Dispersive Equations and Hyperbolic Orbits

Transcription

1 Dispersive Equations and Hyperbolic Orbits H. Christianson Department of Mathematics University of California, Berkeley 4/16/07 The Johns Hopkins University

2 Outline 1 Introduction 3 Applications 2 Main Results General Problem Main Result 3 Sketch of Proof Poincaré Map Quantization of S 4 Further Applications (time allowing) Schrödinger Equation Wave Equation Nonlinear Equations

3 3 Applications 1: Eigenfunction Non-concentration (M, g) compact, Riemannian manifold with or without boundary γ M unstable closed geodesic, only transversal reflections with M, U γ a small neighbourhood Examples: γ X γ Figure: A compact manifold without boundary. Figure: A compact manifold with boundary.

4 3 Applications 1: Eigenfunction Non-concentration (M, g) compact, Riemannian manifold with or without boundary γ M unstable closed geodesic, only transversal reflections with M, U γ a small neighbourhood Examples: γ X γ Figure: A compact manifold without boundary. Figure: A compact manifold with boundary.

5 3 Applications 1: Eigenfunction Non-concentration (M, g) compact, Riemannian manifold with or without boundary γ M unstable closed geodesic, only transversal reflections with M, U γ a small neighbourhood Examples: γ X γ Figure: A compact manifold without boundary. Figure: A compact manifold with boundary.

6 3 Applications Eigenfunction Non-concentration Eigenvalue problem: { ( g λ 2 )u = 0 M u 2 dx = 1 Ergodic case (Zelditch ( 87), Colin de Verdière ( 85), many others...): u 2 dx Area U Area M, λ U

7 3 Applications Eigenfunction Non-concentration Eigenvalue problem: { ( g λ 2 )u = 0 M u 2 dx = 1 Ergodic case (Zelditch ( 87), Colin de Verdière ( 85), many others...): u 2 dx Area U Area M, λ U

8 3 Applications Non-Ergodic Non-ergodic, γ U hyperbolic (C 06): M\U u 2 dx C log λ, λ

9 3 Applications Non-Ergodic Non-ergodic, γ U hyperbolic (C 06): M\U u 2 dx C log λ, λ

10 3 Applications History Generalizes results of 1) Colin de Verdière-Parisse ( 94) for a surface of revolution (sharp) 2) Burq-Zworski ( 04) for real hyperbolic orbits

11 3 Applications Idea of Proof Non-concentration follows from Semiclassical rescaling λ = z/h: (P(h) 1)u = E(h)u, E(h) = O(h) Cutoff estimates: supp χ γ = χu Commutator argument: log(1/h) P(h)χu h supp [P(h),χ] is away from γ

12 3 Applications Idea of Proof Non-concentration follows from Semiclassical rescaling λ = z/h: (P(h) 1)u = E(h)u, E(h) = O(h) Cutoff estimates: supp χ γ = χu Commutator argument: log(1/h) P(h)χu h supp [P(h),χ] is away from γ

13 3 Applications Idea of Proof Non-concentration follows from Semiclassical rescaling λ = z/h: (P(h) 1)u = E(h)u, E(h) = O(h) Cutoff estimates: supp χ γ = χu Commutator argument: log(1/h) P(h)χu h supp [P(h),χ] is away from γ

14 3 Applications 2: Damped Wave Equation Let a(x) C (M), a 0 Damped wave equation { ( 2 t g + 2a(x) t )u = 0 u(x, 0) = 0, u t (x, 0) = f(x) a 0 implies E(t) := t u 2 L 2 (M) + u 2 L 2 (M) f 2 L 2 (M).

15 3 Applications 2: Damped Wave Equation Let a(x) C (M), a 0 Damped wave equation { ( 2 t g + 2a(x) t )u = 0 u(x, 0) = 0, u t (x, 0) = f(x) a 0 implies E(t) := t u 2 L 2 (M) + u 2 L 2 (M) f 2 L 2 (M).

16 3 Applications 2: Damped Wave Equation Let a(x) C (M), a 0 Damped wave equation { ( 2 t g + 2a(x) t )u = 0 u(x, 0) = 0, u t (x, 0) = f(x) a 0 implies E(t) := t u 2 L 2 (M) + u 2 L 2 (M) f 2 L 2 (M).

17 3 Applications Damped Wave Equation a > 0 somewhere implies E (t) < 0 a δ > 0 everywhere or geometric control implies E(t) Ce t/c f 2 L 2 (M)

18 3 Applications Damped Wave Equation a > 0 somewhere implies E (t) < 0 a δ > 0 everywhere or geometric control implies E(t) Ce t/c f 2 L 2 (M)

19 3 Applications Trapped Hyperbolic Orbit γ U hyperbolic trapped orbit = lack of geometric control a > 0 outside U implies (C 06) ǫ > 0 E(t) Ce t/c f 2 H ǫ (M),

20 3 Applications Trapped Hyperbolic Orbit γ U hyperbolic trapped orbit = lack of geometric control a > 0 outside U implies (C 06) ǫ > 0 E(t) Ce t/c f 2 H ǫ (M),

21 3 Applications History Long history: 1) Rauch-Taylor ( 75): geometric control assumption on supp a 2) Lebeau ( 96): surface of revolution 3) Many others...

22 3 Applications Idea of Proof Formally Fourier transform in t: Rescaling τ = z/h: ( τ 2 + iτa(x) )û = f h 2 P(h) = h 2 ( h 2 + i zha(x) z) Apply resolvent estimates

23 3 Applications Idea of Proof Formally Fourier transform in t: Rescaling τ = z/h: ( τ 2 + iτa(x) )û = f h 2 P(h) = h 2 ( h 2 + i zha(x) z) Apply resolvent estimates

24 3 Applications 3: Schrödinger Equation (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

25 3 Applications 3: Schrödinger Equation (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

26 3 Applications 3: Schrödinger Equation (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

27 3 Applications Examples γ Figure: A piece of the catenoid. Figure: R n with two convex bodies removed. γ

28 3 Applications Local Smoothing Effect Non-trapping case: T 0 dist g (x, x 0 ) 1/2 ǫ e it g u 0 2 Trapped hyperbolic orbit (C 06): T 0 H 1/2 dt C u 0 2 L 2. dist g (x, x 0 ) 1/2 ǫ e it g 2 u 0 dt C u 0 2 H 1/2 ǫ L. 2

29 3 Applications Local Smoothing Effect Non-trapping case: T 0 dist g (x, x 0 ) 1/2 ǫ e it g u 0 2 Trapped hyperbolic orbit (C 06): T 0 H 1/2 dt C u 0 2 L 2. dist g (x, x 0 ) 1/2 ǫ e it g 2 u 0 dt C u 0 2 H 1/2 ǫ L. 2

30 3 Applications History and Extensions Studied by 1) Doi ( 96): sharp H 1/2 smoothing effect non-trapping 2) Burq ( 04): Convex bodies removed from Euclidean space Additional applications: Strichartz estimates, nonlinear Schrödinger and wave equations

31 3 Applications History and Extensions Studied by 1) Doi ( 96): sharp H 1/2 smoothing effect non-trapping 2) Burq ( 04): Convex bodies removed from Euclidean space Additional applications: Strichartz estimates, nonlinear Schrödinger and wave equations

32 3 Applications Idea of Proof Formally Fourier transform in t: P(τ)û = ( τ )û = u 0 Resolvent estimates plus interpolation

33 3 Applications Idea of Proof Formally Fourier transform in t: P(τ)û = ( τ )û = u 0 Resolvent estimates plus interpolation

34 General Problem Common Theme Applications follow from cutoff resolvent estimates: χ smooth with compact support χ( g (τ ± iǫ)) 1 χ L 2 L 2 C log(2 + τ ) τ 1/2

35 General Problem Common Theme Applications follow from cutoff resolvent estimates: χ smooth with compact support χ( g (τ ± iǫ)) 1 χ L 2 L 2 C log(2 + τ ) τ 1/2

36 General Problem Common Theme Applications follow from cutoff resolvent estimates: χ smooth with compact support χ( g (τ ± iǫ)) 1 χ L 2 L 2 C log(2 + τ ) τ 1/2

37 General Problem Semiclassical reduction Consider ( g λ 2 )u = f, λ Set λ = z/h, for z [E δ, E + δ] R, 1 h 2( h2 g z)u = f

38 General Problem Semiclassical reduction Consider ( g λ 2 )u = f, λ Set λ = z/h, for z [E δ, E + δ] R, 1 h 2( h2 g z)u = f

39 General Problem Semiclassical reduction Consider ( g λ 2 )u = f, λ Set λ = z/h, for z [E δ, E + δ] R, 1 h 2( h2 g z)u = f

40 General Problem General Assumptions P(h) (second-order), self-adjoint semiclassical pseudodifferential operator For example, P(h) = 1 hd i a ij hd j + a ij hd i hd j + lower order 2 ij Principal symbol real, independent of h p(x,ξ) = ij a ij ξ i ξ j

41 General Problem General Assumptions P(h) (second-order), self-adjoint semiclassical pseudodifferential operator For example, P(h) = 1 hd i a ij hd j + a ij hd i hd j + lower order 2 ij Principal symbol real, independent of h p(x,ξ) = ij a ij ξ i ξ j

42 General Problem Assume p = 0 = dp 0, and ξ C = p ξ 2 /C Assume γ {p = 0} closed hyperbolic orbit for exp th p

43 General Problem Assume p = 0 = dp 0, and ξ C = p ξ 2 /C Assume γ {p = 0} closed hyperbolic orbit for exp th p

44 Main Result Main Estimate Let a C (T X), a 0, a 0 near γ, a 1 away from γ Small complex perturbation: Q(z, h) := P(h) z iha(x, hd), z [ δ,δ] + i[ c 0 h, c 0 h] Theorem (C 06) Q(z, h) 1 L 2 (X) L 2 (X) log 1/h C, z [ δ/2,δ/2] h

45 Main Result Main Estimate Let a C (T X), a 0, a 0 near γ, a 1 away from γ Small complex perturbation: Q(z, h) := P(h) z iha(x, hd), z [ δ,δ] + i[ c 0 h, c 0 h] Theorem (C 06) Q(z, h) 1 L 2 (X) L 2 (X) log 1/h C, z [ δ/2,δ/2] h

46 Main Result Main Estimate Let a C (T X), a 0, a 0 near γ, a 1 away from γ Small complex perturbation: Q(z, h) := P(h) z iha(x, hd), z [ δ,δ] + i[ c 0 h, c 0 h] Theorem (C 06) Q(z, h) 1 L 2 (X) L 2 (X) log 1/h C, z [ δ/2,δ/2] h

47 Main Result Proof Idea a is absorption term: Im Q c 1 h away from γ Main estimate follows from: Q(z, h) 1 u L 2 Ch N u L 2, z [ δ,δ] + i[ c 0 h, c 0 h] if u is concentrated near γ, and three-line theorem from complex analysis (u concentrated near γ means χ C c (T X), supp χ γ, χ(x, hd)u = u + O(h ))

48 Main Result Proof Idea a is absorption term: Im Q c 1 h away from γ Main estimate follows from: Q(z, h) 1 u L 2 Ch N u L 2, z [ δ,δ] + i[ c 0 h, c 0 h] if u is concentrated near γ, and three-line theorem from complex analysis (u concentrated near γ means χ C c (T X), supp χ γ, χ(x, hd)u = u + O(h ))

49 Main Result Proof Idea a is absorption term: Im Q c 1 h away from γ Main estimate follows from: Q(z, h) 1 u L 2 Ch N u L 2, z [ δ,δ] + i[ c 0 h, c 0 h] if u is concentrated near γ, and three-line theorem from complex analysis (u concentrated near γ means χ C c (T X), supp χ γ, χ(x, hd)u = u + O(h ))

50 Poincaré Map Dimension 2 dim M = 2, so dim T M = 4 and dim{p = 0} = 3 Poincaré section N {p = 0}, and Poincaré map S : N S(N): p 1 (0) picture γ p 1 (0) Figure: Poincaré section N and Poincaré map N

51 Poincaré Map Dimension 2 dim M = 2, so dim T M = 4 and dim{p = 0} = 3 Poincaré section N {p = 0}, and Poincaré map S : N S(N): p 1 (0) picture γ p 1 (0) Figure: Poincaré section N and Poincaré map N

52 Poincaré Map Dimension 2 dim M = 2, so dim T M = 4 and dim{p = 0} = 3 Poincaré section N {p = 0}, and Poincaré map S : N S(N): p 1 (0) picture nearby orbit.. γ N p 1 (0) Figure: Poincaré section N and Poincaré map

53 Poincaré Map Hyperbolic Orbit γ hyperbolic means ds(0, 0) is hyperbolic:. Figure: Λ ±

54 Poincaré Map ds(0, 0) = ( µ 0 0 1/µ = exp ( λ 0 0 λ ), µ > 1 ), λ > 0 = S = exp H q, q = λxξ + O((x,ξ) 3 ) Need to control error, and estimate all around γ!

55 Poincaré Map ds(0, 0) = ( µ 0 0 1/µ = exp ( λ 0 0 λ ), µ > 1 ), λ > 0 = S = exp H q, q = λxξ + O((x,ξ) 3 ) Need to control error, and estimate all around γ!

56 Poincaré Map ds(0, 0) = ( µ 0 0 1/µ = exp ( λ 0 0 λ ), µ > 1 ), λ > 0 = S = exp H q, q = λxξ + O((x,ξ) 3 ) Need to control error, and estimate all around γ!

57 Poincaré Map Stable/Unstable Manifolds Rough idea: q does not have a sign, but conjugating Q = Op(q) gives AQA 1 = Q [Q, A]A 1 so we look at commutator [Q, A] General guiding principal: H q (dist 2 (,Λ )) ±dist 2 (,Λ )

58 Poincaré Map Stable/Unstable Manifolds Rough idea: q does not have a sign, but conjugating Q = Op(q) gives AQA 1 = Q [Q, A]A 1 so we look at commutator [Q, A] General guiding principal: H q (dist 2 (,Λ )) ±dist 2 (,Λ )

59 Poincaré Map Change Variables Change variables: { dist 2 x 2 (,Λ ) = The change of variables on the Poincaré section ξ 2 Figure: Λ ± and the change of variables

60 Poincaré Map Change Variables Change variables: { dist 2 x 2 (,Λ ) = The change of variables on the Poincaré section ξ 2.. Figure: Λ ± and the change of variables

61 Poincaré Map Error Control Show (hard...) q(x,ξ) = λ(x,ξ)xξ Then H q (dist 2 (,Λ ) dist 2 (,Λ + )) λ(x,ξ)(x 2 + ξ 2 ) + error, where now error can be controlled

62 Poincaré Map Error Control Show (hard...) q(x,ξ) = λ(x,ξ)xξ Then H q (dist 2 (,Λ ) dist 2 (,Λ + )) λ(x,ξ)(x 2 + ξ 2 ) + error, where now error can be controlled

63 Quantization From S to P z Quantize S as M : L 2 (R n 1 ) L 2 (R n 1 ): M 1 Op (a)m = Op (S a) + O(h 2 ) S varies with energy level z, so M = M(z) Theorem (P z)u L 2 (M) C 1 h (I M(z))Ru L 2 (R n 1 ), R restriction to projection of Poincaré section. Further, h u L 2 (M) Ch Ru L 2 (R n 1 ) + C (P z)u L 2 (M) We need to show (I M(z))Ru h N Ru

64 Quantization From S to P z Quantize S as M : L 2 (R n 1 ) L 2 (R n 1 ): M 1 Op (a)m = Op (S a) + O(h 2 ) S varies with energy level z, so M = M(z) Theorem (P z)u L 2 (M) C 1 h (I M(z))Ru L 2 (R n 1 ), R restriction to projection of Poincaré section. Further, h u L 2 (M) Ch Ru L 2 (R n 1 ) + C (P z)u L 2 (M) We need to show (I M(z))Ru h N Ru

65 Quantization From S to P z Quantize S as M : L 2 (R n 1 ) L 2 (R n 1 ): M 1 Op (a)m = Op (S a) + O(h 2 ) S varies with energy level z, so M = M(z) Theorem (P z)u L 2 (M) C 1 h (I M(z))Ru L 2 (R n 1 ), R restriction to projection of Poincaré section. Further, h u L 2 (M) Ch Ru L 2 (R n 1 ) + C (P z)u L 2 (M) We need to show (I M(z))Ru h N Ru

66 Quantization From S to P z Quantize S as M : L 2 (R n 1 ) L 2 (R n 1 ): M 1 Op (a)m = Op (S a) + O(h 2 ) S varies with energy level z, so M = M(z) Theorem (P z)u L 2 (M) C 1 h (I M(z))Ru L 2 (R n 1 ), R restriction to projection of Poincaré section. Further, h u L 2 (M) Ch Ru L 2 (R n 1 ) + C (P z)u L 2 (M) We need to show (I M(z))Ru h N Ru

67 Quantization A Cheat... For exposition, write M(z) = exp and for G to be determined, e G(x,hDx)/h Me G(x,hDx)/h = exp ( ) i Op (q z), h ( ) i h e G/h Op(q z)e G/h e G/h Op(q)e G/h = Op (q) ihop (H q G) + error with error hard but controllable Op(q) is self-adjoint, hence exp(iop (q)/h) is unitary. If H q G is real and has a definite sign, we re in good shape.

68 Quantization A Cheat... For exposition, write M(z) = exp and for G to be determined, e G(x,hDx)/h Me G(x,hDx)/h = exp ( ) i Op (q z), h ( ) i h e G/h Op(q z)e G/h e G/h Op(q)e G/h = Op (q) ihop (H q G) + error with error hard but controllable Op(q) is self-adjoint, hence exp(iop (q)/h) is unitary. If H q G is real and has a definite sign, we re in good shape.

69 Quantization A Cheat... For exposition, write M(z) = exp and for G to be determined, e G(x,hDx)/h Me G(x,hDx)/h = exp ( ) i Op (q z), h ( ) i h e G/h Op(q z)e G/h e G/h Op(q)e G/h = Op (q) ihop (H q G) + error with error hard but controllable Op(q) is self-adjoint, hence exp(iop (q)/h) is unitary. If H q G is real and has a definite sign, we re in good shape.

70 Quantization A Cheat... For exposition, write M(z) = exp and for G to be determined, e G(x,hDx)/h Me G(x,hDx)/h = exp ( ) i Op (q z), h ( ) i h e G/h Op(q z)e G/h e G/h Op(q)e G/h = Op (q) ihop (H q G) + error with error hard but controllable Op(q) is self-adjoint, hence exp(iop (q)/h) is unitary. If H q G is real and has a definite sign, we re in good shape.

71 Quantization Special Calculus Recall we want G = dist 2 (,Λ ) dist 2 (,Λ + ) = x 2 ξ 2, so H q G x 2 + ξ 2, harmonic oscillator Growth of G is bad, take G = h 2 log ( h + x 2 h + ξ 2 ) Bad calculus! Rescale with new parameter h h: x h 1 2 x, ξ h 2ξ 1 h 1 2 h 1 2

72 Quantization Special Calculus Recall we want G = dist 2 (,Λ ) dist 2 (,Λ + ) = x 2 ξ 2, so H q G x 2 + ξ 2, harmonic oscillator Growth of G is bad, take G = h 2 log ( h + x 2 h + ξ 2 ) Bad calculus! Rescale with new parameter h h: x h 1 2 x, ξ h 2ξ 1 h 1 2 h 1 2

73 Quantization Special Calculus Recall we want G = dist 2 (,Λ ) dist 2 (,Λ + ) = x 2 ξ 2, so H q G x 2 + ξ 2, harmonic oscillator Growth of G is bad, take G = h 2 log ( h + x 2 h + ξ 2 ) Bad calculus! Rescale with new parameter h h: x h 1 2 x, ξ h 2ξ 1 h 1 2 h 1 2

74 Quantization Final Estimate ( ) x 2 H q G hλ 1 + x 2 + ξ2 1 + ξ 2, which is morally like the harmonic oscillator. In rescaled calculus, Op (H q G) h h C, h > 0 fixed

75 Quantization Final Estimate ( ) x 2 H q G hλ 1 + x 2 + ξ2 1 + ξ 2, which is morally like the harmonic oscillator. In rescaled calculus, Op (H q G) h h C, h > 0 fixed

76 Quantization Final Estimate Thus Im e G/h Op(q)e G/h h C, so e G/h Me G/h e 1/C < 1 = (I e G/h Me G/h ) 1 C.

77 Quantization Final Estimate Thus Im e G/h Op(q)e G/h h C, so e G/h Me G/h e 1/C < 1 = (I e G/h Me G/h ) 1 C.

78 Quantization Rescaling... Rescaling back to h calculus, we have the desired estimate: (I M(z))Ru h N Ru

79 Schrödinger Equation Assumptions (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

80 Schrödinger Equation Assumptions (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

81 Schrödinger Equation Assumptions (M, g) non-compact, Riemannian manifold with or without a compact boundary g asymptotically Euclidean scattering metric γ M unstable closed geodesic with only transversal reflections, M non-trapping otherwise

82 Schrödinger Equation Local Smoothing Effect Theorem (1) Let ρ s C (M) satisfy ρ s (x) dist g (x, x 0 ) s, for x outside a compact set and fixed x 0. Suppose V C (M), 0 V C, satisfies V C dist g (x, x 0 ) 1 δ, for some δ > 0. For each s > 1/2 and ǫ > 0, T 0 ρ s e it( g V) u 0 2 H 1/2 ǫ (M) dt C u 0 2 L 2 (M). (4.1)

83 Schrödinger Equation Remark Holds for more general metrics (as studied by Cardoso-Popov-Vodev ( 04)) If M =, (4.1) is global in time if we replace ρ s with ψ C (M) satisfying outside a compact set. ψ exp( dist g (x, x 0 ) 2 ) If (M, g) is equal to R n outside a compact set, (4.1) is global in time with ψ replacing ρ s. Generalizes Constantin-Saut ( 88), Doi ( 96), and Burq ( 04)

84 Schrödinger Equation Remark Holds for more general metrics (as studied by Cardoso-Popov-Vodev ( 04)) If M =, (4.1) is global in time if we replace ρ s with ψ C (M) satisfying outside a compact set. ψ exp( dist g (x, x 0 ) 2 ) If (M, g) is equal to R n outside a compact set, (4.1) is global in time with ψ replacing ρ s. Generalizes Constantin-Saut ( 88), Doi ( 96), and Burq ( 04)

85 Schrödinger Equation Remark Holds for more general metrics (as studied by Cardoso-Popov-Vodev ( 04)) If M =, (4.1) is global in time if we replace ρ s with ψ C (M) satisfying outside a compact set. ψ exp( dist g (x, x 0 ) 2 ) If (M, g) is equal to R n outside a compact set, (4.1) is global in time with ψ replacing ρ s. Generalizes Constantin-Saut ( 88), Doi ( 96), and Burq ( 04)

86 Schrödinger Equation Remark Holds for more general metrics (as studied by Cardoso-Popov-Vodev ( 04)) If M =, (4.1) is global in time if we replace ρ s with ψ C (M) satisfying outside a compact set. ψ exp( dist g (x, x 0 ) 2 ) If (M, g) is equal to R n outside a compact set, (4.1) is global in time with ψ replacing ρ s. Generalizes Constantin-Saut ( 88), Doi ( 96), and Burq ( 04)

87 Wave Equation The Wave Equation Suppose also dim M = n 3 is odd and M is equal to R n outside a compact set Assume e x 2 V(x) = o(1) We study the wave equation ( Dt 2 g + V)u(x, t) = 0, (x, t) M [0, ) u(x, 0) = u 0 H 1 (M), D t u(x, 0) = u 1 L 2 (M). (4.2)

88 Wave Equation The Wave Equation Suppose also dim M = n 3 is odd and M is equal to R n outside a compact set Assume e x 2 V(x) = o(1) We study the wave equation ( Dt 2 g + V)u(x, t) = 0, (x, t) M [0, ) u(x, 0) = u 0 H 1 (M), D t u(x, 0) = u 1 L 2 (M). (4.2)

89 Wave Equation The Wave Equation Suppose also dim M = n 3 is odd and M is equal to R n outside a compact set Assume e x 2 V(x) = o(1) We study the wave equation ( Dt 2 g + V)u(x, t) = 0, (x, t) M [0, ) u(x, 0) = u 0 H 1 (M), D t u(x, 0) = u 1 L 2 (M). (4.2)

90 Wave Equation Local Energy Estimate Theorem (2) For each ǫ > 0 and each pair u 0 C c (M) H 1 (M), and u 1 C c (M) L 2 (M), there is a constant C > 0 such that ψ t u 2 + ψ u 2 dx M Ce t/c ( u 0 2 H 1+ǫ (M) + u 1 2 H ǫ (M) ). Here ψ C (M) satisfies ψ = exp( x 2 ) for large x.

91 Wave Equation Comments on Theorem (2) Analogue of Sharp Huygen s Principle Generalizes results of Morawetz ( 61), Morawetz-Phillips ( 62), Morawetz-Ralston-Strauss ( 77), and Vodev ( 04) The ǫ > 0 loss in Theorems (1-2) comes from a logarithmic loss in high-energy resolvent estimates.

92 Wave Equation Comments on Theorem (2) Analogue of Sharp Huygen s Principle Generalizes results of Morawetz ( 61), Morawetz-Phillips ( 62), Morawetz-Ralston-Strauss ( 77), and Vodev ( 04) The ǫ > 0 loss in Theorems (1-2) comes from a logarithmic loss in high-energy resolvent estimates.

93 Wave Equation Comments on Theorem (2) Analogue of Sharp Huygen s Principle Generalizes results of Morawetz ( 61), Morawetz-Phillips ( 62), Morawetz-Ralston-Strauss ( 77), and Vodev ( 04) The ǫ > 0 loss in Theorems (1-2) comes from a logarithmic loss in high-energy resolvent estimates.

94 Wave Equation The general resolvent estimate Suppose (M, g), V, and ρ s satisfy the assumptions of Theorem 1, and set P = g + V(x). Theorem (3) For each ǫ > 0 sufficiently small and s > 1/2, there is a constant C > 0 such that ρ s (P (τ ± iǫ)) 1 L log(2 + τ ) ρ s C 2 (M) L 2 (M) τ 1/2. (4.3)

95 Wave Equation The general resolvent estimate Suppose (M, g), V, and ρ s satisfy the assumptions of Theorem 1, and set P = g + V(x). Theorem (3) For each ǫ > 0 sufficiently small and s > 1/2, there is a constant C > 0 such that ρ s (P (τ ± iǫ)) 1 L log(2 + τ ) ρ s C 2 (M) L 2 (M) τ 1/2. (4.3)

96 Wave Equation Some comments on Theorem (3) Observe we use τ ± iǫ = λ 2, where the usual resolvent R(λ) = ( g λ 2 ) 1. σ [ C, C] + i(, ǫ] implies ρ s (P σ) 1 L ρ s C/ǫ, 2 (M) L 2 (M) so we need to prove Theorem (3) for τ C. Assumptions of Theorem (1) plus M = imply for σ [ C, C] + i(, 0), ψ(p σ) 1 ψ C L 2 (M) L 2 (M) if ψ exp( dist g (x, x 0 ) 2 ) outside a compact set Assumptions of Theorem (2) imply ψ(p λ 2 ) 1 ψ continues meromorphically to the upper half plane

97 Wave Equation Some comments on Theorem (3) Observe we use τ ± iǫ = λ 2, where the usual resolvent R(λ) = ( g λ 2 ) 1. σ [ C, C] + i(, ǫ] implies ρ s (P σ) 1 L ρ s C/ǫ, 2 (M) L 2 (M) so we need to prove Theorem (3) for τ C. Assumptions of Theorem (1) plus M = imply for σ [ C, C] + i(, 0), ψ(p σ) 1 ψ C L 2 (M) L 2 (M) if ψ exp( dist g (x, x 0 ) 2 ) outside a compact set Assumptions of Theorem (2) imply ψ(p λ 2 ) 1 ψ continues meromorphically to the upper half plane

98 Wave Equation Some comments on Theorem (3) Observe we use τ ± iǫ = λ 2, where the usual resolvent R(λ) = ( g λ 2 ) 1. σ [ C, C] + i(, ǫ] implies ρ s (P σ) 1 L ρ s C/ǫ, 2 (M) L 2 (M) so we need to prove Theorem (3) for τ C. Assumptions of Theorem (1) plus M = imply for σ [ C, C] + i(, 0), ψ(p σ) 1 ψ C L 2 (M) L 2 (M) if ψ exp( dist g (x, x 0 ) 2 ) outside a compact set Assumptions of Theorem (2) imply ψ(p λ 2 ) 1 ψ continues meromorphically to the upper half plane

99 Wave Equation Some comments on Theorem (3) Observe we use τ ± iǫ = λ 2, where the usual resolvent R(λ) = ( g λ 2 ) 1. σ [ C, C] + i(, ǫ] implies ρ s (P σ) 1 L ρ s C/ǫ, 2 (M) L 2 (M) so we need to prove Theorem (3) for τ C. Assumptions of Theorem (1) plus M = imply for σ [ C, C] + i(, 0), ψ(p σ) 1 ψ C L 2 (M) L 2 (M) if ψ exp( dist g (x, x 0 ) 2 ) outside a compact set Assumptions of Theorem (2) imply ψ(p λ 2 ) 1 ψ continues meromorphically to the upper half plane

100 Wave Equation Rescaling Use lower half plane as physical half-plane and write τ iǫ = z h 2, for z [E δ, E + δ] + i(, 0) Rescale: with g + V (τ iǫ) = 1 h2(p(h) z), P(h) = h 2 g + h 2 V z, σ h (P(h)) =: p. Assume exp(th p ) has a closed, hyperbolic orbit γ in {p = E}, and γ = π x ( γ) M is a closed hyperbolic geodesic with only transversal reflections with M.

101 Wave Equation Rescaling Use lower half plane as physical half-plane and write τ iǫ = z h 2, for z [E δ, E + δ] + i(, 0) Rescale: with g + V (τ iǫ) = 1 h2(p(h) z), P(h) = h 2 g + h 2 V z, σ h (P(h)) =: p. Assume exp(th p ) has a closed, hyperbolic orbit γ in {p = E}, and γ = π x ( γ) M is a closed hyperbolic geodesic with only transversal reflections with M.

102 Wave Equation Rescaling Use lower half plane as physical half-plane and write τ iǫ = z h 2, for z [E δ, E + δ] + i(, 0) Rescale: with g + V (τ iǫ) = 1 h2(p(h) z), P(h) = h 2 g + h 2 V z, σ h (P(h)) =: p. Assume exp(th p ) has a closed, hyperbolic orbit γ in {p = E}, and γ = π x ( γ) M is a closed hyperbolic geodesic with only transversal reflections with M.

103 Wave Equation High Energy Proposition Proposition For each s > 1/2, there exist constants C, h 0 > 0 such that for 0 < h h 0 ρ s (P(h) z) 1 L ρ s 2 (M) L 2 (M) Ch 1 log(1/h). (4.4)

104 Wave Equation Comments on the Proposition This is the same estimate as Cardoso-Popov-Vodev with logarithmic loss Observe if Im z c 0 h, then Im (P(h) z)u, u c 0 h u 2, so we assume for the proof that z [E δ, E + δ] + i( c 0 h, 0).

105 Wave Equation Comments on the Proposition This is the same estimate as Cardoso-Popov-Vodev with logarithmic loss Observe if Im z c 0 h, then Im (P(h) z)u, u c 0 h u 2, so we assume for the proof that z [E δ, E + δ] + i( c 0 h, 0).

106 Wave Equation Idea of proof Select cutoffs Apply previous results + non-trapping estimates of Cardoso-Popov-Vodev to P away from γ Use complex absorbing potential to glue together Use precise propagation of singularities to push errors to infinity

107 Wave Equation Idea of proof Select cutoffs Apply previous results + non-trapping estimates of Cardoso-Popov-Vodev to P away from γ Use complex absorbing potential to glue together Use precise propagation of singularities to push errors to infinity

108 Wave Equation Idea of proof Select cutoffs Apply previous results + non-trapping estimates of Cardoso-Popov-Vodev to P away from γ Use complex absorbing potential to glue together Use precise propagation of singularities to push errors to infinity

109 Wave Equation Idea of proof Select cutoffs Apply previous results + non-trapping estimates of Cardoso-Popov-Vodev to P away from γ Use complex absorbing potential to glue together Use precise propagation of singularities to push errors to infinity

110 Nonlinear Equations Semilinear Schrödinger Equation Write F(u) = G ( u 2 )u, where G : R R is at least C 3 and G (k) (t) C k t β k, β 1/2. Assume V satisfies assumptions of Theorem (1) and also V L q, q > 1. Study semilinear Schrödinger equation: { i t u + ( g V(x))u = F(u) on I M; u(0, x) = u 0 (x), where I R, 0 I. (4.5)

111 Nonlinear Equations Semilinear Schrödinger Equation Write F(u) = G ( u 2 )u, where G : R R is at least C 3 and G (k) (t) C k t β k, β 1/2. Assume V satisfies assumptions of Theorem (1) and also V L q, q > 1. Study semilinear Schrödinger equation: { i t u + ( g V(x))u = F(u) on I M; u(0, x) = u 0 (x), where I R, 0 I. (4.5)

112 Nonlinear Equations Semilinear Schrödinger Equation Write F(u) = G ( u 2 )u, where G : R R is at least C 3 and G (k) (t) C k t β k, β 1/2. Assume V satisfies assumptions of Theorem (1) and also V L q, q > 1. Study semilinear Schrödinger equation: { i t u + ( g V(x))u = F(u) on I M; u(0, x) = u 0 (x), where I R, 0 I. (4.5)

113 Nonlinear Equations Semilinear Corollary Corollary Suppose (M, g), V, and γ satisfy the above assumptions and dim M = 2. For each u 0 H0 1 (M), there exists p > 2β 2 such that (4.5) has a unique global solution u C((, ); H 1 0 (M)) Lp ((, ); L (M)). Moreover, the map u 0 (x) u(t, x) C((, ); H 1 0 (M)) is Lipschitz continuous on bounded sets of H 1 0 (M).

114 Nonlinear Equations Quasilinear Wave Equation Assume M = R n \ U, where U R n, n 3 odd, γ M hyperbolic geodesic, is the Dirichlet Laplacian. Suppose satisfies Q(z, w) C (C n C n2 ) i) Q is linear in w, ii) For each w, Q(, w) is a symmetric quadratic form, Consider quasilinear wave equation: { ( D 2 t )u = Q(Du, D 2 u) on M [0, ), u(x, 0) = u 0, D t u(x, 0) = u 1. (4.6)

115 Nonlinear Equations Quasilinear Wave Equation Assume M = R n \ U, where U R n, n 3 odd, γ M hyperbolic geodesic, is the Dirichlet Laplacian. Suppose satisfies Q(z, w) C (C n C n2 ) i) Q is linear in w, ii) For each w, Q(, w) is a symmetric quadratic form, Consider quasilinear wave equation: { ( D 2 t )u = Q(Du, D 2 u) on M [0, ), u(x, 0) = u 0, D t u(x, 0) = u 1. (4.6)

116 Nonlinear Equations Quasilinear Wave Equation Assume M = R n \ U, where U R n, n 3 odd, γ M hyperbolic geodesic, is the Dirichlet Laplacian. Suppose satisfies Q(z, w) C (C n C n2 ) i) Q is linear in w, ii) For each w, Q(, w) is a symmetric quadratic form, Consider quasilinear wave equation: { ( D 2 t )u = Q(Du, D 2 u) on M [0, ), u(x, 0) = u 0, D t u(x, 0) = u 1. (4.6)

117 Nonlinear Equations Quasilinear Wave Corollary The following Corollary follows directly from work of Metcalfe-Sogge ( 05, 06). Corollary Suppose (u 0, u 1 ) C (R n \ U) satisfy a compatibility condition, and if n = 3, the null condition holds. Then there exist ǫ 0 > 0 and N > 0 such that for every ǫ ǫ 0, if x α x α u 0 L 2 + x α +1 x α u 1 L 2 ǫ, α N α N 1 then (4.6) has a unique solution u C ([0, ) R n \ U).

118 Nonlinear Equations Quasilinear Wave Corollary The following Corollary follows directly from work of Metcalfe-Sogge ( 05, 06). Corollary Suppose (u 0, u 1 ) C (R n \ U) satisfy a compatibility condition, and if n = 3, the null condition holds. Then there exist ǫ 0 > 0 and N > 0 such that for every ǫ ǫ 0, if x α x α u 0 L 2 + x α +1 x α u 1 L 2 ǫ, α N α N 1 then (4.6) has a unique solution u C ([0, ) R n \ U).

119 Summary Ergodic, geometric control, or non-trapping assumptions imply good resolvent estimates with many applications. Hyperbolic trapped rays give only a logarithmic loss - almost as good. This is due to the unstable nature of these orbits. We have similar applications with a small loss due to the trapping.

120 Appendix References Definitions Definition Let M be a compact manifold with boundary. A scattering metric on M is a Riemannian metric on the interior of the form g sc = dx 2 x 4 + h x 2, (6.1) where x is a defining function for M. Here h is a smooth symmetric 2-tensor on M such that h M is nondegenerate.

121 Appendix References References I BURQ, N. Smoothing Effect for Schrödinger Boundary Value Problems. Duke Math. Journal. 123, No. 2, 2004, p BURQ, N., GÉRARD, P., AND TZVETKOV, N. On Nonlinear Schrödinger Equations in Exterior Domains. Ann. I H. Poincaré. 21, 2004, p CARDOSO, F., POPOV, G., AND VODEV, G. Semi-classical Resolvent Estimates for the Schrödinger Operator on Non-compact Complete Riemannian Manifolds. Bull. Braz. Math Soc. 35, No. 3, 2004, p CHRISTIANSON, H. Quantum Monodromy and Non-concentration near Semi-hyperbolic Orbits. in preparation.

122 Appendix References References II CONSTANTIN, P. AND SAUT, J. Local smoothing properties of dispersive equations. Journal of the Amer. Math. Soc. 1, 1988, p DOI, S.-I. Smoothing effects of Schrödinger Evolution Groups on Riemannian Manifolds. Duke Mathematical Journal. 82, No. 3, 1996, p HASSELL, A., TAO, T., AND WUNSCH, J. Sharp Strichartz Estimates on Non-trapping Asymptotically Conic Manifolds. preprint METCALFE, J. AND SOGGE, C. Hyperbolic trapped rays and global existence of quasilinear wave equations. Invent. Math. 159 (2005), No. 1, p

123 Appendix References References III MORAWETZ, C. The decay of solutions of the exterior initial-boundary value problem for the wave equation. Comm. Pure Appl. Math. 14 (1961) p MORAWETZ, C. AND PHILLIPS, R. The exponential decay of solutions of the wave equation in the exterior of a star-shaped obstacle. Bull. Amer. Math. Soc. 68 (1962) p MORAWETZ, C., RALSTON, J., AND STRAUSS, W. Decay of solutions of the wave equation outside nontrapping obstacles. Comm. Pure Appl. Math. 30, No. 4, (1977) p

124 Appendix References References IV VODEV, G. Local Energy Decay of Solutions to the Wave Equation for Nontrapping Metrics. Ark. Mat. 42, 2004, p