[PPT] - Asymptotics for (nonlinear) wave propagation Fabrice Planchon 1 1 PowerPoint Presentation

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Asymptotics for (nonlinear) wave propagation

Fabrice Planchon1

1Laboratoire Jean Alexandre Dieudonné,

Université de Nice Sophia-Antipolis et Institut universitaire de France

Monastir, June 2013

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The Cauchy problem Equations The data Local well-posedness Decay : a simple example for NLS Time decay Wave : local energy decay Schrödinger : virial and local smoothing Schrödinger : bilinear virial

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Wave equation

gφ = (∂2

t − ∆g)φ + f(φ, ∂φ) = 0

most simple case : linear wave in Rn, g = Id and f = 0. Cauchy problem : given φ|t=0 = φ0, ∂tφ|t=0 = φ1, characterize φ (existence, uniqueness, “good” estimates). More complicated :

◮ f = 0 : f(φ) = ±|φ|p−1φ, f(φ, ∂φ) = (|∂tφ|2 − |∇φ|2)φ ◮ domain Ω = Rn (with boundary conditions) ◮ variable metric g, possibly g(φ)

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Schrödinger equation

(i∂t + ∆g)ψ + f(ψ, ∂ψ) = 0 Same questions. Likely more difficult (“infinite speed of propagation”) Cauchy problem : given ψ|t=0 = φ0, what are the properties of ψ ?

◮ f = 0 : f(ψ) = ±|ψ|p−1φ ◮ domain Ω = Rn (with boundary conditions) ◮ variable metric g

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How to pick the right (φ0, φ1) or ψ0 ?

◮ If linear and explicit representations exist, anything that fits

in...

◮ If nonlinear (or variable metric), conservation laws suggest

candidates Possible conservation laws (with suitable f)

◮ NLS : mass

ψ2 dx, momentum Im

¯ ψ∇ψ dx, Hamiltonian (energy)

|∇ψ|2 + F(ψ) dx

◮ NLW : Hamiltonian (energy)

|∂tφ|2 + |∇φ|2 + F(φ) dx

strongly suggests Sobolev spaces : derivatives in L2

x, H1 is a

favorite because of the Hamiltonian structure (if present)

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Local theory for cubic NLS in R2 : vector fields

Consider i∂tψ + ∆ψ = |ψ|2ψ with ψ0 ∈ H1 and |x|ψ0 ∈ L2

x

key observation : L = x + 2it∇x has good commutation properties with the equation (linear and nonlinear) Lψ2

L2

x + t2

x |u|4 dx = xψ02 L2

x

◮ provides time decay at the expense of space decay at t = 0 ◮ leads to global in time solutions ◮ strong link to the convexity of

x |x|2|u|2 dx (virial identity)

leads to scattering : existence of asymptotic states φ± such that limt→±∞ ψ − e±it∆ψ± = 0

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Invariance and conformal transform

Solutions u(x, t) to the previous NLS are such that v(x, t) = 1

t u( x t , − 1 t )ei |x|2

4t

is also a solution (pseudo-conformal transform) The cubic wave equation in R3 is truely conformally invariant : with (X, T) = (x, t)/(t2 − |x|2) and |x|φ = |X|Φ T,XΦ ± Φ3 = 0 and deriving the energy identity for Φ means multiplying φ by the conformal vector field, e.g. Kφ = ((t2 + |x|2)∂t + 2tx · ∇x + 2)φ

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local energy decay for compactly supported data

Consider the linear wave equation φ = 0, multiplying by Kφ and integrating between times s = 0, t yields control of

x(|x|2 + t2)(|∇u|2 − | x

|x| · ∇u|2) dx,

x (|x| + t)2(| x

|x|2 · ∇(|x|u)|2 + |∂tu|2) dx,

x (|x| − t)2(| x

|x|2 · ∇(|x|u)|2 − |∂tu|2) dx,

(or the sum of the L2 norms of Lu, Ωu, where Ωij = xi∂j − xj∂i, Li = t∂i + xi∂t and L0 = t∂t + x · ∇x) If boundary, Dirichlet condition φ|∂Ω = 0, additional term like +

∂Ω(x ·

n)| n · ∇xφ|2 dSx

n inner normal to the domain (outer normal to the obstacle)

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Morawetz variant

Consider the nonlinear wave equation φ + φp = 0, and the multiplier M = t∂t + x · ∇x + 1. One can rewrite Mφ(φ + φp) = 0 so that 0 = divt,x(tQ + u∂tu, −tP) +

2 p+1up+1

Q = |∂tu|2+|∇xu|2

2

+ |u|p+1

p+1 + ∂tu( x t · ∇x)u,

P = x

t

|∂tu|2−|∇xu|2

2

− |u|p+1

p+1

+ ∇xu
∂tu + ( x

t · ∇x)u + u t

Boundary terms will come from

n(x) · P integrated on ∂Ω × [0, T]

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Morawetz variant, take two

Let N =

t |x|∂t + x |x| · ∇x + 1 |x|, one may compute

Nφ(φ + φp) = 0 and integrate between time slices. Benefit : Nφ in L2

x is controled by the energy. Let

J = −

∂tφ( x

|x| · ∇xφ + φ |x|) dx

then

d dt J =

Hess(|x|)(∇xφ, ∇xφ) + 2πφ2(0, t) + p−1

p+1 φp+1 |x| dx

If obstacle, additional boundary term +

∂Ω( x

|x| ·

n)| n · ∇xφ|2 dSx

n inner normal to the domain (outer normal to the obstacle)

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Boundaries and boundaries

Terms like x · n are positive for star-shaped boundaries. Other interesting cases :

◮ Obstacles which are illumated from the interior ◮ Obstacles which are illuminated from the exterior ◮ “Almost star-shaped” obstacles ◮ Non-trapping obstacles

All but the last can be treated by suitable multipliers, as far as (linear) local energy decay is concerned.

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Morawetz variant, virial like

Let J = −

∂tφ∇ρ(x) · ∇xφ + ∆ρ(x)

2

φ dx then

d dt J =

Hess(ρ)(∇xφ, ∇xφ) − 1

4(∆2ρ)φ + (∆ρ) p−1 p+1φp+1 dx +

∂Ω(∇xρ ·

n)| n · ∇xφ|2 dSx typical ρ(x) : convex function with suitable level sets Variants on truncated forward or backward cones

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NLS virial on a domain

Let i∂tu + ∆u − ǫ|u|p−1u = 0, with Dirichlet boundary condition u|∂Ω = 0, and Ω is the exterior

f a star-shaped body with smooth boundary.

Mρ(t) =

Ω

|u|2(x, t)ρ(x) dx, d dt Mρ(t) = −2Im

ρ∇ · (¯

u∇u) = 2Im

¯

u∇u · ∇ρ,

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d2 dt2 Mρ(t) = 2Im

(∂t ¯

u∇u + ¯ u∇∂tu) · ∇ρ = −2Im

∂tu (2∇¯

u · ∇ρ + ¯ u∆ρ) = −2Re

(∆u − ǫ|u|p−1u) (2∇¯

u · ∇ρ + ¯ u∆ρ) = −4Re

∆u∇¯

u · ∇ρ + 2

|∇u|2∆ρ + 2Re
¯

u∇u∇∆ρ + 2

ǫ|u|p−1∇(|u|2)∇ρ + 2
ǫ|u|p+1∆ρ

= −4Re

∆u∇¯

u · ∇ρ + 2

|∇u|2∆ρ −
|u|2∆2ρ

+

2ǫ(1 −

2 p + 1)|u|p+1∆ρ .

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Integrating by parts again,

∆u∇¯

u · ∇ρ =

∂Ω

∇¯ u · ∇ρ∂nu −

∇(∇¯

u · ∇ρ) · ∇u, and, as u∂Ω = 0 implies ∂τu∂Ω = 0, 2Re

∆u∇¯

u · ∇ρ = 2

∂Ω

(∂nρ)|∂nu|2 −

∇ρ · ∇(|∇u|2)

− 2

Hess(ρ)(∇u, ∇¯

u) =

∂Ω

(∂nρ)|∂nu|2 +

|∇u|2∆ρ

− 2

Hess(ρ)(∇u, ∇¯

u)

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and finally d2 dt2 Mρ(t) = −

|u|2∆2ρ + 2ǫp − 1

p + 1|u|p+1∆ρ + 2

∂(Rn\Ω)

(∂nρ)|∂nu|2 + 4

Hess(ρ)(∇u, ∇¯

u).

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Canonical choice : ρ(x) = |x|2. Good choice : ρ(x) =

1 + |x|2,

∆ρ = n − 1 ρ(x) + 1 ρ(x)3 , −∆2ρ = (n − 1)(n − 3) ρ(x)3 +6(n − 3) ρ(x)5 + 15 ρ(x)7 , and n(x) · x ≥ 0 if star-shaped, as well as |∇u|2 ρ(x)3 Hess(ρ)(∇u, ∇¯ u), as Hess(ρ) = 1 ρId − 1 ρ3 (xixj)ij,

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1D local smoothing

Consider the domain to be the half-line x > 0, weight ρ(x) = x, Dirichlet u(0, t) = 0, d dt 2Im

x>0

¯ u∂xu = 4|∂xu|2(0) Now, v linear solution on R, let u = v(x) − v(−x), T |∂xv(0, t)|2dt v2

˙ H

1 2 (0) + v2

˙ H

1 2 (T)

eventually sup

x

T |∂xv(x, t)|2dt = C

R

|ξ||ˆ v(ξ)|2 dξ

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Generalizations

◮ works in any dimensions (fix a direction, take the trace on

an hyperplan)

◮ works for the wave (provides local energy decay in free

space)

◮ suggests for nonlinear waves (NLW or NLS), the boundary

term can be controled philosophy : local smoothing/local energy decay “=” control of the normal derivative at a boundary

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Going bilinear : 1D

define the mass density N, the current J and the (one dimensional for now !) “tensor” T, N = |u|2, J = 2Im(¯ u∂xu), T = 4|∂xu|2 − ∆N + 2ε(p − 1 p + 1)N

p+1 2 ,

“local” conservation laws ∂tN + ∂xJ = 0 and ∂tJ + ∂xT = 0, the relation ∂2

t N = ∂2 xT is the virial identity

Let I =

y<x

(x − y)|u|2(x)|u|2(y) dxdy then d2I dt2 = 4

x

(∂x(|u|2))2(x) dx + 4ǫp − 1 p + 1

x

|u|p+3(x) dx.

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Bilinear virial : any D

Now, fix a direction, say xn, In =

yn<xn

(xn − yn)|u|2(x)|u|2(y) dxdy then (ǫ = 1, 0) T

−T

xn
∂n
x′ |u|2(x′, xn, t) dx′

2 dxndt

xn<yn

Im¯ u∂nu(x)|u|2(y) dxdy|T

−T

which is a Radon transform estimate in disguise

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Radon transforms, bilinear on domains

R(f)(s, ω) =

x·ω=s∩Ω

f dµs,ω. Let |ω| = 1, Iω =

x,y∈Ω

|(x − y) · ω||u|2(x)|u|2(y) dxdy Let x = x⊥ + sω

s

|∂s(R(|u|2))(s, ω)|2 + ǫp − 1 p + 1

s

R(|u|2)R(|u|p+1) +

s
x·ω=s
y·ω=s

|u(x⊥+sω)∂su(y⊥+sω)−u(y⊥+sω)∂su(x⊥+sω)|2 −

x∈∂Ω,y∈Ω

|u|2(y)∂n(|(x − y) · ω|)|∂nu|2(x) = ∂2

t Iω.

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Interesting consequences

◮ average over ω ∈ Sn−1 provides |u|22 L2

t ( ˙

H

3−n 2 )

n the RHS

◮ the boundary term as no sign, but in dimension n ≥ 3, can

be controled by the usual virial

◮ in Rn, immediatly provides space-time control of nonlinear

solutions. If 1 + 4

n < p < 1 + 4 n−2, implies scattering ◮ works on domains as well ◮ one may recover Bourgain bilinear improvement over

Strichartz (for linear solutions) in a more geometrical way

◮ “good” convex ρ(x − y) will do (flexible)

Main drawback : do not know how to do the same for the wave

equation. One may form the product of two Schrödinger

solutions in Rn, u(x)u(y), it will be a solution in R2n. Not true for the wave

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What about 2D NLS (exterior to star-shaped) ?

We have a problem with ∆2ρ. Solution :

◮ use ρ1(x, y) = |x − y| + |x + y| ◮ the boundary term has a weight

∂nρ = x − y |x − y| + x + y |x + y| ∼ 2x |y|

◮ use ρ2(x, y) =

x2 + y2 (the 4D “usual” weight)

◮ yields a positive boundary term

∂Ω×Ω

x · n(x) ρ2(x, y)|∂nu|2(x, t)|u|2(y, t) dσ(x)dy

◮ sum both...

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Beyond star-shaped ?

Replace x · n(x) > 0 by (x1, εx2) · n(x) > 0. Previous computation works up to ε ≥ 2/3. All ε ? ? go 8D (F . Abou Shakra, 2013). Call X = (x, y) and Y = (w, z), X ′ = (x, −y)