Second-order hyperbolic Fuchsian systems

Transcription

1 Second-order hyperbolic Fuchsian systems Applications to Einstein vacuum spacetimes Florian Beyer (joint work with P. LeFloch) University of Otago, Dunedin, New Zealand Seminar on MATHEMATICAL GENERAL RELATIVITY, Paris, January 2010 Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

2 Fuchsian theory Fuchsian systems: with D := t t. Du(t,x)+N(x)u(t,x) = t f(t,x,u, x u,du), lim u(t,x) = 0. tց0 What does the Fuchsian theory provide? Theory of singular PDEs which allows to construct solutions with a prescribed singular behavior. Applications in General Relativity: Gowdy vacuum solutions: Kichenassamy-Rendall, Rendall. Polarized T 2 -symmetric vacuum solutions: Isenberg-Kichenassamy. Cosmological models with stiff fluid/scalar field: Andersson-Rendall.... Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

3 Some drawbacks of the current theory Often not Fuchsian equations directly. First need to identify the leading order part of the unknowns at the singularity. Often second-order equations. Numerical scheme for hyperbolic Fuchsian equations and proof of convergence? Aims of the talk today: Tackle these drawbacks for a subset of Fuchsian equations. Present numerical examples. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

4 Second-order Fuchsian equations Class of equations (D D)u(t,x)+2a(x)Du(t,x)+b(x)u(t,x) = f(t,x,u, x u,du, x 2 u, x Du), with 1 u :]0,δ] R R, periodic in space for a δ > 0, 2 a, b are smooth periodic functions on R, 3 f is the source with certain properties, see below. Direct generalizations: Systems. More than one spatial dimensions. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

5 Heuristics and a singular initial value problem Canonical two-term expansion u(t,x)= { u (x)t λ 1 + u (x)t λ 2 + O(t λ 2+α ), a 2 (x) > b(x), u (x)t λ 1 logt + u (x)t λ 1 + O(t λ 1+α ), a 2 (x) = b(x), at t = 0 for α > 0, where λ 1 (x) := a(x)+ a 2 (x) b(x), λ 2 (x) := a(x) a(x) 2 b(x). Singular initial value problem (SIVP) Does there exist a unique solution of the Fuchsian system in a given regularity class which obeys the canonical two-term expansion at t = 0 for given asymptotic data u (x), u (x)? Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

6 Second-order hyperbolic Fuchsian systems Class of hyperbolic equations Consider (D D)u(t,x)+2a(x)Du(t,x)+b(x)u(t,x) = f(t,x,u, x u,du)+t 2 c 2 (t,x) 2 x u(t,x) with the speed of propagation Here, c(t,x) = t β(x) k(t,x). β(x) is a smooth periodic function larger than 1. k(t, x) is a smooth positive spatially periodic function so that all derivatives have finite limits at t = 0. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

7 Examples of 2nd-order hyperbolic Fuchsian equations 1 Euler Poisson Darboux equation: (D D)u λ Du t 2 2 x u = 0, for λ 0. Equivalently t 2u x 2u = 1 t ( λ 1) tu. SIVP: Look for solutions of the form { u (x)+u (x)t λ +... λ > 0, u(t,x) = u (x)logt + u (x)+... λ = 0, 2 (Main evolution part of the) Gowdy vacuum equations: D 2 P t 2 2 x P = e2p (DQ) 2 t 2 e 2P ( x Q) 2, D 2 Q 2k DQ t 2 2 x Q = 2(k + DP)DQ + 2t2 x P x Q. SIVP: Look for solutions of the form P(t,x) = k(x)log t + P (x)+..., Q(t,x) = Q (x)+q (x)t 2k(x) +... Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

8 Function spaces and the SIVP The space X δ,α,1 For a given second-order Fuchsian equation and some δ, α > 0, let X δ,α,1 be the space of functions w C 0 ((0,δ],H 1 (U)) C 1 ((0,δ],L 2 (U)) and w δ,α,1 <, with the standard norm δ,α,1 weighted by the factor t λ 2 α. SIVP We say that u is a solution of the SIVP for given asymptotic data u,u H 1 (U), if u is a weak solution of the 2nd-order hyperbolic Fuchsian eq. and there exists w X δ,α,1 so that u(t,x) = u canonic (t,x)+w(t,x). The function w is called remainder. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

9 Well-Posedness in an important case Consider the case for a given function f 0. Result f(t,x,u, x u,du) = f 0 (t,x) For any asymptotic data u,u H 2 (U), there exists a unique solution of the SIVP with remainder w X δ,α,1 provided: 1 we can choose δ,α > 0 so that the matrix λ 1 λ 2 + α η/2 0 η/2 α t x c x (λ 1 λ 2 )(tc lnt) 0 t x c x (λ 1 λ 2 )(tc lnt) λ 1 λ 2 + α 1 Dc/c is positive semidef. at each (t,x) (0,δ) U for a η > 0. 2 f 0 X δ,α+ε,0 for some ε > 0. 3 α < 2(β(x)+1) (λ 1 (x) λ 2 (x)) for all x U. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

10 Main ingredients of the existence proof 1 Main idea: Solve a sequence of regular initial value problems, i.e. a sequence of approximate solutions (u n ) with initial times (τ n ) going to zero as n. 2 Obtain good energy estimates consistent with the space X δ,α,1 if the energy dissipation matrix is positive semidefinite. 3 Convergence: where G(t) := u n u m δ,α,1 C G(τ n ) G(τ m ), t 0 s 1 s λ 2 α (f 0 (s) L[u canonic ](s)) L 2 (U) ds, and C > 0 independent of n. Under the hypothesis, we have lim t 0 G(t) = 0. Hence (w n ) is a Cauchy sequence in X δ,α,1. 4 Check that the limit function u satisfies the weak equation. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

11 Main ingredients of the existence proof Regular initial value problem (RIVP) We say that ũ is a solution of the RIVP with initial time τ 0 > 0, provided (after mollification) For t (0,τ 0 ) : ũ(t,x) = u canonic (t,x). For t [τ 0,δ) : ũ is a smooth solution of the 2nd-order Fuchsian hyperbolic equation and initial data ũ(τ 0,x) = u canonic (τ 0,x), t ũ(τ 0,x) = t u canonic (τ 0,x). Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

12 Main ingredients of the existence proof 1 Main idea: Solve a sequence of regular initial value problems, i.e. a sequence of approximate solutions (u n ) with initial times (τ n ) going to zero as n. 2 Obtain good energy estimates consistent with the space X δ,α,1 if the energy dissipation matrix is positive semidefinite. 3 Convergence: where G(t) := u n u m δ,α,1 C G(τ n ) G(τ m ), t 0 s 1 s λ 2 α (f 0 (s) L[u canonic ](s)) L 2 (U) ds, and C > 0 independent of n. Under the hypothesis, we have lim t 0 G(t) = 0. Hence (w n ) is a Cauchy sequence in X δ,α,1. 4 Check that the limit function u satisfies the weak equation. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

13 Comments Comments on this result: Main difference to standard Fuchsian theory: approximate by smooth solutions of RIVP with an explicit convergence statement in the norm. Leads to a numerical scheme! Loss of regularity. Generalizations of the rigorous results: 1 General (non-linear) sources: Idea: Iterate over the special case before. Result: If source term is locally Lipschitz in the space X δ,α,1 and δ > 0 is sufficiently small, then obtain a fixed point iteration. Strong convergence in the norm! 2 Increase regularity assumptions to control arbitrarily many derivatives in spaces X δ,α,k with k N. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

14 Comments Consider the case for a given function f 0. Result f(t,x,u, x u,du) = f 0 (t,x) For any asymptotic data u,u H 2 (U), there exists a unique solution of the SIVP with remainder w X δ,α,1 provided: 1 we can choose δ,α > 0 so that the matrix λ 1 λ 2 + α η/2 0 η/2 α t x c x (λ 1 λ 2 )(tc lnt) 0 t x c x (λ 1 λ 2 )(tc lnt) λ 1 λ 2 + α 1 Dc/c is positive semidef. at each (t,x) (0,δ) U for a η > 0. 2 f 0 X δ,α+ε,0 for some ε > 0. 3 α < 2(β(x)+1) (λ 1 (x) λ 2 (x)) for all x U. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

15 Comments Comments on this result: Main difference to standard Fuchsian theory: approximate by smooth solutions of RIVP with an explicit convergence statement in the norm. Leads to a numerical scheme! Loss of regularity. Generalizations of the rigorous results: 1 General (non-linear) sources: Idea: Iterate over the special case before. Result: If source term is locally Lipschitz in the space X δ,α,1 and δ > 0 is sufficiently small, then obtain a fixed point iteration. Strong convergence in the norm! 2 Increase regularity assumptions to control arbitrarily many derivatives in spaces X δ,α,k with k N. Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

16 Example application: Euler-Poisson-Darboux eq. Consider (D D)u λ Du t 2 2 x u = 0, for a constant λ 0, and find solutions of the form { u (x)+u (x)t λ +... λ > 0, u(t,x) = u (x)logt + u (x)+... λ = 0. It turns out: well-posedness theory applies for 0 λ < 2. Periodicity in space explicit solutions in terms of Bessel functions. Result: For 0 λ < 2: Solutions are consistent with the canonical two-term expansion. For λ = 2: Solutions behave like u(t,x) = u (x)+u (x)t λ logt +... Second spatial derivative becomes significant at t = 0! Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

17 Numerics: Euler-Poisson-Darboux eq. General numerical approach, using EPD-equation as a model: 1 Introduce time variable τ := ln t. Then equation becomes 2 τ u λ τu e 2τ 2 x u = 0. Singularity is shifted to τ =. 2 Write the equation for the remainder w, after having fixed asymptotic data u, u. 3 Solve sequence (w n ) of solutions of RIVPs with initial times τ n with data w n (τ n ) = 0, τ w n (τ n ) = 0, using a direct discretization of the second-order equation (as suggested by Kreiss). Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

18 Choice: λ = 1.0, u = cosx, u = e-05 tau_0=-1.0 tau_0=-2.0 tau_0=-3.0 tau_0=-10.0 tau_0=-20.0 exact w at x = 0 1e-10 1e-15 1e-20 1e τ Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

19 Choice: λ = 1.9, u = cosx, u = e-05 tau_0=-1.0 tau_0=-2.0 tau_0=-3.0 tau_0=-10.0 tau_0=-20.0 exact w at x = 0 1e-10 1e-15 1e-20 1e τ Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16

20 Outlook Desired generalizations of the current theory: Mixed first- and second-order hyperbolic Fuchsian systems. Some ideas for applications: Gowdy coupled to a perfect fluid. Interaction of cosmological singularity and shocks? Numerical construction and analysis of solutions with Cauchy horizons or pieces of Cauchy horizons in the Gowdy case? Florian Beyer (Otago) Second-order hyperbolic Fuchsian systems Paris, January / 16