Strauss conjecture on asymptotically Euclidean …xyu/research/11_GSU_StraussConjectureO...Strauss conjecture on asymptotically Euclidean manifolds Xin Yu (Joint with Chengbo Wang)

Strauss conjecture on asymptotically Euclideanmanifolds

Xin Yu (Joint with Chengbo Wang)

Department of Mathematics, Johns Hopkins UniversityBaltimore, Maryland 21218

[email protected]

Mar 12-Mar 13, 2010

Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications

The Problem

We consider the wave equations on asymptocially Euclideanmanifolds (M, g)

(∗)

gu = (∂2t −∆g )u = F (u) on R+ ×M

u(0, ·) = f , ∂tu(0, ·) = g

F (u) ∼ |u|p when u is small.

∆g =∑

ij1√det g

∂i√

det gg ij∂j is the Laplace-Beltramioperator.Assumptions on the metric g

1

∀α ∈ Nn ∂αx (gij − δij) = O(〈x〉−|α|−ρ), (H1)

with δij = δij being the Kronecker delta function.2

g is non-trapping. (H2)


The Problem


(∗)

gu = (∂2t −∆g )u = F (u) on R+ ×M

u(0, ·) = f , ∂tu(0, ·) = g


∆g =∑

ij1√det g

∂i√


1





The Problem


(∗)

gu = (∂2t −∆g )u = F (u) on R+ ×M

u(0, ·) = f , ∂tu(0, ·) = g


∆g =∑

ij1√det g

∂i√


1





The Problem


(∗)

gu = (∂2t −∆g )u = F (u) on R+ ×M

u(0, ·) = f , ∂tu(0, ·) = g


∆g =∑

ij1√det g

∂i√


1





Goals

For small data, we want to set up:

Global existence result (Strauss Conjecture) for n = 3, 4 andp > pc . where pc is the larger root of the equation

(n − 1)p2 − (n + 1)p − 2 = 0.

Local existence result for n = 3 and p < pc with almost sharplife span

Tε = Cεp(p−1)

p2−2p−1+ε′.

Note

pc = 1 +√

2 for n = 3,

pc = 2 for n = 4.


Goals

For small data, we want to set up:

Global existence result (Strauss Conjecture) for n = 3, 4 andp > pc . where pc is the larger root of the equation

(n − 1)p2 − (n + 1)p − 2 = 0.

Local existence result for n = 3 and p < pc with almost sharplife span

Tε = Cεp(p−1)

p2−2p−1+ε′.

Note

pc = 1 +√

2 for n = 3,

pc = 2 for n = 4.


Earlier Work in Minkowski space R+ × Rn

79’ John: n=3, global sol’n for p > 1 +√

2, almost globalsol’n for p < 1 +

√2;

81’ Struss Conjecture: n ≥ 2, global sol’n iff p > pc , where pcis the larger root of

(n − 1)pc − (n + 1)pc − 2 = 0.

81’ Glassey: Verify for n = 2;

87’ Sideris: Blow up for p < pc ;

95’ Zhou: Verify for n = 4;

99’ Georgiev, Lindblad, Sogge and 01’ Tataru: n ≥ 3 andp > pc .





√2;


(n − 1)pc − (n + 1)pc − 2 = 0.









√2;


(n − 1)pc − (n + 1)pc − 2 = 0.






Earlier Work (continued)

On more general domains.

Perturbed by obtacles1 08’ D.M.S.Z: Nontrapping, ∆g = ∆, n = 4, p > pc ;2 08’ H.M.S.S.Z: Nontrapping, n = 3, 4, p > pc ;3 09’ Yu: Trapping (Limited), n = 3, 4, p > pc ; n = 3, p < pc .

10’ Han and Zhou: Star-shaped obstacle and n ≥ 3: Blow upwhen p < pc with an upper bound of life span.

Asymptotically Euclidean metric09’ Sogge and Wang: n = 3, p > pc under symmetric metric.


Earlier Work (continued)

On more general domains.

Perturbed by obtacles1 08’ D.M.S.Z: Nontrapping, ∆g = ∆, n = 4, p > pc ;2 08’ H.M.S.S.Z: Nontrapping, n = 3, 4, p > pc ;3 09’ Yu: Trapping (Limited), n = 3, 4, p > pc ; n = 3, p < pc .

10’ Han and Zhou: Star-shaped obstacle and n ≥ 3: Blow upwhen p < pc with an upper bound of life span.

Asymptotically Euclidean metric09’ Sogge and Wang: n = 3, p > pc under symmetric metric.


Our Result (Global existence part)

Theorem

Suppose (H1) and (H2) hold with ρ > 2. Also assume

2∑i=1

|u|i |∂ iuF (u)|.|u|p.

If n = 3, 4, pc < p < 1 + 4/(n − 1), then there is a global solution(Zαu(t, ·), ∂tZαu(t, ·)) ∈ Hs × Hs−1, |α| ≤ 2, with small data ands = sc − ε.


Sample proof in Minkowski space

Iteration method Let u−1 ≡ 0, uk solves(∂2t −∆g)uk(t, x) = Fp(uk−1(t, x)) , (t, x) ∈ R+ × Ω

uk(0, ·) = f , ∂tuk(0, ·) = g .

Continuity argument. Guaranteed by the Strichartz estimates,

‖|x |(−n2+1−γ)/pu‖Lpt Lpr L2ω.‖(f , g)‖(Hγ ,Hγ−1)+‖|x |

− n2+1−γF‖L1tL1r L2ω

for 1/2− 1/p < γ < n/2− 1/p, and energy estimates ,

‖u‖L∞t Hγx.‖f ‖Hγ + ‖g‖Hγ−1 .


Sample proof in Minkowski space

Iteration method Let u−1 ≡ 0, uk solves(∂2t −∆g)uk(t, x) = Fp(uk−1(t, x)) , (t, x) ∈ R+ × Ω

uk(0, ·) = f , ∂tuk(0, ·) = g .

Continuity argument. Guaranteed by the Strichartz estimates,

‖|x |(−n2+1−γ)/pu‖Lpt Lpr L2ω.‖(f , g)‖(Hγ ,Hγ−1)+‖|x |

− n2+1−γF‖L1tL1r L2ω

for 1/2− 1/p < γ < n/2− 1/p, and energy estimates ,

‖u‖L∞t Hγx.‖f ‖Hγ + ‖g‖Hγ−1 .


Our proof for the case p > pc

Set up the argument.Define the norm X :

‖u(t, ·)‖X = ‖u‖Lsγ (|x |<R) + ‖|x |(−n2+1−γ)/pu‖Lpr L2ω(|x |>R)

Set

Mk =∑|α|≤2

(∥∥Zαuk∥∥L∞t Hγ(R+×Rn)+∥∥∂tZαuk∥∥L∞t Hγ−1(R+×Rn)

+ ‖Zαu‖Lpt X).

GOAL: Show Mk < Cε if∑|α|≤2 ‖Zα(f , g)‖(Hγ ,Hγ−1) < ε.





Set

Mk =∑|α|≤2


+ ‖Zαu‖Lpt X).






Set

Mk =∑|α|≤2


+ ‖Zαu‖Lpt X).



Proof for p > pc , continued

Key Ingredients.

KSS and Strichartz Estimates∑|α|≤2

‖〈x〉−12−s−εZαu‖L2tL2x +‖|x |

n2− n+1

p−s−εZαu‖

Lpt Lp|x|L

2+ηω (|x |>1)

.∑|α|≤2

(‖Zαf ‖Hs + ‖Zαg‖Hs−1

),

Energy Estimates∑|α|≤2

(‖Zαu‖L∞t Hs + ‖∂Zαu‖L∞t Hs−1 + ‖Zαu‖Lpt Lqsx (|x |≤1)

).∑|α|≤2


),

where qs = 2n/(n − 2s).


Transformation on the Equation

Set P = −g∆gg−1. We will prove the estimates if u is the

solution of (∂2 + P)u = F , so that

u(t) = cos(tP12 )f +P−

12 sin(tP

12 )g+

∫ t

0P−

12 sin((t−s)P

12 )F (s)ds .

Equivalence: if v solves (∂2t −∆g )v(t, x) = G (t, x), we haverelation

u = gv , F = gG .


Transformation on the Equation

Set P = −g∆gg−1. We will prove the estimates if u is the

solution of (∂2 + P)u = F , so that

u(t) = cos(tP12 )f +P−

12 sin(tP

12 )g+

∫ t

0P−

12 sin((t−s)P

12 )F (s)ds .

Equivalence: if v solves (∂2t −∆g )v(t, x) = G (t, x), we haverelation

u = gv , F = gG .


Proof of the estimates with order 0

KSS estimates: 08’ Bony, Hafner.

Strichartz estimates: Interpolation between KSS estimatesand angular Sobolev inequality,

‖|x |n2−αe itP

1/2f (x)‖

L∞t,|x|L

2+ηω.‖e itP1/2

f (x)‖L∞t Hαx.‖f ‖Hαx ; (1)

Energy estimates: Equivalence of Ps/2 and ∂s with s ∈ [0, 1];

Local Energy decay (By interpolation between KSS estimates),∥∥βu∥∥L2tH

s . ‖f ‖Hs + ‖g‖Hs−1 .





‖|x |n2−αe itP

1/2f (x)‖

L∞t,|x|L

2+ηω.‖e itP1/2




s . ‖f ‖Hs + ‖g‖Hs−1 .





‖|x |n2−αe itP

1/2f (x)‖

L∞t,|x|L

2+ηω.‖e itP1/2




s . ‖f ‖Hs + ‖g‖Hs−1 .


KSS and Energy estimates with higher order derivatives

Zα = ∂, use relation between ∂ and P1/2.1 ‖u‖Hs ' ‖Ps/2u‖L2

x, for s ∈ [−1, 1];

2 −3/2 ≤ µ1 < µ2 ≤ µ3 ≤ 3/2, then

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.∥∥〈x〉−µ2P1/2u

∥∥L2(Rd )

.n∑`=1

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.

Zα = ∂2, use relation between ∂2 and P.1 For s ∈ [0, 1], we have

‖∂2x f ‖Hs.‖Pf ‖Hs + ‖f ‖Hs .

‖Pf ‖Hs.∑|α|≤2

‖∂αx f ‖Hs .

2 For 0 < µ ≤ 3/2 and k ≥ 2, we have∥∥〈x〉−µ∂2xu∥∥L2x.∥∥〈x〉−µ∂u∥∥

L2x

+∥∥〈x〉−µPu∥∥

L2x.




x, for s ∈ [−1, 1];

2 −3/2 ≤ µ1 < µ2 ≤ µ3 ≤ 3/2, then

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.∥∥〈x〉−µ2P1/2u

∥∥L2(Rd )

.n∑`=1

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.


‖∂2x f ‖Hs.‖Pf ‖Hs + ‖f ‖Hs .

‖Pf ‖Hs.∑|α|≤2

‖∂αx f ‖Hs .


L2x

+∥∥〈x〉−µPu∥∥

L2x.




x, for s ∈ [−1, 1];

2 −3/2 ≤ µ1 < µ2 ≤ µ3 ≤ 3/2, then

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.∥∥〈x〉−µ2P1/2u

∥∥L2(Rd )

.n∑`=1

∥∥〈x〉−µ3 ∂ù∥∥L2(Rd )

.


‖∂2x f ‖Hs.‖Pf ‖Hs + ‖f ‖Hs .

‖Pf ‖Hs.∑|α|≤2

‖∂αx f ‖Hs .


L2x

+∥∥〈x〉−µPu∥∥

L2x.


KSS and Energy estimates with higher order derivatives(continued)

When Zα = Ω or Zα = Ω2, then Zαu solves

(∂2t + P)Zαu = [P,Zα]u,

with initial data (Zαf ,Zαg).

Commutator terms

[P,Ω]u =∑|α|≤2

r2−|α|∂αu.

[P,Ω2]u =∑|α|≤3

r2−|α|∂αu.

where ri ∈ C∞ is such that

∂αx rj(x) = O(〈x〉−ρ−j−|α|

), ∀α ,



When Zα = Ω or Zα = Ω2, then Zαu solves

(∂2t + P)Zαu = [P,Zα]u,

with initial data (Zαf ,Zαg).

Commutator terms

[P,Ω]u =∑|α|≤2

r2−|α|∂αu.

[P,Ω2]u =∑|α|≤3

r2−|α|∂αu.

where ri ∈ C∞ is such that

∂αx rj(x) = O(〈x〉−ρ−j−|α|

), ∀α ,



Techniques to handle commutator terms

Let w solve the wave equation with f = g = 0,

‖〈x〉−1/2−s−εw‖L2tL2x . ‖〈x〉(1/2)+εF‖L2t Hs−1 ;

‖w‖L∞t Hsx.‖〈x〉1/2+εF‖L2t Hs−1

x.

Fractional Lebniz rule. For any s ∈ (−n/2, 0) ∪ (0, n/2),

‖fg‖Hs.‖f ‖L∞∩H|s|,n/|s|‖g‖Hs .

For any s ∈ [0, 1], ε > 0 and |α| = N, we have∑|α|=N

‖〈x〉−(1/2)−ε∂αx u‖L2t Hs−1.‖f ‖HN+s−1∩Hs +‖g‖HN+s−2∩Hs−1 .







x.











x.






Weighted Strichartz estimates with higher order derivatives

∑|α|≤2

‖|x |n2− n+1

p−s−εZαu‖

Lpt Lp|x|L

2+ηω (|x |>1)

.∑|α|≤2


)Interpolation between p = 2 and p =∞

p = 2: KSS estimates;

p =∞:∑|α|≤2

‖|x |n2−sZαu‖

L∞t,|x|L

2+ηω

.∑|α|≤2

‖Zαu‖L∞t Hsx

.∑|α|≤2


)Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications


∑|α|≤2

‖|x |n2− n+1

p−s−εZαu‖

Lpt Lp|x|L

2+ηω (|x |>1)

.∑|α|≤2




p =∞:∑|α|≤2


L∞t,|x|L

2+ηω

.∑|α|≤2

‖Zαu‖L∞t Hsx

.∑|α|≤2




∑|α|≤2

‖|x |n2− n+1

p−s−εZαu‖

Lpt Lp|x|L

2+ηω (|x |>1)

.∑|α|≤2




p =∞:∑|α|≤2


L∞t,|x|L

2+ηω

.∑|α|≤2

‖Zαu‖L∞t Hsx

.∑|α|≤2



Local Energy Decay with higher order derivatives

Interpolation between s = 0 and s = 1.

s = 0,

‖φZαu‖L2t,x . ‖〈x〉−1/2−ε∂xZα−1u‖L2t,x.

∑|α|≤k−1

(‖Zαu0‖H1 + ‖Zαu1‖L2

).

∑|α|≤k

(‖Zαu0‖L2 + ‖Zαu1‖H−1

).

s = 1,

‖φZαu‖L2t H1 . ‖φ ∂xZαu‖L2t,x + ‖φ′ Zαu‖L2t,x. ‖〈x〉−1/2−ε∂xZαu‖L2t,x + ‖〈x〉−3/2−εZαu‖L2t,x.

∑|α|≤k


).


Local Energy Decay with higher order derivatives

Interpolation between s = 0 and s = 1.

s = 0,

‖φZαu‖L2t,x . ‖〈x〉−1/2−ε∂xZα−1u‖L2t,x.

∑|α|≤k−1


).

∑|α|≤k

(‖Zαu0‖L2 + ‖Zαu1‖H−1

).

s = 1,

‖φZαu‖L2t H1 . ‖φ ∂xZαu‖L2t,x + ‖φ′ Zαu‖L2t,x. ‖〈x〉−1/2−ε∂xZαu‖L2t,x + ‖〈x〉−3/2−εZαu‖L2t,x.

∑|α|≤k


).


Our result: Local existence part

Theorem


2∑i=1

|u|i |∂ iuF (u)|.|u|p.

If n = 3, 2 ≤ p < pc = 1 +√

2, then there is an almost globalsolution (Zαu(t, ·), ∂tZαu(t, ·)) ∈ Hs × Hs−1, |α| ≤ 2 with almostsharp life span,

T = c δp(p−1)

p2−2p−1+ε.

with small data and s = sd = 1/2− 1/p.

Idea of Proof. The local result and life span follows if we use thelocal in time KSS estimates for 0 < µ < 1/2 instead of the KSSestimates for µ > 1/2.


Our result: Local existence part

Theorem


2∑i=1

|u|i |∂ iuF (u)|.|u|p.

If n = 3, 2 ≤ p < pc = 1 +√

2, then there is an almost globalsolution (Zαu(t, ·), ∂tZαu(t, ·)) ∈ Hs × Hs−1, |α| ≤ 2 with almostsharp life span,

T = c δp(p−1)

p2−2p−1+ε.

with small data and s = sd = 1/2− 1/p.

Idea of Proof. The local result and life span follows if we use thelocal in time KSS estimates for 0 < µ < 1/2 instead of the KSSestimates for µ > 1/2.


Local in time KSS estimates

For 0 < µ < 1/2,∑|α|≤2

‖〈x〉−µZαu‖L2TL2x.T1/2−µ+ε

∑|α|≤2

(‖Zαf ‖L2 + ‖Zαg‖H−1

).

Proof.

Away from the origin, use the KSS estimates for small perturbationequations.

(1 + T )−2a∥∥|x |−1/2+a(|u′|+ |u|/|x |)

∥∥2L2([0,T ]×Rn)

. ‖u′(0, ·)‖2L2x +

∫ T

0

∫(u′ + u/|x |)(|F |+ (|h′|+ h|x |)/|u′|)dxdt

Near the origin, use the local energy estimates,∑|α|≤k

‖φZαu‖Lpt Hs.∑|α|≤k


).



For 0 < µ < 1/2,∑|α|≤2


∑|α|≤2


).

Proof.


(1 + T )−2a∥∥|x |−1/2+a(|u′|+ |u|/|x |)

∥∥2L2([0,T ]×Rn)

. ‖u′(0, ·)‖2L2x +

∫ T

0

∫(u′ + u/|x |)(|F |+ (|h′|+ h|x |)/|u′|)dxdt




).



For 0 < µ < 1/2,∑|α|≤2


∑|α|≤2


).

Proof.


(1 + T )−2a∥∥|x |−1/2+a(|u′|+ |u|/|x |)

∥∥2L2([0,T ]×Rn)

. ‖u′(0, ·)‖2L2x +

∫ T

0

∫(u′ + u/|x |)(|F |+ (|h′|+ h|x |)/|u′|)dxdt




).


Further Problem

Morawetz est: ‖|x |−1/2−se itD f ‖L2t,x.‖f ‖Hs , 0 < s < n−12 .

Existence theorem for quasilinear wave equations onAsymptotically Euclidean manifolds, with null conditionassumed.

High dimension existence results for semilinear wave equation.


Strauss conjecture on asymptotically Euclidean …xyu/research/11_GSU_StraussConjectureO...Strauss conjecture on asymptotically Euclidean manifolds Xin Yu (Joint with Chengbo Wang)

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