Strichartz Estimates in Domains

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1 Department of Mathematics Johns Hopkins University April 15, 2010

2 Wave equation on Riemannian manifold (M, g) Cauchy problem: 2 t u(t, x) gu(t, x) =0 u(0, x) =f (x), t u(0, x) =g(x) Strichartz estimates: u L q t Lr x (( T,T ) M) f H γ (M) + g H γ 1 (M) Estimates hold on R n or compact manifold without boundary if: Scale invariance: 1 q + n r = n 2 γ 2 Admissibility: q + n 1 r n 1 2

3 Admissibility: The Knapp Example R n e ix ξ it ξ β( ξ /λ)ρ( ξ /λ 1/2 ) dξ, β C 0 ((1, 2)), ρ C 0 ((0, 1)), ξ =(ξ 2, ξ 3,...,ξ n ). For t 1, lives" on x λ 1/2, t x 1 λ 1 Coherent: no dispersion during t 1 Highest possible concentration of Euclidean waves Wave fronts", t = x 1, provide foliation of strip where x λ 1/2

4 Blair-Smith-S, 2008 (M, g)= compact manifold with boundary, Dirichlet or Neumann conditions at M, then the Strichartz estimates hold if 1 q + n r = n 2 γ for 3 q + n 1 r n 1 2, n 4 1 q + 1 r 1 2, n 4 Scale invariant (no-loss), but restriction on admissibility Estimates are subcritical: on R n obtained from critical pair (q, r) followed by Sobolev embedding W s, r L r Above range of q, r certainly not optimal, but includes important estimates.

5 Key step in proof: Eliminate the boundary Geodesic normal coordinates along M : M = {x 2 0} g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1, smooth on x 2 0 Extend g across M to be even in x 2 g = d 2 x 2 + a 11 (x 1, x 2 ) d 2 x 1 Odd extension of Dirichlet solution: is solution for 2 t g x 2 x 2 φ j(x 1, x 2 ) Even extension of Neumann solution: φ j (x 1, x 2 ) is solution for 2 t g

6 Metric g is of special Lipschitz type: d 2 x g δ(x 2) is integrable along non-tangential geodesics.! dx 2 dt θ on γ dx 2 g(γ(t)) dt θ 1 "

7 Smith-S [2007]: Short time parametrix construction Microlocalize data to angle θ: Good parametrix for time t θ. For n = 2, yields sharp square function estimates (i.e., L p xl 2 t ) for wave equation, and hence sharp Lp -eigenfunction estimates. Analogous to Tataru [2002] : Strichartz estimates If 2 t,x g L 1 t L x θ 1, then u L q t Lr x ([ θ,θ] M) f H γ + g H γ 1 for same q, r, γ as smooth manifolds, Euclidean space. Problem: add up u L q t Lr x over time θ 1 slices, lose θ 1/q

8 Gain in bounds from small angle localization K λ,θ (t, x) = λ ξ 2λ ξ n θλ e ix,ξ it ξ dξ K λ,θ (t, x) λ n θ ( 1 + λ t ) n 2 2 ( 1 + λθ 2 t ) 1 2 K λ,θ (t, ) f L q t Lr x λγ θ σ(q,r) f L 2 No-loss Strichartz if σ(q, r) 1/q (i.e, condition in BSS Thm)

9 Saturating non-example on D = { x 1} Gliding wave packet dimensions λ 1 λ 2/3 :!"!!!#!!!$%& Dispersion in direction of travel (non-coherence):

10 Model convex domain (Friedlander Example) Model domain: y 0 Seek egfncts: v = e ixξ f (y) g = 2 y +(1 + y) 2 x g v = e ixξ f (y) ξ 2 (1+y)f (y) Since d 2 dt 2 Ai(t) =t Ai(t), if f (y) =Ai(ξ 2/3 (y b)) Eigenvalue: ξ 2 ( 1 + b ),

11 Solutions for 2 t 2 y (1 + y) 2 x e i(x t 1+b )ξ Ai ξ 2/3 (y b) x - velocity = 1 + b!&'(! )*+ ("!$,,!"#$!"#% Dirichlet: ξ 2/3 b = zero of Airy function { ω k } k=0, ω k =( 3π 2 k)2/3 + O(k 1/3 ). Fixed zero of Ai forces b = ω k ξ 2/3

12 Fix b, ignore boundary condition: u = e iξ(x t 1+b ) Ai ξ 2/3 (y b) ξ 2/3 dξ = e iξ(x t 1+b )+iη(b y) i 1 3 η3 /ξ 2 dξ dη y=b! x=t(1+b) 1/2

13 Boundary conditions: ξ 2/3 b = ω k : ξ k kπ 3 2 b 3/2 Poisson summation: periodize by 4 3 b3/2 v=(1+b) 1/2! y=b y=0 By Poisson summation formula, if ω k ( 3π 2 k)2/3, resulting function would be k eixξ k +it 1+b Ai(ξ 2/3 k (y b))ξ 2/3 k which vanishes when y = 0, by choice of ξ k.

14 Ivanovici s example (Knappovici): For cusps localized to frequency ξ [λ, 2λ] with b = λ 1 2 +!"##!%&'!()*+,##!%!"!!!"#$ u λ λ 2/3 (b (y y(t))) 1/4 φ λ((x x(t))± 2 3 (b (y y(t)))3/2 ), (x(t), y(t)) bouncing ray:

15 Ray optics tracking of cusp wavefront in domain: The two halves of the (moving) cusps each provide foliations of the strip 0 < y < b The coherent λ 2/3 λ 1 wave-packets live on intersection of bouncing ray (x(t), y(t)) and the appropriate leaves of foliation (right half of cusps for left halves of bouncing ray and visa-versa)

16 u λ (t, ) r u λ (0, ) 2 λ 3 4 ( 1 2 r 4 ) r < 4 λ 5 3 ( r ) 1 24 r > 4 Ivanovici [2008] (n = 2) Strichartz estimates fail for 3 q + 1 r > and q > 4 Blair-S-Sogge [2008] (n = 2) Strichartz estimates hold for 3 q + 1 r 1 2

17 Ideal range for L q t Lr x estimates

18 Knappovici rules this out

19 Current world record: [B-S-S]

20 Energy critical wave equation for n = 3 u(t, x) = u 5 (t, x), u Ω = 0, u(0, x) =f (x), t u(0, x) =g(x) Two key Strichartz estimates: u = 0 u L 5 t L 10 x ([0,T ] Ω) + u L 4 t L12 x ([0,T ] Ω) f H 1 (Ω) + g L 2 (Ω)

21 Energy critical wave equation for n = 3 L 5 t L10 x contraction argument small data global existence u 5 L 1 t L 2 x u5 L 5 t L10 x Grillakis-Shatah-Struwe [1992]: global existence for large data uses local L 6 decay: u 6 dx = 0 lim t t 0 x x 0 t 0 t u 5 L 1 t L 2 x u L t L 6 x u4 L 4 t L12 x Burq-Lebeau-Planchon [2008]: obtained existence results, using more involved arguments based on weaker Strichartz estimates (proved by e.f. estimates)

22 Scattering for u = u 5 Bahouri-Gérard-Shatah [ ]: Ω = R 3 There exists free solutions v ± = 0 such that lim E(u v ±)(t) =0 t ± Key step: lim u 6 (t, x) dx = 0 t ± R 3 Blair-Smith-S [2008]: Ω = exterior to star-shaped obstacle. Strichartz estimates global in time on exterior to non-trapping obstacle by Smith-S [2000]

23 Energy critical wave equation for n = 4 u(t, x) = u 3 (t, x), u Ω = 0, u(0, x) =f (x), t u(0, x) =g(x) Key Strichartz estimate: u = 0 u L 3 t L 6 x ([0,T ] Ω) f H 1 (Ω) + g L 2 (Ω) Contraction argument + energy conservation small data global existence, large data global existence for sub-critical powers. Large data global existence for critical power requires an estimate with q < 3, but q = 3 is endpoint for [B-S-S].

24 Exterior domains Schrödinger equation on Riemannian manifold (M, g) Cauchy problem: i t u(t, x) g u(t, x) =0 Strichartz estimates: u(0, x) =f (x) u L q t Lr x (( T,T ) M) f H γ (M) + g H γ 1 (M) Estimates hold for M = R n if: Scale invariance: 2 q + n r = n 2 γ Admissibility: 2 q + n r n 2 Loss of derivatives on compact manifolds...

25 Exterior domains Semiclassical / Short time parametrices Staffilani-Tataru, Burq-Gérard-Tzvetkov For û λ (t, ξ) localized to ξ [λ, 2λ], good parametrices for time t λ 1 u λ L q t Lr x ([0,λ 1 ] M) λγ f λ L 2 LeBeau [1992]: Rescale t λt i t λ 1 g has bicharacteristics of speed 1 for ξ λ.

26 Exterior domains For data localized to angle θ from tangent, good parametrices for time λ 1 θ : gain θ σ(q,r) in Strichartz estimates. Blair-S.-Sogge, manifolds wth boundary For û λ (t, ξ) localized to ξ [λ, 2λ], scale invariant estimates u λ L q t Lr x ([0,λ 1 ] M) λγ f λ L 2 hold for the range 3 q + n r n 2 n = 2 1 q + 1 r 1 2 n 3 Add over time intervals: u λ L q t Lr x ([0,1] M) λγ+1/q f λ L 2 (i.e., 1/q loss derivatives)

27 Exterior domains Nontrapping exterior domains: No-loss estimates Burq-Gérard-Tzvetkov [2004]: Local Smoothing If Ω = R n \K is nontrapping, then for χ C c (Ω), s [0, 1] χu L 2 t H s+ 1 2 f H s Using this, above, and arguments going back to Journé-Soffer-S and Staffilani-Tataru, can prove no-loss Strichartz estimates for nontrapping domains when 3 q + n r n 2 for n = 2, 3... Similar applications to nonlinear Schrödinger (small data global existence for energy critical equation, etc.)