Evolution equations on symmetric spaces...The Legacy of Joseph Fourier after 250 years Conference TSIMF, Sanya, China 17{21 December 2018 Evolution equations on symmetric spaces Jean{Philippe

The Legacy of Joseph Fourier after 250 years

ConferenceTSIMF, Sanya, China17–21 December 2018

Evolution equations on symmetric spaces

Jean–Philippe Anker (Universite d’Orleans)

Survey of collaborations with

Lizhen Ji (Ann Arbor) Stefano Meda (Milano)Patrick Ostellari (Nancy) Alaa Jamal Eddine (Orleans)Vittoria Pierfelice (Orleans) Yannick Sire (Baltimore)Maria Vallarino (Torino) Hongwei Zhang (Orleans)

Jean–Philippe Anker (Universite d’Orleans) Sanya, 19 December 2018

Joseph Fourier (1768 –1830)


Equations

Heat {∂tu(t, x)−∆xu(t, x) = F (t, x)

u(0, x)=f(x)(1)

Wave {∂ 2t u(t, x)−∆xu(t, x) = F (t, x)

u(0, x)=f0(x), ∂t|t=0u(t, x)=f1(x)(2)

Schrodinger {i ∂tu(t, x) + ∆xu(t, x) = F (t, x)

u(0, x)=f(x)(3)


Equations (continued)

Heat {∂tu(t, x)−∆xu(t, x) = F (t, x)

u(0, x)=f(x)(1)

Half–wave {i ∂tu(t, x) + (−∆x)1/2 u(t, x) = F (t, x)

u(0, x)=f(x)(4)

Schrodinger for the fractional Laplacian (0<α≤2){i ∂tu(t, x) + (−∆x)α/2 u(t, x) = F (t, x)

u(0, x)=f(x)(5)


Spaces

X=G/K Riemannian symmetric space of noncompact type

G noncompact semisimple Lie group (connected, finite center)K maximal compact subgroup

Cartan decomposition:

G =K(exp a+)K X=K(exp a+)��K/��K

The rank of G/K is the dimension ` of a

Classification in rank `=1

X Hn(R) Hn(C) Hn(H) H2(O)

G SO0(n,1) SU(n,1) Sp(n,1) F4(−20)

K SO(n) S[U(n)×U(1)] Sp(n) SO(9)

d n 2n 4n 16

ρ n−12 n 2n+1 11


Plan

1 Heat kernel

rank 1higher rank

2 Schrodinger equation

rank 1(higher rank)

3 Schrodinger equation for the fractional Laplacian in rank 1

4 Wave equations in rank 1

5 Homogeneous trees


Heat kernel on G/K

Heat semigroup:e t∆xKf(xK) = f ∗

Ght(x)

Inverse spherical Fourier transform:

ht(KxK) = const.

∫ae−t(|λ|

2+ |ρ|2)ϕλ(x) |c(λ)|−2 dλ

Estimate in rank 1 [Davies–Mandouvalos]

ht(r) � t−d2 (1+ t+ r)

d−32 e−ρ

2 t−ρ r− r24t

for every t>0 and r≥0.

Estimate in higher rank [A–Ji, A–Ostellari]

ht(expx) � t−d2

{∏α∈R+

red

(1+ t+ |〈α,x〉|)mα+m2α

2−1}×

× e−|ρ|2 t−〈ρ,x〉− |x|

2

4t

for every t>0 and x∈a+.


Applications

Martin compactification of symmetric spaces X=G/K

Spectral gap λ0 of locally symmetric spaces Y = Γ\G/KRank 1 [Elstrodt, Patterson, Sullivan, Corlette]

λ0 =

{ρ2 if 0≤ δ≤ρρ(2ρ− δ ) if ρ≤ δ≤2ρ

Higher rank [Leuzinger, Weber]λ0 = |ρ|2 if 0≤ δ≤ ρ|ρ|2−(δ−ρ)2≤λ0≤|ρ|2 if ρ≤ δ≤ρ|ρ|2−(δ−ρ)2≤λ0≤|ρ|2−(δ−|ρ|)2 if ρ≤ δ≤2ρ

Here δ denotes the critical exponent of convergence for thePoincare series

Pz(xK, yK) =∑

γ∈Γe−zd(xK,γ yK)

and ρ = minx∈a+, |x|=1

〈ρ,x〉 ≤ |ρ|


Schrodinger equation on X=G/K

Schrodinger equation:{i ∂tu(t, x) + ∆xu(t, x) = F (t, x)

u(0, x)=f(x)(3)

Formal solution (Duhamel’s formula):

u(t, x) = eit∆xf(x)− i∫ t

0ei(t−s)∆xF (s, x)ds

Schrodinger propagator:

eit∆xKf(xK) = f ∗Gst(x)


st(KxK) = const.

∫ae−it(|λ|

2+ |ρ|2)ϕλ(x) |c(λ)|−2 dλ


Schrodinger equation in rank 1

Kernel estimate [A–Pierfelice, Ionescu–Staffilani]

|st(r)| .{|t|− d2

� j(r)− 12︷︸︸︷

(1+ r)d−12 e−ρr if |t|≤1+r

|t|− 32 (1+ r)e−ρr︸︷︷︸� ϕ0(r)

if |t|≥1+ r


Schrodinger equation in rank 1 (continued)

Dispersive estimate

For every 2<q, q≤∞ and t∈R∗,

‖e it∆‖Lq′→Lq .{|t|−d max{ 1

2− 1q, 12− 1q} if |t| is small

|t|− 32 if |t| is large

Main tool for |t| large = following version ofthe Kunze–Stein phenomenon

Lemma

Assume that g is a radial (measurable) function on X. Then,for every 2≤ q <∞,

‖f ∗ g‖Lq(X)

≤ ‖f‖Lq′(X)

{∫Xdxϕ0(x)|g(x)|q2

}2q

=⇒ Lq′(G/K)∗Lq(K\G/K)⊂Lq(G/K) ∀ q

2≤ q < q


Schrodinger equation in rank 1 (continued)

Strichartz inequality

‖u(t, x)‖Lpt∈IL

qx

. ‖f‖L2 + ‖F (t, x)‖

Lp′t∈IL

q′x

for solutions u(t, x) of (3) on I×X (I interval 30).Here (1

p ,1q ) and (1

p ,1q ) are any couple in the following triangle

0

1

1

12

12

1p

1q

12 − 1

d

Euclidean line

local

1


Application: nonlinear Schrodinger equation in rank 1

Consider power–like nonlinearities F (t, x) = F (u(t, x))

i.e.

{|F (u)| . |u|γ|F (u)−F (v)| . {|u|γ−1+ |v|γ−1} |u−v |

for some γ >1

Theorem

γ ≤1+ 4d : global well–posedness for small L2 data.

γ <1+ 4d : local well–posedness for arbitrary L2 data.

Moreover, global well–posedness for arbitrary L2 dataif F (u) = cu|u|γ−1 with c∈R.

Remarks :

On Rd, first result holds only in the critical case γ = 1+ 4d

Additional smoothness larger powers γ

L2(X) Hσ(X) Sobolev space1+ 4

d 1+ 4d−2σ

(0<σ< d2)


Schrodinger kernel on general X=G/K

Conjectural kernel estimate

For every t∈R∗ and x∈G,

|st(x)| . |t|− d2 e−〈ρ,x+〉××∏

α∈R+red

(1+〈α, x+〉

)(1+ |t|+〈α, x+〉

)mα+m2α2

−1

where x+∈a+ denotes the radial component of xin the Cartan decomposition G=K(exp a+)K

OK in rank `=1 or if G is complex

Weaker result [A–Meda–Pierfelice–Vallarino]

|st(x)| .{|t|−(d− `

2) (1+ |x+|)N e−〈ρ,x+〉 if |t| is small

|t|−D2 (1+ |x+|)N e−〈ρ,x+〉 if |t| is large

where D= `+ 2|R+red| is the dimension at infinity


Schrodinger equation for the fractional Laplacian on G/K

Assume that 0<α<1 or 1<α<2

Schrodinger equation for the fractional Laplacian:{i ∂tu(t, x) + (−∆x)α/2 u(t, x) = F (t, x)

u(0, x)=f(x)(5)

Duhamel’s formula:

u(t, x) = eit(−∆x)α/2f(x)− i∫ t

0ei(t−s)(−∆x)α/2F (s, x)ds

Need additional smoothness σ≥0

Operator:

(−∆xK)−σ/2 eit(−∆xK)α/2f(xK) = f ∗Gkσt (x)


kσt (KxK) = const.

∫a

(|λ|2+ |ρ|2)−σ/2 e it(|λ|2+ |ρ|2)α/2

×ϕλ(x) dλ|c(λ)|2 dλ


Schrodinger fractional in rank 1

Kernel estimate [A–Sire]

Assume that σ= (2−α) d2 . Then, for every t∈R∗ and r≥0,

|kσt (r)| .{|t|−n2 (1+ r)

d−12 e−ρr if |t|≤1+ r

|t|− 32 (1+ r) e−ρr if |t|≥1+ r

Dispersive estimate [A–Sire]

Let 2<q, q≤∞ and σ= (2−α)dmax{1

2−1q ,

12−

1q

}.

Then, for every t∈R∗,∥∥(−∆)−σ/2 e i t(−∆)α/2∥∥Lq′→Lq

.

{|t|−d max{ 1

2− 1q, 12− 1q} if t is small

|t|− 32 if t is large


Schrodinger fractional in rank 1 (continued)

Strichartz inequality [A–Sire]

‖(−∆)−σ/4x u(t, x)‖

Lpt∈IL

qx

. ‖f‖L2 + ‖(−∆)

σ/4x F (t, x)‖

Lp′t∈IL

q′x

for solutions u(t, x) of (3) on I×X, I being any interval 30.Here (1

p ,1q ), (1

p ,1q ) are couples in the same triangle as before

and σ= (2−α)d(

12− 1

q

), σ= (2−α)d

(12− 1

q

).


Half–wave equation in rank 1

Different behavior in the limit case α=1

Restrict to d≥3

Analytic family of operators

W σt = eσ

2

Γ(n+12−σ)

(I−∆)−σ/2 eit(−∆)α/2

in the strip 0≤Reσ≤ d+12

Kernel estimate [A–Pierfelice]

Assume that Reσ= d+12 . Then, for every t∈R∗ and r≥0,

|wσt (r)| ≤ C

|t|− d−1

2 if 0< |t|≤1 and 0≤ r≤1

r e−ρr if r≥max{1, |t|}|t|− 3

2 (1+ r)2 e−ρr if |t|≥max{1, r}where the constant C>0 doesn’t depend on σ, t, r


Half–wave equation in rank 1 (continued)

Dispersive estimate [A–Pierfelice]

Let 2<q, q<∞ and σ= (d+1) max{1

2−1q ,

12−

1q

}.

Then, for every t∈R∗,∥∥(−∆)−σ/2 e i t(−∆)α/2∥∥Lq′→Lq

.

{|t|−(d−1) max{ 1

2− 1q, 12− 1q} if t is small

|t|− 32 if t is large



Strichartz inequality [A–Pierfelice]

‖(−∆)−σ/4x u(t, x)‖

Lpt∈IL

qx

. ‖f‖L2 + ‖(−∆)

σ/4x F (t, x)‖

Lp′t∈IL

q′x

for solutions u(t, x) of (3) on I×X, I being any interval 30.Here (1

p ,1q ), (1

p ,1q ) are couples in the following triangle

and σ= d+12

(12− 1

q

), σ= d+1

2

(12− 1

q

).

0

1

1

12

12

1p

1q

12 − 1

d−1

1



Similar results for the wave equation[Metcalfe–Taylor in 3D, A–Pierfelice in rank 1]

Local/global wellposedness of the nonlinear wave equation

No blow–up for small powers γcontrarily to the Euclidean setting [John, Strauss, . . . ]


Homogeneous trees

T= TQ homogeneous tree with Q+1≥3 edges

Example : Q= 3

0

1

Discrete rank 1 symmetric space

Combinatorial Laplacian on (the vertices of) T :

∆f(x) = 1Q+1

∑d(y,x)=1

f(y) − f(x)

Heat kernel

Discrete time ∼ simple random walkContinuous time [Setti]


Schrodinger equation for the fractional Laplacian on T

Fractional Laplacian (−∆)α2 with 0<α≤2

Schrodinger equation with continuous time on T{i ∂tu(t, x) + (−∆x)

α2 u(t, x) = F (t, x)

u(0, x)=f(x)(6)

α= 2 : standard Schrodinger equation [Jamal Eddine]

α= 1 : half–wave equation [Medolla–Setti]

Schrodinger propagator :

e it(−∆)α/2f(x) =∑

y∈Tf(y) kαt (d(x, y))︸︷︷︸f ∗ kαt (x)


Schrodinger fractional on T (continued)

Inverse spherical Fourier transform

kαt (r) = const.

∫ πlogQ

0dλ |c(λ)|−2 ϕλ(x) e it [1−γ(λ)]α/2

where γ(λ) = Q iλ+Q−iλ

Q1/2 +Q−1/2

c(λ) = 1

Q1/2 +Q−1/2

Q1/2+ iλ−Q−1/2− iλ

Q iλ−Q−iλ

ϕλ(r) = c(λ)Q(−1/2+iλ)r+ c(−λ)Q(−1/2−iλ)r

Kernel estimate [A–Sire]

|kαt (r)| . Q−r2 ∀ t∈R∗, ∀ r∈N.

Moreover there exists a constant C>0 such that

|kαt (r)| . |t|− 32 (1+ r)Q−

r2

if 1+ r≤C |t|.


Schrodinger fractional on T (continued)

Dispersive estimate

Let 0<α≤2 and 2<q, q≤∞. Then

‖e it(−∆)α/2‖`q′→`q . (1+ |t|)− 32 ∀ t∈R∗

Strichartz inequality

‖u(t, x)‖Lpt `qx

. ‖f‖`2 + ‖F (t, x)‖

Lp′t `q′x

for all admissible pairs (1p ,

1q ) and (1

p ,1q ) in the following square

0

1

112

12

1p

1q

1


THE END

Thank you for your attention !


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Evolution equations on symmetric spaces...The Legacy of Joseph Fourier after 250 years Conference TSIMF, Sanya, China 17{21 December 2018 Evolution equations on symmetric spaces Jean{Philippe