N + S h ;N H = def Λ/3

Geometric energy methods Goal: geo-analytic structures useful coercive quantities Ex: L = 1 2 (g 1 ) αβ α φ β φ L s Euler-Lagrange eqn: (g 1 ) αβ α β φ = I µν def T = 2 L + (g 1 ) µν L g µν = (g 1 ) µα (g 1 ) νβ α φ β φ 1 2 (g 1 ) µν (g 1 ) αβ α φ β φ Lemma encodes the almost conservation laws {}}{ α T αµ = I(g 1 ) µα α φ

Vectorfield method of (John), Klainerman, Christodoulou X ν = multiplier vectorfield (X) J µ [ φ, φ] = (X) J µ = T µ νx ν = compatible current µ (X) J µ = IX α α φ + T µ ν µ X ν

The divergence theorem ξ Σ t Ω ξ Integral identity - one for each X! Σ 0 energy { }}{ (X) J α ξ α = Σ t (X) J α ξ α + Σ 0 ξ = oriented unit normal to Σ Ω IX α α φ + T µ ν µ X ν,

Coerciveness example for g = diag( 1, 1, 1, 1) Σ t = {(τ, x) τ = t} ξ µ = ( 1, 0, 0, 0) X = (r t) 2 L + (r + t) 2 L, conformal Killing (Morawetz) L = t r, L = t + r {(t r) 2 ( L φ) 2 + (t + r) 2 ( L φ) 2 + (t 2 + r 2 ) φ 2} (X) J µ ξ µ = T µ νξ µ X ν = 1 2 } {{ } positivity dominant energy condition Morawetz-type coercive energy estimate (with φ 2 correction ) { (t r) 2 L φ 2 + (t + r) 2 L φ 2 + (t 2 + r 2 ) φ 2 + φ 2} dx Σ t { r 2 L φ 2 + r 2 L φ 2 + r 2 φ 2 + φ 2} dx Σ 0 }{{} Data { } + I (r t) 2 L φ + (r + t) 2 L φ dxdt Ω

Overview FLRW Geometric Energies Recent Research Regular Hyperbolicity Geo-Electro Dynamics Future Directions Null condition t L = t r t L = t + r r= co angular directions ns t g = diag( 1, 1, 1, 1) gαβ Lα Lβ = 0 = gαβ Lα Lβ gαβ Lα Lβ = 2 null vectorfields (g 1 )αβ α φ β φ = L φ L φ + 6 φ 2 null condition

Geometric methods: recent applications Proofs of local well-posedness: Wong, Smulevici, J.S. Global existence and almost global existence for nonlinear wave and wave-like equations: Katayama, Klainerman, Lindblad, Morawetz, Sideris, Metcalfe/Nakamura/Sogge, Keel/Smith/Sogge, Chae/Huh, Sterbenz, Yang, J.S. The analysis of singular limits in hyperbolic PDEs: J.S., J.S./Strain Geometric continuation principles: Klainerman/Rodnianski, Kommemi, Shao, Wang Wave maps: Ringström

Geometric methods: recent applications The decay of waves on black hole backgrounds & the stability of the Kerr black hole family: Dafermos/Rodnianski, Baskin, Blue, Andersson/ Blue, Soffer/Blue, Holzegel, Schlue, Aretakis, Finster/Kamran/Smoller/Yau, Luk, Tataru/Tohaneanu Black hole uniqueness (in smooth class): Klainerman, Ionescu, Chrusciel, Alexakis, Nguyen, Wong The evolutionary formation of trapped surfaces for the vacuum Einstein equations: Christodoulou, Klainerman/Rodnianski, Yu The formation of shocks for the 3 dimensional Euler equations: Christodoulou The proof of Strichartz-type inequalities for quasilinear wave equations: Klainerman

Regular hyperbolicity à la Christodoulou φ = (φ 1,, φ n ) : R m R n h αβ AB α β φ B = I A assume : h αβ AB = hβα BA assume : hyperbolicity Theorem (J.S. & W. Wong, in preparation) Under the above conditions, one can completely identify all of the ξ µ, X ν that lead to a coercive (in the integrated sense) integral identity. Counterexamples if h αβ AB hβα BA. Canonical stress: Q µ ν = def h (µλ) λφ B ν φ A 1 2 δµ ν h (κλ) κφ A λ φ B T µ ν AB AB

Overview FLRW Geometric Energies Recent Research Regular Hyperbolicity Geo-Electro Dynamics Future Directions Geometry of the equations X Jp Ip ξ (a) Cp def Cp = Characteristic subset of Tp Rm def αβ = {ζ det hab ζα ζβ = 0} def Cp = Characteristic subset of Tp Rm ; is the dual (b) Cp

The Einstein-nonlinear electromagnetic equations M = R 1+3 In Maxwell s theory: M (Maxwell) µν = F µν Ric µν 1 2 Rg µν = T µν (df) λµν = 0 (dm) λµν = 0 null condition {}}{ T µν (Maxwell) = Fµ κ F νκ 1 4 g µν F κλ F κλ For many other Lagrangian theories: M µν = M (Maxwell) µν + quadratic error terms T µν = T µν (Maxwell) + cubic error terms

The stability of Minkowski spacetime Theorem (J.S. 2010; upcoming PNAS article) The Minkowski spacetime solution g = diag( 1, 1, 1, 1), F = 0 is globally stable. The result holds for a large family of electromagnetic models that reduce to Maxwell s theory in the weak-field limit. Proof: vectorfield method + regular hyperbolicity Stability: Christodoulou-Klainerman, Lindblad-Rodnianski, Bieri, Zipser, Choquet-Bruhat/Chrusciel/Loizelet Electromagnetism: Bialynicki-Birula, Boillat, Kiessling, Tahvildar-Zadeh, Carley, study e.g. the Maxwell-Born-Infeld model ( minimal surface equation) Above results the alternate electromagnetic models are physically reasonable

Future research directions MGHD - AIM workshop on shocks (with S. Klainerman, G. Holzegel, W. Wong, P. Yu) Sub-exponential expansion rates Wave equations on expanding Lorentzian manifolds Alternative matter model: the relativistic Boltzmann equation (with R. Strain)

Future research directions continued Singularity formation in general relativity Regular hyperbolicity (with W. Wong) classification of integral identities for nonlinear electromagnetism

I Great open problem Maximal Globally Hyperbolic Developments Weak Cosmic Censorship Hypothesis (R. Penrose, 1969) Generic fi s spherically singularities symmetric, are then hidden its projectio inside black n to holes Q will Singularity r = 0 I + r = 2M Black hole Σ Far-away observer r = 2M I + D I r = 0 Figure: Schwarzschild solution of mass M (M. Dafermos)

Wanted list Wanted: work on solving the Einstein constraint equations for some of the matter models: Euler-Einstein with Λ > 0 Einstein-nonlinear electromagnetic Einstein-Boltzmann Wanted: numerical evidence for the cases c 2 s 1/3 Wanted: blow-up results away from the small-data regime

Thank you

The Divergence Problem Electric field surrounding an electron: E (Maxwell) = ˆr r 2 Electrostatic energy: e (Maxwell) = E(Maxwell) R 2 dvol 3 = 4π 1 r=0 dr = r 2 Lorentz force: F (Lorentz) = qe (Maxwell) = Undefined (at the electron s location) Analogous difficulties appear in QED M. Born: The attempts to combine Maxwell s equations with the quantum theory (...) have not succeeded. One can see that the failure does not lie on the side of the quantum theory, but on the side of the field equations, which do not account for the existence of a radius of the electron (or its finite energy=mass).