On Strichartz estimates from decoupling and applications
aa r X i v : . [ m a t h . A P ] J a n ON STRICHARTZ ESTIMATES FROM ℓ -DECOUPLING ROBERT SCHIPPA
Abstract.
Strichartz estimates are derived from ℓ -decoupling for phase func-tions satisfying a curvature condition. Bilinear refinements without loss in thehigh frequency are discussed. Estimates are established from uniform curvaturegeneralizing Galilean invariance or from transversality in one dimension. Thebilinear refinements are utilized to prove local well-posedness for generalizedcubic nonlinear Schr¨odinger equations. Introduction
We point out how ℓ -decoupling implies Strichartz estimates for non-degeneratephase functions on tori T n = ( R / π Z ) n . These estimates apply to solutions tolinear dispersive PDE(1) (cid:26) i∂ t u + ϕ ( ∇ /i ) u = 0 , ( t, x ) ∈ R × T ,u (0) = u , where ϕ ∈ C ( R n , R ).The eigenvalues of D ϕ ( ξ ) are denoted by { γ ( ξ ) , . . . , γ n ( ξ ) } and we set σ ϕ ( ξ ) = min( { neg.γ i ( ξ ) , pos.γ i ( ξ ) } )The non-degeneracy hypothesis we assume reads as follows:There is ψ : 2 N → R > such that min( | γ i ( ξ ) | ) ∼ max( | γ i ( ξ ) | ) ∼ ψ ( N ) , | ξ | ∈ [ N, N )( E σ ϕ ( ψ ))and σ ϕ ( ξ ) is independent of ξ. By P N we denote the frequency projector( P N f ) b ( ξ ) = ( [ N, N ) ( | ξ | ) ˆ f ( ξ ) , N ∈ N [0 , ( | ξ | ) ˆ f ( ξ ) , N = 0The Strichartz estimates we will prove read(2) k P N e itϕ ( ∇ /i ) u k L p ( I × T n ) . | I | /p N s ( ϕ ) k P N u k L To prove (2) we will use ℓ -decoupling (cf. [2, 3]), more precisely, (variants of) thediscrete L -restriction theorem. This was carried out in [2, 3] in the special casesof ϕ ( ξ ) = P ni =1 α i ξ i , α i ∈ R \
0. The proof of the more general estimate will clarifythe role of the asymptotic behaviour of the eigenvalues of D ϕ , i.e., the curvatureof the characteristic surface of (1). The following proposition is proved: Key words and phrases. dispersive equations, decoupling, Strichartz estimates, NLS.Financial support by the German Science Foundation (IRTG 2235) is gratefully acknowledged.
Proposition 1.1.
Suppose that ϕ satisfies ( E k ( ψ )) and let I ⊆ R be a compactinterval. Then, we find the following estimates to hold for any ε > : (3) k P N e itϕ ( ∇ /i ) u k L p ( I × T n ) . ε | I | /p N ( n − n +2 p ) + ε (min( ψ ( N ) , /p , n + 2 − k ) n − k ≤ p < ∞ . Recall that certain Strichartz estimates from [1–3] are known to be sharp upto endpoints. With the above proposition being a generalization, the Strichartzestimates proved above are also sharp in this sense. We shall also consider theexample ϕ ( ξ ) = | ξ | a , < a < , a = 1, where the proposition gives an additionalloss of derivatives due to decreased curvature compared to the Schr¨odinger case.When we consider the associated nonlinear Schr¨odinger equation we shall see whythis additional loss does probably not admit relaxation. Moreover, as in [2, 3] thereare estimates for 2 ≤ p ≤ n +2 − k ) n − k which follow from interpolation.In fact, as p = 2, Proposition 1.1 does not yield Strichartz estimates without lossof derivatives. When we aim to apply these estimates to prove well-posedness ofgeneralized cubic nonlinear Schr¨odinger equations(4) (cid:26) i∂ t u + ϕ ( ∇ /i ) u = ±| u | u, ( t, x ) ∈ R × T n ,u (0) = u ∈ H s ( T n ) , we will use orthogonality considerations to prove bilinear L t,x -estimates for High × Low → High -interaction without loss of derivatives in the high frequency. In[4, Theorem 3, p. 193] was proved the following proposition to derive well-posednessto cubic Schr¨odinger equations on compact manifolds:
Proposition 1.2.
Let u , v ∈ L ( T n ) , K, N ∈ N . If there exists s > such that (5) k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T n ) . | I | / min( N, K ) s k P N u k L k P K v k L , where I ⊆ R is a compact time interval with | I | & , then the Cauchy problem (4) is locally well-posed in H s for s > s . For ϕ = P ni =1 α i ξ (5) follows from almost orthogonality and the Galilean trans-formation (cf. [1, 11]). It turns out that it is enough to require ( E σ ϕ ( ψ )) to hold forsome uniform constant:( E σ ϕ ( C ϕ ))There is C ϕ > ξ ∈ R n we have min( | γ i ( ξ ) | ) ∼ max( | γ i ( ξ ) | ) ∼ C ϕ . This will be sufficient to generalize the Galilean transformation and prove the fol-lowing:
Proposition 1.3.
Suppose that ϕ ∈ C ( R n , R ) satisfies ( E k ( C ϕ )) . Then, there is s ( n, k ) such that we find the estimate (6) k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T n ) . C ϕ ,s K s | I | / k P N u k L k P K v k L to hold for s > s ( n, k ) , where I ⊆ R denotes a compact time interval, | I | & . This bilinear improvement can also stem from transversality: In [8–10] short-time bilinear Strichartz estimates were discussed and the following transversality
TRICHARTZ ESTIMATES FROM DECOUPLING 3 condition played a crucial role in the derivation of the estimates:There is α > |∇ ϕ ( ξ ) ± ∇ ϕ ( ξ ) | ∼ N α , whenever | ξ | ∼ K, | ξ | ∼ N, ( T α ) K ≪ N, K, N ∈ N . The corresponding shorttime estimate reads(7) k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ([0 ,N − α ] ,L x ( T )) . ϕ N − α/ k P N u k L k P K v k L This is sufficient to derive an L t,x -estimate for finite times by just gluing togetherthe short time intervals: Proposition 1.4.
Suppose that ϕ satisfies ( T α ) and let K ≪ N, K, N ∈ N . Then,we find the following estimate to hold: (8) k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T ) . ϕ | I | / k P N u k L k P K v k L whenever I ⊆ R is a compact time interval with | I | & N − α .Proof. Let I = S j I j , | I j | ∼ N − α , where the I j are disjoint. Then, k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T ) . X I j k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I j × T ) . ( I j ) N − α k P N u k L k P K v k L and the claim follows from I j ∼ | I | N α . (cid:3) Invoking Proposition 1.2 together with Propositions 1.3 or 1.4 the below theoremfollows:
Theorem 1.5.
Suppose that ϕ ∈ C ( R n , R ) satisfies ( E k ( C ϕ )) . Then, there is s ( n, k ) such that (4) is locally well-posed for s > s ( n, k ) .Let n = 1 and suppose that ϕ satisfies ( T α ) . Then, there is s = s ( ϕ ) such that (4) is locally well-posed for s > s ( ϕ ) . Finally, we give examples: In one dimension we treat the fractional Schr¨odingerequation(9) (cid:26) i∂ t u + D a u = ±| u | u, ( t, x ) ∈ R × T ,u (0) = u ∈ H s ( T ) , where D = ( − ∆) / .Theorem 1.5 yields uniform local well-posedness for s > − a , 1 < a <
2, which isprobably sharp up to endpoints as discussed in [5], where the endpoint s = − a wascovered by resonance considerations.For 0 < a < s > − a , which seems to be a new local well-posedness result.We also discuss hyperbolic Schr¨odinger equations. The well-posedness result from[7, 11] is recovered for the hyperbolic nonlinear Schr¨odinger equation in two dimen-sions, which is known to be sharp up to endpoints. Generalizing this example tohigher dimensions indicates that there is only a significant difference between hy-perbolic and elliptic Schr¨odinger equations in low dimensions.This note is structured as follows: In Section 2 we prove linear Strichartz estimatesutilizing ℓ -decoupling, in Section 3 we discuss bilinear Strichartz estimates without R. SCHIPPA loss in the high frequency and in Section 4 the implied well-posedness results forgeneralized cubic nonlinear Schr¨odinger equations are discussed.2.
Linear Strichartz Estimates
We prove Proposition 1.1 utilizing ℓ -decoupling. This generalizes the proofsfrom [2, 3]: Proof of Proposition 1.1:
Without loss of generality let I = [0 , T ]. First, let p ≥ n +2 − k ) n − k and compute k P N e itϕ ( ∇ /i ) u k pL p ( I × T n ) = Z ≤ x ,...,x n ≤ π, ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ |∼ N e i ( x.ξ + tϕ ( ξ )) ˆ u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dxdt ∼ N − ( n +2) ψ ( N ) Z ≤ x ,...,x n ≤ N, ≤ t ≤ T N ψ ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ |∼ ,ξ ∈ Z n /N e i ( x.ξ + tN ψ ( N ) ϕ ( Nξ )) ˆ u ( N ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dxdt We distinguish between ψ ( N ) ≪ ψ ( N ) &
1. In the latter case, we useperiodicity in space to find ∼ N − ( n +2) ( T N ψ ( N )) n ψ ( N ) Z ≤ x ,...,x n ≤ T N ψ ( N ) , ≤ t ≤ T N ψ ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ |∼ ,ξ ∈ Z n /N ˆ u ( N ξ ) e i ( x.ξ + tN ψ ( N ) ϕ ( Nξ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dxdt This expression is amenable to the discrete L -restriction theorem [2, Theorem 2.1, p. 354]or the variant for hyperboloids because T N ψ ( N ) & N and the frequency pointsare separated of size N and the eigenvalues of ϕ ( N · ) N ψ ( N ) are approximately one.Hence, we have the following estimate uniform in ϕ (the dependence is encoded in ψ ( N ), which drops out in the ultimate estimate) . ε N − ( n +2) ( T N ψ ( N )) n ψ ( N ) ( T N ψ ( N )) n +1 N ( n − n +2 p ) p + ε k P N u k p . T N ( n − n +2 p ) p + ε k P N u k p Next, suppose that ψ ( N ) ≪
1. In this case we can not avoid loss of derivatives ingeneral. Following along the above lines we find for p ≥ n +2 − k ) n − k k P N e itϕ ( ∇ /i ) u k pL p ( I × T n ) ∼ N − ( n +2) ψ ( N ) Z ≤ x ,...,x n ≤ N, ≤ t ≤ T N ψ ( N ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ |∼ ,ξ ∈ Z n /N e i ( x.ξ + t ϕ ( Nξ ) N ψ ( N ) ) ˆ u ( N ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) p dxdt . N − ( n +2) ( N T ) n ψ ( N ) Z ≤ x ,...,x n ≤ T N , ≤ t ≤ T N (cid:12)(cid:12)(cid:12)(cid:12)X e i ( x.ξ + tϕ ( Nξ ) N ψ ( N ) ) ˆ u ( N ξ ) (cid:12)(cid:12)(cid:12)(cid:12) p dxdt . ε Tψ ( N ) N ( n − n +2 p ) p + ε k P N u k p , which yields the claim. (cid:3) TRICHARTZ ESTIMATES FROM DECOUPLING 5
As an example consider Strichartz estimates for the free fractional Schr¨odingerequation(10) (cid:26) i∂ t u + D a u = 0 , ( t, x ) ∈ R × T n ,u (0) = u , The phase function ϕ ( ξ ) = | ξ | a , < a < , a = 1 is elliptic and the lack of higherdifferentiability in the origin is not an issue because low frequencies can always betreated with Bernstein’s inequality. ϕ satisfies ( E ( ψ )) with ψ ( N ) = N a − , hencewe find by virtue of Proposition 1.1(11) k e itD a u k L t,x ( I × T n ) . s | I | / k u k H s , s > s = ( − a , n = 1 , − a + (cid:0) n − n +24 (cid:1) , elseTo find the L t,x -estimate in one dimension we interpolate the L t,x -endpoint estimatewith the trivial L t,x -estimate. In one dimension in case 1 < a < Bilinear Strichartz estimates and transversality
The argument from Section 2 admits bilinearization provided that the disper-sion relation satisfies ( E σ ϕ ( C ϕ )). This generalizes Galilean invariance which waspreviously used to infer a bilinear estimate with no loss in the high frequency (cf.[1, 11]). Proof of Proposition 1.3:
Let P N = P K R K , where R K projects to cubes of side-length K . Then, by means of almost orthogonality k P N e itϕ ( ∇ /i ) u P K e itϕ ( ∇ /i ) v k L t,x ( I × T n ) . X K k R K e itϕ ( ∇ /i ) u P K e itϕ ( ∇ /i ) v k L t,x ( I × T n ) After applying H¨older’s inequality we are left with estimating two L t,x -norms.Clearly, by Proposition 1.1 k P K e itϕ ( ∇ /i ) v k L t,x ( I × T n ) . ϕ,s K s k P K v k L provided that s > s ( n, σ ϕ ).To treat the other term let ξ denote the center of the cube Q K onto which R K is projecting in frequency space and following along the above lines we write k R K e itϕ ( ∇ /i ) u k L t,x ( I × T n ) = Z ≤ x ,...,x n ≤ π, ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X ξ ∈ Q K e i ( x.ξ + tϕ ( ξ )) ˆ u ( ξ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdt = Z ≤ x ,...,x n ≤ π, ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ ′ |≤ K ˆ u ( ξ + ξ ′ ) e i ( x. ( ξ + ξ ′ )+ tϕ ( ξ + ξ ′ )) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdt = Z ≤ x ,...,x n ≤ π, ≤ t ≤ T (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X | ξ ′ |≤ K e i (( x + t ∇ ϕ ( ξ )) .ξ ′ + tψ ξ ( ξ ′ )) ˆ w ( ξ ′ ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) dxdt = k P ≤ K e itψ ξ ( ∇ /i ) w ( x + t ∇ ϕ ( ξ )) k L ( I × T n ) After breaking k P ≤ K e itψ ξ ( ∇ /i ) w k L t,x ( I × T n ) ≤ P ≤ L ≤ K k P L e itψ ξ ( ∇ /i ) w k L thesingle expressions are amenable to Proposition 1.1. Indeed, the size of the moduli R. SCHIPPA of the eigenvalues of D ψ ξ are approximately independent of the frequencies andhence, k P L e itψ ξ ( ∇ /i ) w k L t,x ( I × T n ) . ε,C ϕ L s ( n,k )+ ε k P L w k L and from carrying out the sum and the relation of u and w we find k P ≤ K e itψ ξ ( ∇ /i ) w k L ( I × T n ) . ε,ϕ K s ( n,k )+ ε k R K u k L . The claim follows from almost orthogonality, i.e., X K k R K u k L ! / . k P N u k L (cid:3) In one dimension (and for certain phase functions also in higher dimensions, see[10]) transversality ( T α ) of the phase function allows us to derive Proposition 1.4which improves the above estimate.4. Local well-posedness of generalized cubic Schr¨odinger equation
Deploying Proposition 1.2 by making use of the estimates from Section 2 and 3we can conclude the proof of Theorem 1.5:
Proof.
First, suppose that ϕ satisfies ( E σ ϕ ( C ϕ )). In case K ≪ N Proposition 1.3yields the estimate(12) k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T n ) . ε,ϕ | I | / K s ( n,σ ϕ )+ ε k P N u k L k P K v k L For K ∼ N follows after applying H¨older’s inequality and Proposition 1.1. FromProposition 1.2 we find (4) to be locally well-posed provided that s > s ( n, σ ϕ ).In case ϕ satisfies ( E ( ψ ( N ))) and ( T α ) we have the improved bilinear bound k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T ) . ϕ | I | / k P N u k L k P K v k L due to Proposition 1.4, so that the only loss stems from High × High → High -interaction, where K ∼ N : By means of Proposition 1.1 and H¨older’s inequality wederive k P N e ± itϕ ( ∇ /i ) u P K e ± itϕ ( ∇ /i ) v k L t,x ( I × T ) . ϕ K s | I | / k P N u k L k P K v k L and by Proposition 1.2 we find (4) to be locally well-posed for s > s ( ϕ ). (cid:3) We turn to examples: As discussed in Section 2 the phase functions ϕ ( ξ ) = | ξ | a (0 < a < , a = 1) do not satisfy ( E ( C ϕ )), but ( T a − ) for 1 < a < < a < K ≪ N , | I | & k P N e ± itD a u P K e ± itD a v k L t,x ( I × T ) . | I | / K − a k P N u k L k P K v k L which can be proved like in [9, 10].Consequently, by (the proof of) Theorem 1.5 we find (9) to be locally well-posed for s > − a . As discussed in [5] this is likely to be the threshold of uniform local well-posedness which indicates that the linear Strichartz estimates from Section 2 arein this case sharp up to endpoints. Although the linear Strichartz estimates mightwell be sharp in higher dimensions, satisfactory bilinear L t,x -Strichartz estimatesappear to be beyond the methods of this paper so that we can not prove non-trivialwell-posedness results in higher dimensions. TRICHARTZ ESTIMATES FROM DECOUPLING 7
For hyperbolic phase functions Theorem 1.5 recovers the results from [7, 11] whereessentially sharp local well-posedness of(13) (cid:26) i∂ t u + ( ∂ x − ∂ x ) u = ±| u | u, ( t, x ) ∈ R × T ,u (0) = u , was proved for s > /
2. Notably, due to subcriticality of the L t,x -Strichartz estimatealready for the hyperbolic equations(14) (cid:26) i∂ t u + ( ∂ x − ∂ x + ∂ x ) u = ±| u | u, ( t, x ) ∈ R × T ,u (0) = u , and(15) (cid:26) i∂ t u + ( ∂ x − ∂ x + ∂ x − ∂ x ) u = ±| u | u, ( t, x ) ∈ R × T ,u (0) = u , the (essentially sharp) Strichartz estimates yield the same well-posedness results asfor the elliptic counterparts:Firstly, recall the counterexample from [11] which showed C -ill-posedness of (13)for s < /
2. As initial data consider φ N ( x ) = N − / X | k |≤ N e ikx e − ikx , which satisfies k φ N k H s ∼ N s and S [ φ N ]( t ) := e it ( ∂ x − ∂ x ) φ N = φ N . This implies (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T | S [ φ N ]( s ) | S [ φ N ]( s ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s = T k| φ N | φ N k H s & T N s For details regarding this estimate see [11]. The veracity of the estimate (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T ds | S [ φ N ]( s ) | S [ φ N ]( s ) (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s . k φ N k H s ( T . s ≥ / s ≥ / C -well-posedness of(14). This regularity is reached up to the endpoint by Theorem 1.5.When considering (15) we modify the above example to φ N ( x ) = N − X | k | , | k |≤ N e ik x e − ik x e ik x e − ik x , which again satisfies k φ N k H s ∼ N s .Carrying out the estimate for the first Picard iterate with the necessary modifica-tions yields (cid:13)(cid:13)(cid:13)(cid:13)(cid:13)Z T | S [ φ N ]( s ) | S [ φ N ]( s ) ds (cid:13)(cid:13)(cid:13)(cid:13)(cid:13) H s = T k| φ N | φ N k H s & T N s , which implies C -ill-posedness unless s ≥
1. This regularity is again obtained upto the endpoint by Theorem 1.5.Apparently, for other hyperbolic Schr¨odinger equations the L t,x -Strichartz estimatealso coincides with the elliptic L t,x -estimate and modifications of the above coun-terexample yield lower thresholds than in the elliptic case. This indicates that thedifference between elliptic and hyperbolic Schr¨odinger equations is only significantat lower dimensions. R. SCHIPPA
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