Strauss conjecture on asymptotically Euclidean manifolds

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1 Strauss conjecture on asymptotically Euclidean manifolds Xin Yu (Joint with Chengbo Wang) Department of Mathematics, Johns Hopkins University Baltimore, Maryland Mar 12-Mar 13, 2010

2 The Problem We consider the wave equations on asymptocially Euclidean manifolds (M, g) ( ) { g u = ( 2 t g )u = F (u) on R + M u(0, ) = f, t u(0, ) = g F (u) u p when u is small. g = 1 ij det g i det gg ij j is the Laplace-Beltrami operator. Assumptions on the metric g 1 2 α N n α x (g ij δ ij ) = O( x α ρ ), (H1) with δ ij = δ ij being the Kronecker delta function. g is non-trapping. (H2)

3 The Problem We consider the wave equations on asymptocially Euclidean manifolds (M, g) ( ) { g u = ( 2 t g )u = F (u) on R + M u(0, ) = f, t u(0, ) = g F (u) u p when u is small. g = 1 ij det g i det gg ij j is the Laplace-Beltrami operator. Assumptions on the metric g 1 2 α N n α x (g ij δ ij ) = O( x α ρ ), (H1) with δ ij = δ ij being the Kronecker delta function. g is non-trapping. (H2)

4 The Problem We consider the wave equations on asymptocially Euclidean manifolds (M, g) ( ) { g u = ( 2 t g )u = F (u) on R + M u(0, ) = f, t u(0, ) = g F (u) u p when u is small. g = 1 ij det g i det gg ij j is the Laplace-Beltrami operator. Assumptions on the metric g 1 2 α N n α x (g ij δ ij ) = O( x α ρ ), (H1) with δ ij = δ ij being the Kronecker delta function. g is non-trapping. (H2)

5 The Problem We consider the wave equations on asymptocially Euclidean manifolds (M, g) ( ) { g u = ( 2 t g )u = F (u) on R + M u(0, ) = f, t u(0, ) = g F (u) u p when u is small. g = 1 ij det g i det gg ij j is the Laplace-Beltrami operator. Assumptions on the metric g 1 2 α N n α x (g ij δ ij ) = O( x α ρ ), (H1) with δ ij = δ ij being the Kronecker delta function. g is non-trapping. (H2)

6 Goals For small data, we want to set up: Note Global existence result (Strauss Conjecture) for n = 3, 4 and p > p c. where p c is the larger root of the equation (n 1)p 2 (n + 1)p 2 = 0. Local existence result for n = 3 and p < p c with almost sharp life span T ɛ = Cɛ p(p 1) p 2 2p 1 +ɛ. p c = for n = 3, p c = 2 for n = 4.

7 Goals For small data, we want to set up: Note Global existence result (Strauss Conjecture) for n = 3, 4 and p > p c. where p c is the larger root of the equation (n 1)p 2 (n + 1)p 2 = 0. Local existence result for n = 3 and p < p c with almost sharp life span T ɛ = Cɛ p(p 1) p 2 2p 1 +ɛ. p c = for n = 3, p c = 2 for n = 4.

8 Earlier Work in Minkowski space R + R n 79 John: n=3, global sol n for p > 1 + 2, almost global sol n for p < 1 + 2; 81 Struss Conjecture: n 2, global sol n iff p > p c, where p c is the larger root of (n 1)p c (n + 1)p c 2 = Glassey: Verify for n = 2; 87 Sideris: Blow up for p p c.

9 Earlier Work in Minkowski space R + R n 79 John: n=3, global sol n for p > 1 + 2, almost global sol n for p < 1 + 2; 81 Struss Conjecture: n 2, global sol n iff p > p c, where p c is the larger root of (n 1)p c (n + 1)p c 2 = Glassey: Verify for n = 2; 87 Sideris: Blow up for p p c.

10 Earlier Work in Minkowski space R + R n 79 John: n=3, global sol n for p > 1 + 2, almost global sol n for p < 1 + 2; 81 Struss Conjecture: n 2, global sol n iff p > p c, where p c is the larger root of (n 1)p c (n + 1)p c 2 = Glassey: Verify for n = 2; 87 Sideris: Blow up for p p c.

11 Earlier Work (continued) On more general domains. Perturbed by obtacles 1 08 D.M.S.Z: Nontrapping, g =, n = 4, p > p c ; 2 08 H.M.S.S.Z: Nontrapping, n = 3, 4, p > p c ; 3 09 Yu: Trapping (Limited), n = 3, 4, p > p c ; n = 3, p p c under symmetric metric.

12 Earlier Work (continued) On more general domains. Perturbed by obtacles 1 08 D.M.S.Z: Nontrapping, g =, n = 4, p > p c ; 2 08 H.M.S.S.Z: Nontrapping, n = 3, 4, p > p c ; 3 09 Yu: Trapping (Limited), n = 3, 4, p > p c ; n = 3, p p c under symmetric metric.

13 Our Result (Global existence part) Theorem Suppose (H1) and (H2) hold with ρ > 2. Also assume 2 u i uf i (u) u p. i=1 If n = 3, 4, p c < p < 1 + 4/(n 1), then there is a global solution (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣ s 1, α 2, with small data and s = s c ɛ.

14 Sample proof in Minkowski space Iteration method Let u 1 0, u k solves { ( 2 t g )u k (t, x) = F p (u k 1 (t, x)), (t, x) R + Ω u k (0, ) = f, t u k (0, ) = g. Continuity argument. Guaranteed by the Strichartz estimates, x ( n 2 +1 γ)/p u L p t Lp r L 2 ω (f, g) (Ḣγ,Ḣγ 1 ) + x n 2 +1 γ F L 1 t L 1 r L2 ω for 1/2 1/p < γ < n/2 1/p, and energy estimates, u L t Ḣ γ x f Ḣ γ + g Ḣγ 1.

15 Sample proof in Minkowski space Iteration method Let u 1 0, u k solves { ( 2 t g )u k (t, x) = F p (u k 1 (t, x)), (t, x) R + Ω u k (0, ) = f, t u k (0, ) = g. Continuity argument. Guaranteed by the Strichartz estimates, x ( n 2 +1 γ)/p u L p t Lp r L 2 ω (f, g) (Ḣγ,Ḣγ 1 ) + x n 2 +1 γ F L 1 t L 1 r L2 ω for 1/2 1/p < γ < n/2 1/p, and energy estimates, u L t Ḣ γ x f Ḣ γ + g Ḣγ 1.

16 Our proof for the case p > p c Set up the argument.define the norm X : Set u(t, ) X = u L sγ ( x <R) + x ( n 2 +1 γ)/p u L p r L 2 ω( x >R) M k = ( Z α u L k t Ḣ γ (R + R n ) + t Z α u L k t Ḣ γ 1 (R + R n ) + Z α u L p t X ). GOAL: Show M k < Cɛ if Z α (f, g) ( Ḣ γ,ḣγ 1 ) < ɛ.

17 Our proof for the case p > p c Set up the argument.define the norm X : Set u(t, ) X = u L sγ ( x <R) + x ( n 2 +1 γ)/p u L p r L 2 ω( x >R) M k = ( Z α u L k t Ḣ γ (R + R n ) + t Z α u L k t Ḣ γ 1 (R + R n ) + Z α u L p t X ). GOAL: Show M k < Cɛ if Z α (f, g) ( Ḣ γ,ḣγ 1 ) < ɛ.

18 Our proof for the case p > p c Set up the argument.define the norm X : Set u(t, ) X = u L sγ ( x <R) + x ( n 2 +1 γ)/p u L p r L 2 ω( x >R) M k = ( Z α u L k t Ḣ γ (R + R n ) + t Z α u L k t Ḣ γ 1 (R + R n ) + Z α u L p t X ). GOAL: Show M k < Cɛ if Z α (f, g) ( Ḣ γ,ḣγ 1 ) < ɛ.

19 Proof for p > p c, continued Key Ingredients. KSS and Strichartz Estimates x 1 2 s ɛ Z α u L 2 t L 2+ x n 2 n+1 p s ɛ Z α u x L p Energy Estimates t Lp x L2+η ω ( Z α f Ḣs + Z α ) g Ḣs 1, ( ) Z α u L t Ḣ + Z α u s L t Ḣ + Z α u s 1 L p t Lqs x ( x 1) where q s = 2n/(n 2s). ( Z α f Ḣs + Z α ) g Ḣs 1, ({ x >1})

20 Transformation on the Equation Set P = g g g 1. We will prove the estimates if u is the solution of ( 2 + P)u = F, so that u(t) = cos(tp 1 2 )f +P 1 2 sin(tp 1 2 )g + t 0 P 1 2 sin((t s)p 1 2 )F (s)ds. Equivalence: if v solves ( 2 t g )v(t, x) = G(t, x), we have relation u = gv, F = gg.

21 Transformation on the Equation Set P = g g g 1. We will prove the estimates if u is the solution of ( 2 + P)u = F, so that u(t) = cos(tp 1 2 )f +P 1 2 sin(tp 1 2 )g + t 0 P 1 2 sin((t s)p 1 2 )F (s)ds. Equivalence: if v solves ( 2 t g )v(t, x) = G(t, x), we have relation u = gv, F = gg.

22 Proof of the estimates with order 0 KSS estimates: 08 Bony, Häfner. Strichartz estimates: Interpolation between KSS estimates and angular Sobolev inequality, x n 2 α e itp1/2 f (x) eitp1/2 L t, x L 2+η f (x) ω L t Ḣ f x α Ḣ ; (1) x α Energy estimates: Equivalence of P s/2 and s with s [0, 1]; Local Energy decay (By interpolation between KSS estimates), βu L 2 f t H s Ḣs + g Ḣs 1.

23 Proof of the estimates with order 0 KSS estimates: 08 Bony, Häfner. Strichartz estimates: Interpolation between KSS estimates and angular Sobolev inequality, x n 2 α e itp1/2 f (x) eitp1/2 L t, x L 2+η f (x) ω L t Ḣ f x α Ḣ ; (1) x α Energy estimates: Equivalence of P s/2 and s with s [0, 1]; Local Energy decay (By interpolation between KSS estimates), βu L 2 f t H s Ḣs + g Ḣs 1.

24 Proof of the estimates with order 0 KSS estimates: 08 Bony, Häfner. Strichartz estimates: Interpolation between KSS estimates and angular Sobolev inequality, x n 2 α e itp1/2 f (x) eitp1/2 L t, x L 2+η f (x) ω L t Ḣ f x α Ḣ ; (1) x α Energy estimates: Equivalence of P s/2 and s with s [0, 1]; Local Energy decay (By interpolation between KSS estimates), βu L 2 f t H s Ḣs + g Ḣs 1.

25 KSS and Energy estimates with higher order derivatives Z α =, use relation between and P 1/2. 1 u Ḣs P s/2 u L 2 x, for s [ 1, 1]; 2 3/2 µ 1 < µ 2 µ 3 3/2, then x µ 3 l u L 2 (R d ) x µ 2 P 1/2 u n L 2 (R d ) x µ 3 l u L 2 (R d ). Z α = 2, use relation between 2 and P. 1 For s [0, 1], we have 2 x f Ḣs Pf Ḣs + f Ḣs. Pf Ḣs x α f Ḣs. 2 For 0 < µ 3/2 and k 2, we have l=1 x µ 2 x u L 2 x x µ u L 2 x + x µ Pu L 2 x.

26 KSS and Energy estimates with higher order derivatives Z α =, use relation between and P 1/2. 1 u Ḣs P s/2 u L 2 x, for s [ 1, 1]; 2 3/2 µ 1 < µ 2 µ 3 3/2, then x µ 3 l u L 2 (R d ) x µ 2 P 1/2 u n L 2 (R d ) x µ 3 l u L 2 (R d ). Z α = 2, use relation between 2 and P. 1 For s [0, 1], we have 2 x f Ḣs Pf Ḣs + f Ḣs. Pf Ḣs x α f Ḣs. 2 For 0 < µ 3/2 and k 2, we have l=1 x µ 2 x u L 2 x x µ u L 2 x + x µ Pu L 2 x.

27 KSS and Energy estimates with higher order derivatives Z α =, use relation between and P 1/2. 1 u Ḣs P s/2 u L 2 x, for s [ 1, 1]; 2 3/2 µ 1 < µ 2 µ 3 3/2, then x µ 3 l u L 2 (R d ) x µ 2 P 1/2 u n L 2 (R d ) x µ 3 l u L 2 (R d ). Z α = 2, use relation between 2 and P. 1 For s [0, 1], we have 2 x f Ḣs Pf Ḣs + f Ḣs. Pf Ḣs x α f Ḣs. 2 For 0 < µ 3/2 and k 2, we have l=1 x µ 2 x u L 2 x x µ u L 2 x + x µ Pu L 2 x.

28 KSS and Energy estimates with higher order derivatives (continued) When Z α = Ω or Z α = Ω 2, then Z α u solves ( t 2 + P)Z α u = [P, Z α ]u, with initial data (Z α f, Z α g). Commutator terms [P, Ω]u = r 2 α α u. [P, Ω 2 ]u = r 2 α α u. α 3 where r i C is such that x α r j (x) = O ( x ρ j α ), α,

29 KSS and Energy estimates with higher order derivatives (continued) When Z α = Ω or Z α = Ω 2, then Z α u solves ( t 2 + P)Z α u = [P, Z α ]u, with initial data (Z α f, Z α g). Commutator terms [P, Ω]u = r 2 α α u. [P, Ω 2 ]u = r 2 α α u. α 3 where r i C is such that x α r j (x) = O ( x ρ j α ), α,

30 KSS and Energy estimates with higher order derivatives (continued) Techniques to handle commutator terms Let w solve the wave equation with f = g = 0, x 1/2 s ɛ w L 2 t L 2 x x (1/2)+ɛ F L 2 t Ḣ s 1 ; w L t Ḣx s x 1/2+ɛ F L 2 t Ḣx s 1. Fractional Lebniz rule. For any s ( n/2, 0) (0, n/2), fg Ḣs f L Ḣ s,n/ s g Ḣs. For any s [0, 1], ɛ > 0 and α = N, we have x (1/2) ɛ x α u L 2 t Ḣ f s 1 ḢN+s 1 Ḣs + g ḢN+s 2. Ḣs 1 α =N

31 KSS and Energy estimates with higher order derivatives (continued) Techniques to handle commutator terms Let w solve the wave equation with f = g = 0, x 1/2 s ɛ w L 2 t L 2 x x (1/2)+ɛ F L 2 t Ḣ s 1 ; w L t Ḣx s x 1/2+ɛ F L 2 t Ḣx s 1. Fractional Lebniz rule. For any s ( n/2, 0) (0, n/2), fg Ḣs f L Ḣ s,n/ s g Ḣs. For any s [0, 1], ɛ > 0 and α = N, we have x (1/2) ɛ x α u L 2 t Ḣ f s 1 ḢN+s 1 Ḣs + g ḢN+s 2. Ḣs 1 α =N

32 KSS and Energy estimates with higher order derivatives (continued) Techniques to handle commutator terms Let w solve the wave equation with f = g = 0, x 1/2 s ɛ w L 2 t L 2 x x (1/2)+ɛ F L 2 t Ḣ s 1 ; w L t Ḣx s x 1/2+ɛ F L 2 t Ḣx s 1. Fractional Lebniz rule. For any s ( n/2, 0) (0, n/2), fg Ḣs f L Ḣ s,n/ s g Ḣs. For any s [0, 1], ɛ > 0 and α = N, we have x (1/2) ɛ x α u L 2 t Ḣ f s 1 ḢN+s 1 Ḣs + g ḢN+s 2. Ḣs 1 α =N

33 Weighted Strichartz estimates with higher order derivatives x n 2 n+1 p s ɛ Z α u L p t Lp x L2+η ω ({ x >1}) ( Z α f Ḣs + Z α g Ḣs 1) Interpolation between p = 2 and p = p = 2: KSS estimates; p = : x n 2 s Z α u L t, x L 2+η ω Z α u L t Ḣ s x ( Z α f Ḣs + Z α g Ḣs 1)

34 Weighted Strichartz estimates with higher order derivatives x n 2 n+1 p s ɛ Z α u L p t Lp x L2+η ω ({ x >1}) ( Z α f Ḣs + Z α g Ḣs 1) Interpolation between p = 2 and p = p = 2: KSS estimates; p = : x n 2 s Z α u L t, x L 2+η ω Z α u L t Ḣ s x ( Z α f Ḣs + Z α g Ḣs 1)

35 Weighted Strichartz estimates with higher order derivatives x n 2 n+1 p s ɛ Z α u L p t Lp x L2+η ω ({ x >1}) ( Z α f Ḣs + Z α g Ḣs 1) Interpolation between p = 2 and p = p = 2: KSS estimates; p = : x n 2 s Z α u L t, x L 2+η ω Z α u L t Ḣ s x ( Z α f Ḣs + Z α g Ḣs 1)

36 Local Energy Decay with higher order derivatives Interpolation between s = 0 and s = 1. s = 0, φz α u L 2 t,x x 1/2 ɛ x Z α 1 u L 2 t,x ( Z α u 0 Ḣ1 + Z α u 1 L 2) s = 1, α k 1 α k ( Z α u 0 L 2 + Z α ) u 1 Ḣ 1. φz α u L 2 t Ḣ 1 φ x Z α u L 2 t,x + φ Z α u L 2 t,x x 1/2 ɛ x Z α u L 2 t,x + x 3/2 ɛ Z α u L 2 t,x ( Z α u 0 Ḣ1 + Z α ) u 1 L 2. α k

37 Local Energy Decay with higher order derivatives Interpolation between s = 0 and s = 1. s = 0, φz α u L 2 t,x x 1/2 ɛ x Z α 1 u L 2 t,x ( Z α u 0 Ḣ1 + Z α u 1 L 2) s = 1, α k 1 α k ( Z α u 0 L 2 + Z α ) u 1 Ḣ 1. φz α u L 2 t Ḣ 1 φ x Z α u L 2 t,x + φ Z α u L 2 t,x x 1/2 ɛ x Z α u L 2 t,x + x 3/2 ɛ Z α u L 2 t,x ( Z α u 0 Ḣ1 + Z α ) u 1 L 2. α k

38 Our result: Local existence part Theorem Suppose (H1) and (H2) hold with ρ > 2. Also assume 2 u i uf i (u) u p. i=1 If n = 3, 2 p < p c = 1 + 2, then there is an almost global solution (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣ s 1, α 2 with almost sharp life span, T = c δ p(p 1) p 2 +ɛ 2p 1. with small data and s = s d = 1/2 1/p. Idea of Proof. The local result and life span follows if we use the local in time KSS estimates for 0 < µ < 1/2 instead of the KSS estimates for µ > 1/2.

39 Our result: Local existence part Theorem Suppose (H1) and (H2) hold with ρ > 2. Also assume 2 u i uf i (u) u p. i=1 If n = 3, 2 p < p c = 1 + 2, then there is an almost global solution (Z α u(t, ), t Z α u(t, )) Ḣ s Ḣ s 1, α 2 with almost sharp life span, T = c δ p(p 1) p 2 +ɛ 2p 1. with small data and s = s d = 1/2 1/p. Idea of Proof. The local result and life span follows if we use the local in time KSS estimates for 0 < µ < 1/2 instead of the KSS estimates for µ > 1/2.

40 Local in time KSS estimates For 0 < µ < 1/2, x µ Z α 1/2 µ+ɛ u L 2 T L 2 T x ( Z α f L 2 + Z α ) g Ḣ 1. Proof. Away from the origin, use the KSS estimates for small perturbation equations. (1 + T ) 2a x 1/2+a ( u + u / x ) 2 L 2 ([0,T ] R n ) T u (0, ) 2 L + (u + u/ x )( F + ( h + h x )/ u )dxdt 2 x α k 0 Near the origin, use the local energy estimates, φz α u L p ( Z α f t Ḣs Ḣs + Z α ) g Ḣs 1. α k

41 Local in time KSS estimates For 0 < µ < 1/2, x µ Z α 1/2 µ+ɛ u L 2 T L 2 T x ( Z α f L 2 + Z α ) g Ḣ 1. Proof. Away from the origin, use the KSS estimates for small perturbation equations. (1 + T ) 2a x 1/2+a ( u + u / x ) 2 L 2 ([0,T ] R n ) T u (0, ) 2 L + (u + u/ x )( F + ( h + h x )/ u )dxdt 2 x α k 0 Near the origin, use the local energy estimates, φz α u L p ( Z α f t Ḣs Ḣs + Z α ) g Ḣs 1. α k

42 Local in time KSS estimates For 0 < µ < 1/2, x µ Z α 1/2 µ+ɛ u L 2 T L 2 T x ( Z α f L 2 + Z α ) g Ḣ 1. Proof. Away from the origin, use the KSS estimates for small perturbation equations. (1 + T ) 2a x 1/2+a ( u + u / x ) 2 L 2 ([0,T ] R n ) T u (0, ) 2 L + (u + u/ x )( F + ( h + h x )/ u )dxdt 2 x α k 0 Near the origin, use the local energy estimates, φz α u L p ( Z α f t Ḣs Ḣs + Z α ) g Ḣs 1. α k

43 Further Problem Morawetz est: x 1/2 s e itd f L 2 t,x f Ḣs, 0 < s < n 1 2. Existence theorem for quasilinear wave equations on Asymptotically Euclidean manifolds, with null condition assumed. High dimension existence results for semilinear wave equation.

Strauss conjecture for nontrapping obstacles

Strauss conjecture for nontrapping obstacles Chengbo Wang Joint work with: Hart Smith, Christopher Sogge Department of Mathematics Johns Hopkins University Baltimore, Maryland 21218 wangcbo@jhu.edu November 3, 2010 1 Problem and Background Problem