Strichartz inequalities on non compact manifolds Jean-Marc Bouclet - PowerPoint PPT Presentation

Strichartz inequalities on non compact manifolds Jean-Marc Bouclet Institut de Mathématiques de Toulouse Rencontre Nosevol #3, 9 avril 2014, Rennes

What are Strichartz inequalities ? Schrödinger-Strichartz estimates i ∂ t u = ∆ u = ⇒ || u || L p ([ 0 , T ] , L q ) � || u ( 0 ) || L 2 if p , q ≥ 2 satisfy the admissibilty condition 2 p + n q = n p , q ≥ 2 , 2 . Wave-Strichartz estimates ∂ 2 t u = ∆ u = ⇒ || u || L p ([ 0 , T ] , L q ) � || u ( 0 ) || H γ + || ∂ t u ( 0 ) || H γ − 1 under the (sufficient) condition on p , q ≥ 2 that � 1 � 2 p + n − 1 = n − 1 γ = n + 1 2 − 1 , q 2 2 q [Strichartz,Ginibre-Velo]

An explicit example Consider a wave packet centered at ( y , ζ ) � i � h ζ · ( x − y ) − | x − y | 2 G h ( x ) = π − n / 4 h − κ n 2 exp 2 h 2 κ By explicit computation: � � � � h κ n − | x − y − 2 t ζ/ h | 2 � � � = π − n 4 h − n κ � e it ∆ G h 4 exp 2 2 ( h 2 κ + 4 t 2 h − 2 κ ) ( h 4 κ + 4 t 2 ) n and � � � � � h 2 κ + 4 t 2 h − 2 κ � n � � � � � � q − 1 1 � e it ∆ G h � � � x ) = c qn 2 2 L q ( R n 2 q − n n n 4 ( 2 / q ) 2 q . Using the admissibility condition: where c qn = π � T � 2 Th − 2 κ � � � � p 1 � � � � L q dt = c p � e it ∆ G h 1 + τ 2 d τ. � � � qn 0 0

Why are they useful ? Non linear Cauchy problem at low regularity , e.g. i ∂ t u + ∆ u = ±| u | ν − 1 u , u | t = 0 = u 0 ∈ L 2 ( R 2 ) , 1 < ν < 3 . Rewrite it as an integral equation � t u ( t ) = e it ∆ u 0 ∓ i e i ( t − s )∆ | u ( s ) | ν − 1 u ( s ) ds 0 and use a fixed point argument in a suitable closed ball of p = 2 ν + 2 X T := C ([ 0 , T ] , L 2 ) ∩ L p ([ 0 , T ] , L q ) , ν − 1 , q = ν + 1 . Strichartz inequalities allow to show that e it ∆ u 0 ∈ X T , and that � t e i ( t − s )∆ | v ( s ) | ν − 1 v ( s ) ds is a contraction v �→ 0 for T small enough (this uses inhomogeneous inequalities).

Estimates in non Euclidean geometries Wave equation: weaker dispersion but finite propagation speed 1. M smooth with positive injectivity radius : same estimates (local in time) as on R n [Kapitanski] 2. M with boundary : Additional losses in general [Ivanovici-Lebeau-Planchon]. Unavoidable at least if if � � q > 4 and n ∈ { 2 , 3 , 4 } (additional loss of 1 1 4 − 1 6 q [Ivanovici]) 3. low regularity metrics : additional losses in general below C 2 regularity [Bahouri-Chemin, Tataru, Smith-Tataru]

Estimates in non Euclidean geometries (continued) Schrödinger equation: one expects possible losses || u || L p ([ 0 , T ] , L q ( M )) � || u ( 0 ) || H σ ( M ) := || ( 1 − ∆) σ/ 2 u ( 0 ) || L 2 ( M ) (infinite propagation speed!) 1. M closed : σ = 1 p [Burq-Gérard-Tzvetkov] (optimal on S 3 ), but for M = T 2 and p = q = 4, any σ > 0 [Bourgain]! 2. M compact with boundary : Additional losses in general 3 4 ( σ = 2 p [Anton], 3 p [Blair,Smith,Sogge]) 3. M non compact with large ends : No loss if no (or little) trapping; either for M asymp. flat or hyperbolic (including: outside a convex [Ivanovici] or polygonal obstacles [Baskin-Marzuola-Wunsch])

About the proof of Strichartz estimates The classical strategy is to prove L 1 → L ∞ estimates for the evolution and use the following type of abstract result. Proposition. Assume � �� � �� � U h ( t ) � ≤ B h , | t | ≤ T L 2 → L 2 � �� � U h ( t ) U h ( s ) ∗ � �� D h � ≤ | t − s | δ , | t | , | s | ≤ T L 1 → L ∞ Then, if p > 2, q ≥ 2 and � 1 � 2 − 1 = 1 δ p , q we have � �� �� � 1 2 − 1 2 � U h ( · ) f � q 1 q L p ([ 0 , T ] , L q ) � B h D || f || L 2 h

About the proof of Strichartz estimates (continued) Up to a Littlewood-Paley argument, to localize spectrally the problem (with ϕ ∈ C ∞ 0 ( 0 , + ∞ ) ), the usual estimates follow from: Schrödinger � �� � ϕ ( − h 2 ∆) e i ( t − s )∆ � �� L 1 ( M ) → L ∞ ( M ) � | t − s | − n � 2 Wave � � − ∆ � � √ � � � � L 1 ( M ) → L ∞ ( M ) � h − n + 1 2 | t − s | − n − 1 � ϕ ( − h 2 ∆) e i ( t − s ) � � � 2 on suitable time scales. Typically, if ̺ inj = injectivity radius, | t | , | s | � ̺ inj (Wave) | t | , | s | � h × ̺ inj (Schrödinger)

Problem: what happens if ̺ inj vanishes ? ◮ are there still Strichartz estimates ? ◮ if yes, are there additional losses ? ◮ if yes, are they unavoidable ? We address these questions for (smooth) surfaces with cusps .

Surfaces with cusps ◮ Model for the cusp: G 0 = dr 2 + e − 2 φ ( r ) d θ 2 , S 0 = [ r 0 , ∞ ) × A , A = a union of circles and � ∞ e − φ ( r ) dr < ∞ i . e . aera ( S 0 ) < ∞ r 0 We also assume that φ ( j ) is bounded for all j ≥ 1. ◮ More generally, we can consider ( S , G ) with ◦ ◦ S = K⊔ S 0 , with K compact and G = G 0 on S 0 . Example: S = R × S 1 with G = dr 2 + d θ 2 / cosh 2 ( r )

Operators and measures on S 0 ∆ 0 = ∂ 2 ∂ r 2 − φ ′ ( r ) ∂ ∂ r + e 2 φ ( r ) ∆ A , d vol 0 = e − φ ( r ) drd A ∆ 0 is symmetric on L 2 G 0 := L 2 ( S 0 , d vol 0 ) . We also let � �� �� � �� � � �� � ψ � � ( 1 − ∆ 0 ) σ/ 2 ψ � = H σ L 2 G 0 G 0 To use the standard Lebesgue measure, it is useful to consider G 0 ∋ ψ �→ u := U ψ = e − φ ( r ) / 2 ψ ∈ L 2 := L 2 ( S 0 , drd A ) . U : L 2 P := U ( − ∆ 0 ) U ∗ = − ∂ 2 ∂ r 2 − e 2 φ ( r ) ∆ A + w ( r ) , where w = ( φ ′ 2 − 2 φ ′′ ) / 4. P is symmetric on L 2 . Note also that � �� �� � � � 1 2 − 1 � e φ ( r ) � || ψ || L q G 0 = u q L q

Projection away from zero modes We let π 0 = orthogonal projection on Ker L 2 ( A ) (∆ A ) and define Π c = I ⊗ ( I − π 0 ) Π = I ⊗ π 0 , seen as operators (orthogonal projections) on both L 2 (( r 0 , ∞ ) , dr ) ⊗ L 2 ( A , d A ) L 2 ≈ L 2 (( r 0 , ∞ ) , e − φ ( r ) dr ) ⊗ L 2 ( A , d A ) L 2 ≈ G 0 If e 0 , . . . , e k 0 − 1 is an orthonormal basis of Ker L 2 ( A ) (∆ A ) , �� � � Π ψ = e k ( α ) ψ ( r , α ) d A ⊗ e k A k < k 0

Zero angular modes ⇒ No Strichartz estimates Theorem 1 Let p ≥ 1, q > 2 and σ ≥ 0. 1. There is a sequence ( ψ n ) n ≥ 0 in H σ G 0 ∩ Ran (Π) such that || ψ n || L q G 0 sup = + ∞ . || ψ n || H σ n ≥ 0 G 0 2. There is a sequence ( ψ n ) n ≥ 0 of in H σ G 0 ∩ Ran (Π) such that || cos ( t √− ∆ 0 ) ψ n || L p ([ 0 , 1 ] t ; L q G 0 ) sup = + ∞ . || ψ n || H σ n ≥ 0 G 0 3. Consider e φ ( r ) = e r and r 0 = 0. There is a sequence ( ψ n ) n ≥ 0 in H σ G 0 ∩ Ran (Π) such that || e it ∆ ψ n || L p ([ 0 , 1 ] t ; L q G 0 ) = + ∞ . sup || ψ n || H σ n ≥ 0 G 0

Wave-Strichartz estimates at infinity away from zero angular modes Let r 1 > r 0 and 1 [ r 1 , ∞ ) ( r ) be a localization inside the cusp. Theorem 2 Let ( p , q ) be sharp wave admissible in dimension two 2 p + 1 q = 1 2 and set � 1 � σ w = 3 2 − 1 . 2 q Then, if we set √ √ − ∆) ψ 0 + sin ( t − ∆) Ψ( t ) = cos ( t √ ψ 1 , − ∆ we have � �� �� � � Π c 1 [ r 1 , ∞ ) ( r )Ψ � G 0 ) � || ψ 0 || H σ w + || ψ 1 || H σ w − 1 L p ([ 0 , 1 ]; L q G G

Schrödinger-Strichartz estimates at infinity away from zero angular modes Theorem 3 Let ( p , q ) be Schrödinger admissible � 1 � 1 p + 1 q = 1 σ S = 1 2 − 1 = 1 2 , 2 q 2 p Fix ϕ ∈ C ∞ 0 ( R ) . Then, if we set Ψ h ( t ) = e it ∆ ϕ ( − h 2 ∆) ψ we have � �� �� � � Π c 1 [ r 1 , ∞ ) ( r )Ψ h � G 0 ) � || ψ || H σ S L p ([ 0 , h ]; L q G Corollary Let ( p , q ) be a Schrödinger admissible pair. If we set Ψ( t ) = e it ∆ ψ we have �� � � �� � Π c 1 [ r 1 , ∞ ) ( r )Ψ � G 0 ) � || ψ || L p ([ 0 , 1 ]; L q 3 2 p H G

Separation of variables Using an orthonormal eigenbasis ( e k ) k ≥ 0 of ∆ A , ∆ A e k = − µ 2 k e k we have a unitary equivalence � � L 2 � � L 2 ∋ u �→ ( u k ) k ∈ ( r 0 , ∞ ) , dr , u k ( r ) = e k ( α ) u ( r , α ) d A k ≥ 0 Through this mapping, for any bounded Borel function f , we have � f ( P ) u = f ( p k ) u k ⊗ e k k where k e 2 φ ( r ) + w ( r ) . p k = − ∂ 2 r + µ 2

Elliptic estimates away from zero angular modes Proposition Let χ ∈ C ∞ 0 ( R ) such that χ ≡ 1 near r 0 . Then for any N > 0 � � r Π c ( 1 − χ ( r ))( 1 − ∆ 0 ) − N � � � � � � � ( e 2 φ ( r ) ∆ A ) N 1 ∂ N 2 < ∞ � � � L 2 G 0 → L 2 G 0 provided that 2 N 1 + N 2 ≤ 2 N . In particular, for N large enough � � � e N φ ( r ) Π c ( 1 − ∆ 0 ) − N � � � � � � < ∞ � � � L 2 G 0 → L ∞ G 0

Localization in frequency: Littlewood-Paley decomposition Consider a dyadic partition of unity � ϕ ( − h 2 ∆ 0 ) I = ϕ 0 ( − ∆ 0 ) + h 2 = 2 − n with ϕ 0 ∈ C ∞ 0 ( R ) , ϕ ∈ C ∞ 0 ( 0 , + ∞ ) Proposition. For all q ∈ [ 2 , ∞ ) and χ ∈ C ∞ 0 ( R ) such that χ ≡ 1 near r 0 , �� � 1 � �� � �� 2 � 2 || Π c ( 1 − χ ) ψ || L q � Π c ( 1 − χ ) ϕ ( − h 2 ∆ 0 ) ψ G 0 � + || ψ || L 2 L q G 0 G 0 h