Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping

Local smoothing and Strichartz estimates for manifolds with degenerate hyperbolic trapping H. Christianson partly joint work with J. Wunsch (Northwestern) Department of Mathematics University of North Carolina - Chapel Hill April 23, 2012

Outline 1 Introduction: Dispersive estimates and trapping

Outline 1 Introduction: Dispersive estimates and trapping 2 Main results

Outline 1 Introduction: Dispersive estimates and trapping 2 Main results 3 Sketch of proofs

What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0,

What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ

What is dispersion? In R n, solutions to Schrödinger equation: { (D t )u = 0, u(0, x) = u 0, look like e ixξ it ξ 2 û 0 (ξ)dξ If û 0 is localized at a frequency ξ 0, stationary phase says x = 2tξ 2tξ 0, so speed depends on frequency. Practical consequences: on average in time solution is nicer than initial conditions (local smoothing/strichartz estimates).

Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work!

Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work! We will look at some examples which suggest the following:

Moral of talk Trapping: if some geodesics do not go to infinity, this doesn t work! We will look at some examples which suggest the following: Conjecture Local smoothing estimates depend quantitatively on the nature of the trapping. Strichartz estimates depend on the dimension of the trapping.

What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation).

What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing:

What is local smoothing? In R n, Schrödinger propagator e it is unitary on H s spaces (energy conservation). However, on average in time, and locally in space, propagator is smoothing: T 0 x 1/2 ǫ e it u 0 2 H 1/2 dt C u 0 2 L 2. First local smoothing result for KdV by Kato ( 83) Generalized for dispersive equations by Sjölin ( 87) and Constantin-Saut ( 88)

What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives.

What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives. In R n, the scale-invariant estimate holds: e it u 0 L p ([0,T])L q (R n ) C T u 0 L 2 (R n ), for 2 p + n q = n 2.

What are Strichartz estimates? Strichartz estimates express a smoothing effect in terms of integrability rather than honest derivatives. In R n, the scale-invariant estimate holds: for e it u 0 L p ([0,T])L q (R n ) C T u 0 L 2 (R n ), 2 p + n q = n 2. Picture for trapping geometry is much more complicated: you may lose derivatives, but possibly not.

Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood:

Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping

Trapping: local smoothing What about trapping geometries? Previously only thin fractal trapped sets were understood: Doi ( 96): sharp H 1/2 smoothing effect non-trapping Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), ǫ > 0 loss. C ( 08) One periodic geodesic, boundary value problems, thin fractal sets, ǫ > 0 loss. Datchev ( 09): Thin hyperbolic trapped set (different infinite ends ), ǫ > 0 loss.

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

Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations:

Trapping: Strichartz estimates Can use local smoothing to glue together Strichartz estimates on semiclassical timescales. This works well in non-trapping situations. In trapping situations: Burq-Gérard-Tzvetkov ( 04) and Burq ( 04): Convex bodies removed from Euclidean space (Ikawa s example), 1/p +ǫ > 0 loss, 1/p due to boundary. C ( 08): One periodic geodesic or thin fractal set without boundary ǫ > 0 loss. Burq-Guillarmou-Hassell ( 10): using work of Anantharaman, extend parametrix logarithmically leading to lossless estimates.

Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss?

Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes...

Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1).

Obvious questions Question 1: Is there something in between ǫ > 0 loss and total (1/2 derivatve) loss? Answer: Yes... Can lose 1/2 1/(m+1). Question 2: Is there something between no loss and 1/p loss Strichartz estimates?

Degenerate hyperbolic orbits Let X = R x R θ /2πZ, with g = dx 2 +(1+x 2m ) 1/m dθ 2, m Z +. X is asymptotically Euclidean: (1+x 2m ) 1/m x 2 as x

Degenerate hyperbolic orbits Let X = R x R θ /2πZ, with g = dx 2 +(1+x 2m ) 1/m dθ 2, m Z +. X is asymptotically Euclidean: (1+x 2m ) 1/m x 2 as x X has one periodic geodesic at min of (1+x 2m ) 1/m, and no other trapping Min is of order x 2m at x = 0, which is degenerate for m > 1. The Gaussian curvature K 0, K 0 as x ±, K = 0 to order 2m 2 at x = 0.

Examples γ Figure: A piece of the manifold X and the periodic geodesic γ. The Gaussian curvature is K = A /A = (2m 1)x 2m 2 (1+x 2m ) 2.

The Schrödinger equation on X Theorem (Local Smoothing) Suppose X is as above for m 2, and assume u solves { (D t )u = 0 in R X, u t=0 = u 0 H s for some s m/(m+1). Then for any T <, there exists a constant C > 0 such that T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ).

Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic.

Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect.

Remarks The case m = 1 was already known in great detail, and the loss there is actually logarithmic. Shows a relationship between the curvature near the trapped set and the local smoothing effect. The weak semiclassical timescale is a frequency dependent timescale, which also depends on m. The choice of infinite ends is somewhat irrelevant

A sharp resolvent estimate Theorem Fix m 2, and assume X = R 2 for x C. For any χ C c (X), there exists a constant C = C m,χ > 0 such that for λ 0, χ( g (λ i0) 2 ) 1 χ L 2 L 2 Cλ 2/(m+1). Moreover, this estimate is sharp, in the sense that no better polynomial rate of decay holds true.

n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential.

n-dimensional extension Smoothing theorem relies on separation of variables to produce 1-dim l semiclassical Schrödinger operator with potential. If X = R x S n 1, with g = dx 2 +(1+x 2m ) 1/m g S n 1, m Z +, (or any other compact (n 1)-dim l manifold), (n 1)-dim l trapped sphere, but same theorem holds: T 0 x 3/2 u 2 H 1 (X) dt C( D θ m/(m+1) u 0 2 L 2 + D x 1/2 u 0 2 L 2 ).

Strichartz estimates with trapping Theorem (C ( 11)) For our manifold X, if u 0 H k localized along a single harmonic subspace, for any ǫ > 0, e it u 0 L p T Lq C ǫ D θ (n 2)/2n+ǫ u 0 L 2,

Strichartz estimates with trapping Theorem (C ( 11)) For our manifold X, if u 0 H k localized along a single harmonic subspace, for any ǫ > 0, e it u 0 L p T Lq C ǫ D θ (n 2)/2n+ǫ u 0 L 2, provided 2 p + n q = n 2.

Remarks ǫ > 0 loss is probably an artifact of the method. Fractional loss is sharp.

Remarks ǫ > 0 loss is probably an artifact of the method. Fractional loss is sharp. Suggests that Strichartz estimates depend only on dimension of trapped set, but local smoothing depends only on degeneracy. Fractional loss is (n 2)/2n < 1/2 derivatives, which is always better than the (endpoint) Burq-Gérard-Tzvetkov estimate.

The Schrödinger equation in polar coordinates I In R 2 with polar coordinates, have { (D t 2 r r 1 r r 2 2 θ )u = 0, u t=0 = u 0.

The Schrödinger equation in polar coordinates I In R 2 with polar coordinates, have { (D t r 2 r 1 r r 2 θ 2 )u = 0, u t=0 = u 0. Using equation, for B self-adjoint, 0 = 2i Im = = T 0 T 0 T 0 B(D t )u, u dt B(D t )u, u u, B(D t )u dt [B, ]u, u dt + i Bu, u T 0.

The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 0

The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 Then 0 i[b,( 2 r r 1 r r 2 2 θ )] = 2 r 2 2 r + 2 arctan rr 3 2 θ + l.o.t.,

The Schrödinger equation in polar coordinates II If B = arctan(r)d r, last term is bounded using energy estimates by Bu, u T C u 0 2. H 1/2 Then 0 i[b,( r 2 r 1 r r 2 θ 2 )] = 2 r 2 r 2 + 2 arctan rr 3 θ 2 + l.o.t., IBP in r and θ, T 0 ( r 1 r u 2 L 2 + r 1/2 r 1 θ u 2 L 2 )dt C u 0 2 H 1/2.

Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t.,

Positive commutator again On X, conjugate to flat problem. With Tu(x,θ) = (1+x 2m ) 1/4m u(x,θ), T : L 2 (dvol) L 2 (dxdθ), := T T 1 = 2 x A 2 (x) 2 θ + l.o.t., Use B = arctan(x) x and commutator idea: [, B] = 2 x 2 2 x + 2A A 3 arctan(x) 2 θ + l.o.t. Coefficient A arctan(x) vanishes to order 2m at x = 0.

Estimate near 0 Separate variables: u(t, x,θ) = k e ikθ u k (t, x), and u 0 (x,θ) = k e ikθ u 0,k (x). Estimate on each mode u k.

Estimate near 0 Separate variables: u(t, x,θ) = k e ikθ u k (t, x), and u 0 (x,θ) = k e ikθ u 0,k (x). Estimate on each mode u k. Suffices to show T 0 χ(x)ku k 2 L 2 (R) dt C( k m/(m+1) u 0,k 2 L 2 + u 0,k 2 H 1/2 ) for χ C c (R), χ(x) 1 near x = 0.

Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate:

Reductions By TT argument, energy cutoff, and Fourier transform t τ, suffices to show cutoff resolvent estimate: ϕ(x, D/ k )(τ + Q) 1 ϕ(x, D/ k ) L 2 x L 2 x Ck 2/(m+1) with ϕ 1 near (0, 0) with compact support. Rescale: h = k 1 : Q = (hd x ) 2 + A 2 z + l.o.t., z 1 and theorem follows if we can prove: Proposition Let ϕ Φ 0 have wavefront set sufficiently close to (0, 0), and let m 2. Then for each ǫ > 0 sufficiently small, there exists a constant C > 0 such that ϕ(x, hd) Q 1 ϕ(x, hd) L 2 L 2 Ch 2m/m+1, z [1 ǫ, 1+ǫ].

Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1.

Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1. Idea: Escape function. Cook up a function G which increases along the Hamiltonian flow: H q G > 0.

Local dynamics Near (0, 0), symbol q of Q has Taylor expansion q ξ 2 1 m x 2m + 1 z, z 1. Idea: Escape function. Cook up a function G which increases along the Hamiltonian flow: H q G > 0. Nondegenerate case (m = 1): q = ξ 2 x 2. Hamiltonian flow: {ẋ = 2ξ, ξ = 2x,

Stable/Unstable Manifolds Λ ± = {ξ = x} is the stable/unstable manifold. ξ ξ = x x ξ = x Figure: The case non-degenerate hyperbolic case m = 1 with the stable/unstable manifolds ξ = x.

Stable/Unstable Manifolds II Idea: G ± = dist 2 (,Λ ± ) = (ξ ± x) 2 satisfies H q G ± = const.g ±. Then G = G G + satisfies H q G = 4(ξ 2 + x 2 ) is symbol of quantum harmonic oscillator (bounded below by h).

Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1,

Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, ξ = 0.

Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, degenerate critical point. ξ = 0.

Local dynamics For m 2, Hamiltonian system for q is {ẋ = 2ξ, ξ = 2x 2m 1, which has linearization about zero: {ẋ = 2ξ, degenerate critical point. Factorizing q: ξ = 0. (ξ ± m 1/2 x m ) = ±2m 1/2 x m 1 (ξ ± m 1/2 x m ),

Weakly stable/unstable manifolds ξ ξ = m 1/2 x m x ξ = m 1/2 x m Figure: The weakly stable/unstable manifolds Λ = {ξ = m 1/2 sgn(x) x m }.

Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then

Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ),

Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below.

Escape function for the degenerate case If G = dist 2 (,Λ ) dist 2 (,Λ + ), then H q G = H q (4m 1/2 ξsgn(x) x m = 8m 1/2 x m 1 (ξ 2 + x 2m ), no longer bounded below. Homogeneity try same escape function as m = 1:

Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus.

Inhomogeneous blowup calculus Idea: We are working in marginal (h 1/(m+1), h m/(m+1) ) calculus. Rescaling in phase space (x,ξ) (h 1/(m+1) x, h m/(m+1) ξ) is unitary if we switch from h-calculus to 1-calculus. q q 1 = h 2m/(m+1) (ξ 2 1 m x 2m ) takes h-dependent operator into scale-invariant calculus Escape function G xξ near (0, 0) (better control away) H q1 G = 2h 2m/(m+1) (ξ 2 +x 2m )

Complex scaling Model operator P = h 2 2 x x 2m near x = 0.

Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u.

Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u. ( e 2iθ h 2 2 x e 2imθ x 2m )w = E 0 w, w = T 1 u. If θ = π/(2m+2), get ( h 2 2 x + x 2m )w = e 2iθ E 0 w.

Complex scaling Model operator P = h 2 2 x x 2m near x = 0. Complex scaling. Let T be defined: Tu(x) = u(e iθ x), θ [0,π/2). T 1 PTT 1 u = E 0 T 1 u. ( e 2iθ h 2 x 2 e 2imθ x 2m )w = E 0 w, w = T 1 u. If θ = π/(2m+2), get ( h 2 x 2 + x 2m )w = e 2iθ E 0 w. Anharmonic oscillator e 2iθ E 0 = c 0 h 2m/(m+1), so Im E 0 = αh 2m/(m+1) for some α > 0.

Quasimodes Complex WKB quasimodes u satisfying: ( h 2 x 2 x 2 )u = E 0 u +O(h 2m/(m+1) u L 2), (3.2) and supp u { x ǫh 1/(m+1) }.

Quasimodes Complex WKB quasimodes u satisfying: ( h 2 2 x x 2 )u = E 0 u +O(h 2m/(m+1) u L 2), (3.2) and supp u { x ǫh 1/(m+1) }. Theorem Let ϕ 0 (x,θ) = e ikθ u(x), with u C c (R) as in (3.2), and h 1 = k > 0. Suppose ψ solves { (D t + )ψ = 0, ψ t=0 = ϕ 0. Then C 0 > 0 independent of k such that k 2/(m+1) /C 0 0 D θ ψ 2 L 2 dt C 1 0 D θ m/(m+1) ϕ 0 2 L 2.

Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold.

Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion

Idea of Proof of Strichartz estimates Away from trapping, perfect Strichartz estimates hold. Near trapping, try to construct parametrix and use stationary phase to compute dispersion Rescale and again look at semiclassical Schrödinger operator hd t h 2 2 x + V(x). dispersion on h-dependent timescales. Use local smoothing to glue together Strichartz estimates on semiclassical time scales.

Rough picture ξ ξ = 1 V(x) x ξ = 1 V(x) Figure: The total phase plane. The weakly unstable/stable manifolds are given by Λ ± = {ξ sgn(x) 1 V(x) = 0}.

Rough picture ξ ξ = 1 V(x) x ξ = 1 V(x) Figure: The total phase plane. The weakly unstable/stable manifolds are given by Λ ± = {ξ sgn(x) 1 V(x) = 0}. Near x = 0, V 1 x 2m.

Phase construction Hamiltonian ODEs for phase construction: {ẋ = 2ξ, so that x(t) = x 0 + 2tξ 0 + ξ = 2mx 2m 1, t 0 (t s)2mx 2m 1 (s)ds.

Phase construction Hamiltonian ODEs for phase construction: {ẋ = 2ξ, so that x(t) = x 0 + 2tξ 0 + Need to invert x 0 x(t). ξ = 2mx 2m 1, t 0 (t s)2mx 2m 1 (s)ds.

Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1).

Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate.

Dyadic decomposition Break up { x ǫ} into O(log(1/h)) regions where x 2 j h 1/(m+1). Good stationary phase estimates up to time h (1 m)/(m+1) dispersion estimate. Prove wave packets propagate out of {x 2 j h 1/(m+1) } also in time h (1 m)/(m+1). O(h ) cross correlations, at least for timescales h (1 m)/(m+1) ǫ/2. Prove approximate solution is close to actual solution for timescales h (1 m)/(m+1)+ǫ/2.