Strichartz estimates for the wave equation in convex domains Oana - - PowerPoint PPT Presentation

Strichartz estimates for the wave equation in convex domains

Oana Ivanovici Gilles Lebeau Fabrice Planchon

Laboratoire Jean-Alexandre Dieudonn´ e Universit´ e de Nice Sophia-Antipolis

Monastir June 2013

The wave equation

(Ω, g) = Riemannian manifold of dimension d ≥ 2, ∆g = Laplace-Beltrami operator on (Ω, g). (W) ∂2

t u − ∆gu = 0,

(−T, T) × Ω, u|t=0 = u0, ∂tu|t=0 = u1. If ∂Ω = ∅, Dirichlet boundary condition: (D) u|(−T,T)×∂Ω = 0. (in fact Neumann works as well..)

◮ Ω = Rd: Strichartz estimates for (W):

(S) uLq(R,Lr(Rd)) ≤ C

• u0Hγ(Rd) + u1Hγ−1(Rd)
• ,

(q, r) = d - admissible, i.e. q, r ≥ 2, (q, r, d) = (2, ∞, 3),

d=3 1/r 1/q 1/r 1/q 1/2 1/4 1/2 1/2 d=2

Here γ = γ(d, q, r) := d( 1

2 − 1 r ) − 1 q (scaling condition)

d=3 1/q 1/r 1/2 1/2

d=3 Energy estimate: ∂t,xuL∞

t L2 x ∇xu0L2 x + u1L2 x

Strichartz endpoint: uL

2+ t

L

∞− x

u0 ˙

H1

x + u1L2 x

◮ On average (in time !), a solution to the wave

equation has better integrability (almost L∞ rather than L6) than a given H1 function

◮ Or, to have u ∈ L∞− (almost everywhere in time), we

just need u ∈ H1 rather than u ∈ H3/2

◮ Outside the admissibility set: Knapp counterexample

(a correctly sized wave packet holds its shape on the correct time scale)

Dispersive estimates in Rd

◮ Strichartz follows from energy conservation,

interpolation and dispersion:

◮ A fundamental property of the half-wave operator

χ(−λ−2∆Rd)eit√

−∆Rd L1(Rd)→L∞(Rd) λ

d+1 2 |t|− d−1 2 .

?? non flat background and/or boundaries ??

Boundary points: different scenarios

transversal rays diffractive ray gliding ray highly!multiply reflected rays

Dispersion in convex domains

(Ω, g) strictly convex, a ∈ Ω, δa = the Dirac function: χ(λ−1Dt)eit√

−∆g(δa)L∞(Ω) λ

(d+1) 2 |t|− (d−1) 2 (λ|t|) 1 4

◮ Loss of 1 4 power of λt compared to the Euclidian case ◮ The result is optimal because of the presence of

swallowtail singularities in the wave front set.

◮ Currently: the Friedlander model case in full details (ILP)

Theorem The above dispersive estimate holds in the Friedlander model case (actually, better bounds on the Green function that we shall use for Strichartz).

Strichartz (derived from previous dispersion) compared to preceding results: ILP to the left, Blair-Smith-Sogge to the right:

◮ d = 3

d=3 1/r 1/2 1/2 1/3 3/8 1/r 1/2 1/2 3/8 1/3 1/q 1/q

◮ d = 4

1/6 1/r 1/2 1/2 1/r 1/2 1/q 1/q d=4 1/2 1/10 1/10 1/6

We do better than that: optimal loss for Strichartz

= loss of 1/6(1/4−1/r) derivatives = loss is unavoidble (O.I. ’08,’09) = sharp Stricharz ( I−L−P ’12) = GAP to be filled 1/q

d=2

1/6 1/8 1/4 1/2 5/24 1/4 1/r

d=3

?

1/4 1/4 1/3 5/12 1/2 1/2

?

1/q

◮ open: (partially) fill the GAP (conjecture: it should be

filled entirely !)

Theorem The above Strichartz estimates hold in the Friedlander model case. There is no loss (compared to the flat case) for (q, r, d) = (24/5, ∞, 2), (q, r, d) = (12/5, ∞, 3). In fact, we have optimal results for any d ≥ 2. Counterexamples are optimal at these endpoints.

◮ Requires a finer analysis to average out singularities ◮ d = 2 is already known to hold (for any smooth

boundary) by previous work of Blair-Smith-Sogge

Model for convex boundaries

Disk: r ≤ 1 ∆disk = ∂2

r + 1 r 2∂2 θ

Model domain: x ≥ 0, y ∈ R ∆g = ∂2

x + (1 + x)∂2 y Disk Model domain

Same to first order under x = 1 − r, y = θ.

Parametrix for waves starting at t = 0 from δ(x=a,y=0), a ∈ Ω small A detailed description of the ”sphere” of center a and radius t, i.e. the set of points that can be reached following all the optical rays starting from a of length t.

two main cases

◮ a λ−4/7 : use gallery modes (e.g. spectral decomposition); ◮ a > λ−4/7 : write the flow as a superposition of waves

essentially supported between two consecutive reflections; the only case which involves swallowtails

In the last case, the parametrix is a sum of degenerate

• scillatory integrals

◮ Non-tangential directions |ξ| ≫ λ−1/4: at worst, cusp-like

singularities

◮ Main contribution comes from |ξ| λ−1/4, where the swallowtail

appears between every two consecutive reflexions

The Green function

Explicit representation for the half-wave initial value problem with a Dirac at (a, b) as initial condition at time s: G((x, y, t), (a, b, s)) =

• k≥1
• R

e±i(t−s)√

λk(η)ei(y−b)ηek(x, η)ek(a, η) dη

where λk(η) = η2 + η4/3ωk, with ωk ≃ ( 3kπ

2 )2/3(1 + O( 1 k )) a zero of

the Airy function and ek(x, η) = fk η1/3 k1/6 Ai(η

2 3 x − ωk),

(fk normalization constants) (ek)k≥1 is an L2 orthonormal basis (gallery modes) which provides a spectral decomposition of our Laplacian −∂2

x + (1 + x)η2

Useful reductions

◮ reduce to d = 2 by rotational symmetry ◮ spectral localization at

• λk(η) ∼ λ:

insert ψ(

• λk(η)/λ) = ψ(Dt/λ)

◮ localization to tangent directions η ∼ λ:

insert ψ(η/λ) this induces that k ≤ λ/100, as for the gallery modes (microlocally) (η/λ)4/3λ−2/3ωk ∼ (ξ/λ)2 + x(η/λ)2 and therefore k small compared to λ is equivalent to ξ small compared to η (at t = 0, x = a)

◮ G is symmetric in x and a:

we may restrict the computation of any L∞ norm over (x, a) to the set {(0 ≤)x ≤ a}

Initial data very close to ∂Ω (the case a small compared to λ−4/7)

ua,λ(t, x, y) = e−it√

−∆gψ2(

• −∆g/λ)ψ1(Dy/λ)δx=a,y=0

=

• 1≤k≤λ/100

λ 2π

• eiλ(yη−tη√

1+ωk(ηλ)−2/3)ψ(η)ek(a, ηλ)ek(x, ηλ)dη .

Proposition ILP 2013, a λ−4/7 1x≤aua,λ(t, x, y)L∞

x,y ≤ Cλ2 min(1,

1 (λ|t|)1/3 ) .

◮ the geometry is irrelevant when a is very small (too many

singularities in the WF).

◮ Finer analysis on the sum of gallery modes (inspired by

exponential sum method).

◮ Why 4/7? for technical reasons not so clear yet..

Small time estimates

ua,λ(t) = λ2λ−1/3

• k≃a3/2λ
• eiλyη ˜

ψ(η)f(k, t, x, a, ηλ)dη+O(λ−∞) , f(k, .) = 1 k1/3χ( ω3/2

k

ηλa3/2)Ai((ηλ)2/3x − ωk)Ai((ηλ)2/3a − ωk), where χ ∈ C∞ is supported near 1. Lemma |

• k≃L

f(k)| L1/3.

◮ For values L ≤ 1 t (where L = a3/2λ) ⇒ OK ◮ It remains to study t > 1 a3/2λ.

Poisson formula

k f(k) = N ˆ

f(−2πN), ζ =

ω3/2

k

ηλa3/2

ˆ f(−2πN) = aλ2/3

• eiλη( 4

3a3/2Nζ−t√

1+aζ2/3)χ(ζ)

× Ai(a(ηλ)2/3(1 − ζ2/3))Ai(a(ηλ)2/3(x a − ζ2/3))dζ.

◮ Stationary phase with ζ1/3 c

≃

t 4N√a ≃ 1 yields:

ua,λ(t) = λ2λ−1/6

• a

t

• |4N√a−t|≤ct
• eiλη(y−t− 2

3 t3 64N2 )ψ(η)

× Ai(a(ηλ)2/3(1 − ζ2/3

c

))Ai(a(ηλ)2/3(x a − ζ2/3

c

))dη

◮ For a2λ < t we bound each Airy factor by a constant ◮ For t > 1 a3/2λ the supports in y are disjoint! ◮ The condition t > 1 a3/2λ ≥ a2λ gives a λ−4/7.

Initial data localized at distance a > λ−4/7

vN(t, x, y, λ) = λ2 (2π)2

• eiληyuN(t, x, λη)χ0(η) dη ,

uN(t, x, ηλ) =(−i)Nλη 2π

• eiλη((t−t′)ζ+ψa(t′)+s(x+1−ζ2)+s3/3−4N/3(ζ2−1)

3 2 )

× χ(ζ)σN(t′, λη) dt′dsdζ , v(t, x, y, λ) =

• 0≤N≤C0/√a

vN(t, x, y, λ). Proposition There exists C0, σ0 such that the following holds true:

• 1. v is a solution to ✷v = 0 for x > −1;
• 2. its trace on the boundary, v(t ∈ [0, 1], x = 0) is O(λ−∞);
• 3. at time t = 0, we have

v(0, x, y, λ)−( λ 2π )2

• eiλ(ηy+(x−a)ξ)χ0(η)χ1(ξ/η) dηdξ = O(λ−∞).

Key points on the parametrix

◮ y (or η) is irrelevant. One solves the boundary value problem in

(x, t), as x = 0 is non characteristic. σ0 is chosen as to get the Dirac in x at t = 0

◮ By construction, the trace at x = 0 is the sum of the N = 0,

N = Nmax terms (and then stationary phase to get λ−∞)

◮ the vN do not overlap too much: the supremum of the sum will

be the supremum over N

◮ each vN is analyzed by degenerate stationary phase: let

φa,N = y+(t−t′)ζ+s(x+1−ζ2)+s3/3−4N/3(ζ2−1)

3 2 +ψa(t′)

then vN =

λ2 (2π)2

• eiληφa,Nχ0(η)χ(ζ)σN(t′, λη)dt′dsdζdη

◮ if N < min(a−1/2, a1/2λ1/3), one critical point in t′ of order

exactly 3 (hence order of caustic = 1

4!) while integration in η

yields a swallowtail type picture; we know its exact location in space-time (optimality!). Otherwise, at most cusps (of order 1

6)...

Sharp estimates near the swallowtail points t ≃ 4N√a, x ≃ a

Proposition (a > λ−4/7) 1x≤avN(t, x, y)L∞

x,y ≤ Cλ2

a1/8λ−1/4 N1/4 + (√aλ)−1/12(t2

N − t2)1/6,

for t ∈ (tN −

√a N , tN + √a N ) and where tN = 4N√a. ◮ Follows directly from the stationnary phase argument

which yields dispersion

◮ allows to track precisely where the usual TT ⋆

argument will fail

Main result: Strichartz for 2D waves

Let f(a, b, s) be spectrally localised at frequency λ Theorem Inhomogeneous Strichartz for 2D waves If G(f)(x, y, t) =

• s<t
• a,b G(x, y, t, a, b, s)f(a, b, s)dadbds

then G(f)L6

t L∞ x,y C(λ)fL6/5 s

L1

a,b.

We summarize:

◮ the swallowtail occurs only at tN = 4N√a, x = a,

yN := tN + O(a3/2N);

◮ they have an effect on IN := (tN − √a N , tN + √a N ). ◮ outside IN only cusps with (λt)1/6 loss.