Strauss conjecture on asymptotically Euclidean …xyu/research/11_GSU_StraussConjectureO...Strauss...

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.


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