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Strauss conjecture on asymptotically Euclideanmanifolds
Xin Yu (Joint with Chengbo Wang)
Department of Mathematics, Johns Hopkins UniversityBaltimore, Maryland 21218
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)
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)
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)
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)
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
),
where qs = 2n/(n − 2s).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 ∂`u∥∥L2(Rd )
.∥∥〈x〉−µ2P1/2u
∥∥L2(Rd )
.n∑`=1
∥∥〈x〉−µ3 ∂`u∥∥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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 ∂`u∥∥L2(Rd )
.∥∥〈x〉−µ2P1/2u
∥∥L2(Rd )
.n∑`=1
∥∥〈x〉−µ3 ∂`u∥∥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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 ∂`u∥∥L2(Rd )
.∥∥〈x〉−µ2P1/2u
∥∥L2(Rd )
.n∑`=1
∥∥〈x〉−µ3 ∂`u∥∥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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 ,
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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 ,
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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−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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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−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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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−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 .
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
)Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαu0‖H1 + ‖Zαu1‖L2
).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαu0‖H1 + ‖Zαu1‖L2
).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
Our result: Local 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, 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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
Our result: Local 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, 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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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
(‖Zαf ‖Hs + ‖Zαg‖Hs−1
).
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications
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.
Xin Yu (Joint with Chengbo Wang), 2011 AMS Sectional Meeting , Georgia Southern UniversityStrichartz Estimates and Applications