Asymptotics for (nonlinear) wave propagation Fabrice Planchon 1 1 Laboratoire Jean Alexandre Dieudonné, Université de Nice Sophia-Antipolis et Institut universitaire de France Monastir, June 2013
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
Wave equation � g φ = ( ∂ 2 t − ∆ g ) φ + f ( φ, ∂φ ) = 0 most simple case : linear wave in R n , g = Id and f = 0. Cauchy problem : given φ | t = 0 = φ 0 , ∂ t φ | t = 0 = φ 1 , characterize φ (existence, uniqueness, “good” estimates). More complicated : ◮ f � = 0 : f ( φ ) = ±| φ | p − 1 φ , f ( φ, ∂φ ) = ( | ∂ t φ | 2 − |∇ φ | 2 ) φ ◮ domain Ω � = R n (with boundary conditions) ◮ variable metric g , possibly g ( φ )
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 = φ 0 , what are the properties of ψ ? ◮ f � = 0 : f ( ψ ) = ±| ψ | p − 1 φ ◮ domain Ω � = R n (with boundary conditions) ◮ variable metric g
How to pick the right ( φ 0 , φ 1 ) or ψ 0 ? ◮ 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 ) � ¯ ψ 2 dx , momentum I m ◮ NLS : mass � ψ ∇ ψ dx , Hamiltonian |∇ ψ | 2 + F ( ψ ) dx � (energy) | ∂ t φ | 2 + |∇ φ | 2 + F ( φ ) dx ◮ NLW : Hamiltonian (energy) � x , H 1 is a strongly suggests Sobolev spaces : derivatives in L 2 favorite because of the Hamiltonian structure (if present)
Local theory for cubic NLS in R 2 : vector fields Consider i ∂ t ψ + ∆ ψ = | ψ | 2 ψ with ψ 0 ∈ H 1 and | x | ψ 0 ∈ L 2 x key observation : L = x + 2 it ∇ x has good commutation properties with the equation (linear and nonlinear) x | u | 4 dx = � x ψ 0 � 2 � L ψ � 2 x + t 2 � L 2 L 2 x ◮ provides time decay at the expense of space decay at t = 0 ◮ leads to global in time solutions x | x | 2 | u | 2 dx (virial identity) ◮ strong link to the convexity of � leads to scattering : existence of asymptotic states φ ± such that lim t →±∞ � ψ − e ± it ∆ ψ ± � = 0
Invariance and conformal transform Solutions u ( x , t ) to the previous NLS are such that t ) e i | x | 2 v ( x , t ) = 1 t u ( x t , − 1 4 t is also a solution (pseudo-conformal transform) The cubic wave equation in R 3 is truely conformally invariant : with ( X , T ) = ( x , t ) / ( t 2 − | x | 2 ) and | x | φ = | X | Φ � T , X Φ ± Φ 3 = 0 and deriving the energy identity for Φ means multiplying � φ by the conformal vector field, e.g. K φ = (( t 2 + | x | 2 ) ∂ t + 2 tx · ∇ x + 2 ) φ
local energy decay for compactly supported data Consider the linear wave equation � φ = 0, multiplying by K φ and integrating between times s = 0 , t yields control of x ( | x | 2 + t 2 )( |∇ u | 2 − | x | x | · ∇ u | 2 ) dx , � | x | 2 · ∇ ( | x | u ) | 2 + | ∂ t u | 2 ) dx , x ( | x | + t ) 2 ( | x � | x | 2 · ∇ ( | x | u ) | 2 − | ∂ t u | 2 ) dx , x ( | x | − t ) 2 ( | x � (or the sum of the L 2 norms of Lu , Ω u , where Ω ij = x i ∂ j − x j ∂ i , L i = t ∂ i + x i ∂ t and L 0 = t ∂ t + x · ∇ x ) If boundary, Dirichlet condition φ | ∂ Ω = 0, additional term like n · ∇ x φ | 2 dS x � ∂ Ω ( x · � n ) | � + � n inner normal to the domain (outer normal to the obstacle)
Morawetz variant Consider the nonlinear wave equation � φ + φ p = 0, and the multiplier M = t ∂ t + x · ∇ x + 1. One can rewrite M φ ( � φ + φ p ) = 0 so that p + 1 u p + 1 2 0 = div t , x ( tQ + u ∂ t u , − tP ) + Q = | ∂ t u | 2 + |∇ x u | 2 + | u | p + 1 p + 1 + ∂ t u ( x t · ∇ x ) u , 2 � | ∂ t u | 2 −|∇ x u | 2 − | u | p + 1 � � � P = x ∂ t u + ( x t · ∇ x ) u + u + ∇ x u t 2 p + 1 t Boundary terms will come from � n ( x ) · P integrated on ∂ Ω × [ 0 , T ]
Morawetz variant, take two | x | ∂ t + x t | x | · ∇ x + 1 Let N = | x | , one may compute N φ ( � φ + φ p ) = 0 and integrate between time slices. Benefit : N φ in L 2 x is controled by the energy. Let ∂ t φ ( x | x | · ∇ x φ + φ � J = − | x | ) dx then φ p + 1 Hess ( | x | )( ∇ x φ, ∇ x φ ) + 2 πφ 2 ( 0 , t ) + p − 1 d � dt J = | x | dx p + 1 If obstacle, additional boundary term n · ∇ x φ | 2 dS x ∂ Ω ( x � | x | · � n ) | � + � n inner normal to the domain (outer normal to the obstacle)
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.
Morawetz variant, virial like Let ∂ t φ ∇ ρ ( x ) · ∇ x φ + ∆ ρ ( x ) � J = − φ dx 2 then p + 1 φ p + 1 dx + 4 (∆ 2 ρ ) φ + (∆ ρ ) p − 1 d Hess ( ρ )( ∇ x φ, ∇ x φ ) − 1 � dt J = n · ∇ x φ | 2 dS x � ∂ Ω ( ∇ x ρ · � n ) | � typical ρ ( x ) : convex function with suitable level sets Variants on truncated forward or backward cones
NLS virial on a domain Let i ∂ t u + ∆ u − ǫ | u | p − 1 u = 0 , with Dirichlet boundary condition u | ∂ Ω = 0, and Ω is the exterior of a star-shaped body with smooth boundary. � | u | 2 ( x , t ) ρ ( x ) dx , M ρ ( t ) = Ω d � � ρ ∇ · (¯ ¯ dt M ρ ( t ) = − 2 I m u ∇ u ) = 2 I m u ∇ u · ∇ ρ,
d 2 � ( ∂ t ¯ u ∇ u + ¯ dt 2 M ρ ( t ) = 2 I m u ∇ ∂ t u ) · ∇ ρ � ∂ t u ( 2 ∇ ¯ u · ∇ ρ + ¯ = − 2 I m u ∆ ρ ) � (∆ u − ǫ | u | p − 1 u ) ( 2 ∇ ¯ u · ∇ ρ + ¯ = − 2 R e u ∆ ρ ) � � � |∇ u | 2 ∆ ρ + 2 R e ∆ u ∇ ¯ ¯ = − 4 R e u · ∇ ρ + 2 u ∇ u ∇ ∆ ρ � � ǫ | u | p − 1 ∇ ( | u | 2 ) ∇ ρ + 2 ǫ | u | p + 1 ∆ ρ + 2 � � � |∇ u | 2 ∆ ρ − | u | 2 ∆ 2 ρ ∆ u ∇ ¯ = − 4 R e u · ∇ ρ + 2 � 2 p + 1 ) | u | p + 1 ∆ ρ . + 2 ǫ ( 1 −
Integrating by parts again, � � � ∆ u ∇ ¯ ∇ ¯ ∇ ( ∇ ¯ u · ∇ ρ = u · ∇ ρ∂ n u − u · ∇ ρ ) · ∇ u , ∂ Ω and, as u ∂ Ω = 0 implies ∂ τ u ∂ Ω = 0, � � � ( ∂ n ρ ) | ∂ n u | 2 − ∇ ρ · ∇ ( |∇ u | 2 ) ∆ u ∇ ¯ 2 R e u · ∇ ρ = 2 ∂ Ω � Hess ( ρ )( ∇ u , ∇ ¯ − 2 u ) � � ( ∂ n ρ ) | ∂ n u | 2 + |∇ u | 2 ∆ ρ = ∂ Ω � Hess ( ρ )( ∇ u , ∇ ¯ − 2 u )
and finally d 2 � | u | 2 ∆ 2 ρ + 2 ǫ p − 1 p + 1 | u | p + 1 ∆ ρ dt 2 M ρ ( t ) = − � ( ∂ n ρ ) | ∂ n u | 2 + 2 ∂ ( R n \ Ω) � Hess ( ρ )( ∇ u , ∇ ¯ + 4 u ) .
Canonical choice : ρ ( x ) = | x | 2 . � 1 + | x | 2 , Good choice : ρ ( x ) = ∆ ρ = n − 1 ρ ( x ) 3 , − ∆ 2 ρ = ( n − 1 )( n − 3 ) 1 + 6 ( n − 3 ) + 15 ρ ( x ) + ρ ( x ) 7 , ρ ( x ) 3 ρ ( x ) 5 and n ( x ) · x ≥ 0 if star-shaped, as well as |∇ u | 2 u ) , as Hess ( ρ ) = 1 ρ Id − 1 ρ ( x ) 3 � Hess ( ρ )( ∇ u , ∇ ¯ ρ 3 ( x i x j ) ij ,
1D local smoothing Consider the domain to be the half-line x > 0, weight ρ ( x ) = x , Dirichlet u ( 0 , t ) = 0, d � u ∂ x u = 4 | ∂ x u | 2 ( 0 ) ¯ dt 2 I m x > 0 Now, v linear solution on R , let u = v ( x ) − v ( − x ) , � T | ∂ x v ( 0 , t ) | 2 dt � � v � 2 2 ( 0 ) + � v � 2 2 ( T ) 1 1 ˙ ˙ H H 0 eventually � T � v ( ξ ) | 2 d ξ | ∂ x v ( x , t ) | 2 dt = C | ξ || ˆ sup x 0 R
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
Going bilinear : 1D define the mass density N , the current J and the (one dimensional for now !) “tensor” T , u ∂ x u ) , T = 4 | ∂ x u | 2 − ∆ N + 2 ε ( p − 1 p + 1 N = | u | 2 , J = 2 I m (¯ 2 , p + 1 ) N “local” conservation laws ∂ t N + ∂ x J = 0 and ∂ t J + ∂ x T = 0 , the relation ∂ 2 t N = ∂ 2 x T is the virial identity Let � ( x − y ) | u | 2 ( x ) | u | 2 ( y ) dxdy I = y < x then d 2 I � ( ∂ x ( | u | 2 )) 2 ( x ) dx + 4 ǫ p − 1 � | u | p + 3 ( x ) dx . dt 2 = 4 p + 1 x x
Bilinear virial : any D Now, fix a direction, say x n , � ( x n − y n ) | u | 2 ( x ) | u | 2 ( y ) dxdy I n = y n < x n then ( ǫ = 1 , 0) � T �� 2 � � �� x ′ | u | 2 ( x ′ , x n , t ) dx ′ ∂ n dx n dt � − T x n � u ∂ n u ( x ) | u | 2 ( y ) dxdy | T I m ¯ − T x n < y n which is a Radon transform estimate in disguise
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