Strichartz estimates from decoupling and applications

The situation is less clear in the periodic case. Below, we show how the essen-tially sharp `²-decoupling results from [BD15, BD17a] imply Strichartz estimates for more general phase functions.

In the Bourgain-Demeter works this was pointed out for Schr¨odinger dispersion relation. The following modest generalization of the argument clarifies the role of curvature. We point out how `²-decoupling implies Strichartz estimates for non-degenerate phase functions on tori Tⁿ = (R/2πZ)ⁿ. These estimates apply to solutions to linear dispersive PDE

i∂tu+ϕ(∇/i)u = 0, (t, x)∈R×T,

u(0) =u0, (3.23)

whereϕ∈C²(Rⁿ,R).

The eigenvalues ofD²ϕ(ξ) are denoted by{γ1(ξ), . . . , γn(ξ)}and we set σϕ(ξ) = min({#neg.γi(ξ),#pos.γi(ξ)}).

The non-degeneracy hypothesis we assume reads as follows forψ: 2^N0→R^>0: min(|γi(ξ)|)∼max(|γi(ξ)|)∼ψ(N), |ξ| ∈[N,2N), σ_ϕ(ξ)≡k. (E^k(ψ)) The Strichartz estimates we prove below read

kP_Ne^itϕ(∇/i)u₀k_Lp(I×Tⁿ).|I|^1/pN^s(ϕ)kP_Nu₀k_L2. (3.24) To prove (3.24), we will use`²-decoupling (cf. [BD15, BD17a]), more precisely, (variants of) the discreteL²-restriction theorem.

Proposition 3.2.1. Suppose that ϕsatisfies (E^k(ψ)) and let I ⊆R be a compact interval with|I|&1. Then, we find the following estimates to hold for any ε >0

kPNe^itϕ(∇/i)u₀kL^p(I×Tⁿ).^ε|I|^1/p N(ⁿ2−ⁿ⁺²_p )^+ε

(min(ψ(N),1))^1/pkPNu₀kL² (3.25) provided that ^2(n+2−k)_n−k ≤p <∞.

Proof. Without loss of generality letI= [0, T]. First, letp≥ ^2(n+2−k)_n−k and compute

lhs(3.25)^p= Z

0≤x1,...,x_n≤2π, 0≤t≤T

X

|ξ|∼N

ei(x.ξ+tϕ(ξ))uˆ0(ξ)

p

dxdt

∼ N⁻⁽ⁿ⁺²⁾ ψ(N)

Z

0≤x1,...,xn≤N, 0≤t≤T N²ψ(N)

X

|ξ|∼1,ξ∈Zⁿ/N

e^i(x.ξ+

t

N2ψ(N)ϕ(N ξ))

ˆ u0(N ξ)

p

dxdt.

We distinguish between ψ(N) 1 and ψ(N) & 1. In the latter case, we use periodicity in space to find

∼ N⁻⁽ⁿ⁺²⁾ (T N ψ(N))ⁿψ(N)

Z

0≤x1,...,xn≤T N²ψ(N), 0≤t≤T N²ψ(N)

| X

|ξ|∼1, ξ∈Zⁿ/N

ˆ

u₀(N ξ)e^{i(x.ξ+N2ψ(N)t ϕ(N ξ))}|^pdxdt.

This expression is amenable to the discreteL²-restriction theorem

[BD15, Theorem 2.1, p. 354] or the variant for hyperboloids becauseT N²ψ(N)&

N²and the frequency points are separated of size _N¹ and the eigenvalues of _N^ϕ(N2ψ(N)^·)

are approximately one.

Hence, we have the following estimate uniform inϕ(the dependence is encoded inψ(N), which drops out in the ultimate estimate)

.^ε N⁻⁽ⁿ⁺²⁾

(T N ψ(N))ⁿψ(N)(T N²ψ(N))ⁿ⁺¹N(ⁿ₂−ⁿ⁺²_p )^p+εkPNu0k^p₂ .^εT N(ⁿ₂−ⁿ⁺²_p )^p+εkPNu0k^p₂.

Next, suppose that ψ(N)1. In this case we can not avoid loss of derivatives in general. Following along the above lines, we find forp≥ ^2(n+2−k)_n−k

lhs(3.25)^p∼ N⁻⁽ⁿ⁺²⁾ ψ(N)

Z

0≤x1,...,x_n≤N, 0≤t≤T N²ψ(N)

X

|ξ|∼1, ξ∈Zⁿ/N

e^i(x.ξ+t

ϕ(N ξ) N2ψ(N))

ˆ u₀(N ξ)

p

dxdt

. N⁻⁽ⁿ⁺²⁾ (N T)ⁿψ(N)

Z

0≤x1,...,xn≤T N², 0≤t≤T N²

Xe^i(x.ξ+

tϕ(N ξ) N2ψ(N))

ˆ u₀(N ξ)

p

dxdt

.^ε T

ψ(N)N(ⁿ₂−ⁿ⁺²_p )^p+εkPNu0k^p₂, which yields the claim.

Recall that certain Strichartz estimates from [Bou93a, BD15, BD17a] are known to be sharp up to endpoints. Since the proposition is a generalization, the Strichartz estimates proved above are also sharp in this sense. Moreover, as in [BD15, BD17a]

there are estimates for 2 ≤ p ≤ ^2(n+2−k)_n−k , which follow from interpolation. As an example, we consider Strichartz estimates for the free fractional Schr¨odinger equation

i∂_tu+D^au = 0, (t, x)∈R×Tⁿ,

u(0) =u₀. (3.26)

The phase function ϕ(ξ) = |ξ|^a, 0 < a <2, a 6= 1 is elliptic, and the lack of differentiability at the origin is not an issue because low frequencies can always be treated with Bernstein’s inequality. ϕsatisfies (E⁰(ψ)) withψ(N) =N^a−2. Hence, we find by virtue of Proposition 3.2.1

ke^itDau0k_L4

t,x(I×Tⁿ).^n,a,s|I|^1/4ku0kH^s, s > s0= (_2−a

8 , n= 1,

2−a

4 + ⁿ₂ −ⁿ⁺²₄

, else.

(3.27) To find the L⁴_t,x-estimate in one dimension, we interpolate the L⁶_t,x-endpoint estimate with the trivialL²_t,x-estimate.

In casen= 1, 1< a <2 this recovers the Strichartz estimates from [DET16], and for 0< a <1, this estimate was proved in [Din17].

For n ≥ 2, 1 < a < 2, the estimates seem to be new. In [Din17] short-time arguments were used to derive Strichartz estimates on arbitrary compact manifolds.

These estimates we can improve on tori for 1< a <2 because we do not have to sum up over frequency dependent time intervals.

However, forp6= 2, Proposition 3.2.1 does not yield Strichartz estimates without loss of derivatives. When we want to apply these estimates to prove well-posedness of generalized cubic nonlinear Schr¨odinger equations

i∂_tu+ϕ(∇/i)u =±|u|²u, (t, x)∈R×Tⁿ,

u(0) =u₀∈H^s(Tⁿ), (3.28)

we will use orthogonality considerations to prove refined bilinearL²_t,x-estimates for High×Low→High-interaction. These estimates have no loss of derivatives in the high frequency, thus allowing us to close the contraction argument.

In [BGT05, Theorem 3, p. 193] was proved the following proposition to derive well-posedness to cubic Schr¨odinger equations on compact manifolds:

Proposition 3.2.2. Let u0, v0 ∈ L²(Tⁿ), K, N ∈2^N. If there exists s0 >0 such that

kPNe^{±itϕ(∇/i)}u0PKe^{±itϕ(∇/i)}v0k_L2 t,x(I×Tⁿ)

.|I|^1/2min(N, K)^s0kPNu0k_L2kPKv0k_L2,

(3.29) where I ⊆ R is a compact time interval with |I| & 1, then the Cauchy problem (3.28)is locally well-posed inH^s fors > s0.

For ϕ= Pn

i=1αiξ² (3.29) follows from almost orthogonality and the Galilean transformation (cf. [Bou93a, Wan13]). It turns out that it is enough to require (E^k(ψ)) to hold for some uniform constantC_ϕ>0:

∀ξ∈Rⁿ: min(|γ_i(ξ)|)∼max(|γ_i(ξ)|)∼C_ϕ, σ_ϕ(ξ)≡k. (E^k(C_ϕ)) This will be sufficient to generalize the Galilean transformation and prove the fol-lowing:

Proposition 3.2.3. Suppose thatϕ ∈C²(Rⁿ,R) satisfies (E^k(Cϕ)). Then, there iss(n, k) such that we find the estimate

kPNe^{±itϕ(∇/i)}u0PKe^{±itϕ(∇/i)}v0k_L2

t,x(I×Tⁿ).^Cϕ,sK^2s|I|^1/2kPNu0kL²kPKv0kL²

(3.30) to hold fors > s(n, k), whereI⊆Rdenotes a compact time interval,|I|&1.

Proof. Partition PN = P

K1RK₁, where RK projects to cubes of sidelength K.

Then, by means of almost orthogonality, lhs(3.30)².X

K1

kRK₁e^itϕ(∇/i)u0PKe^itϕ(∇/i)v0k²_L2

t,x(I×Tⁿ).

After applying H¨older’s inequality, we are left with estimating two L⁴_t,x-norms.

Clearly, by Proposition 3.2.1

kPKe^itϕ(∇/i)v0k_L4

t,x(I×Tⁿ).^ϕ,sK^skPKv0k_L2

provided thats > s(n, σ_ϕ).

To treat the other term, let ξ0 denote the center of the cube QK₁, onto which RK₁ is projecting in frequency space, and following along the above lines, we write

kRK₁e^itϕ(∇/i)u0k⁴_L4 t,x(I×Tⁿ)

= Z

0≤x₁,...,x_n≤2π, 0≤t≤T

X

ξ∈QK1

ei(x.ξ+tϕ(ξ))uˆ0(ξ)

4

dxdt

= Z

0≤x1,...,xn≤2π, 0≤t≤T

X

|ξ⁰|≤K

ˆ

u₀(ξ+ξ⁰)e^{i(x.(ξ0+ξ0)+tϕ(ξ0+ξ0))}

4

dxdt

= Z

0≤x1,...,xn≤2π, 0≤t≤T

X

|ξ⁰|≤K

e^{i((x+t∇ϕ(ξ0)).ξ0+tψξ0(ξ0))}wˆ₀(ξ⁰)

4

dxdt

=kP≤K1e^{itψξ0(∇/i)}w₀(x+t∇ϕ(ξ0))k⁴_L4(I×Tⁿ), whereψξ₀(ξ⁰) =ϕ(ξ0+ξ⁰)−ϕ(ξ0)− ∇ϕ(ξ0).ξ⁰.

After breaking kP_≤Ke^{itψξ0(∇/i)}w0k_L4

t,x(I×Tⁿ) ≤ P

1≤L≤KkPLe^{itψξ0(∇/i)}w0k_L4, the single expressions are amenable to Proposition 3.2.1. Indeed, the size of the moduli of the eigenvalues ofD²ψ_ξ0 are approximately independent of the frequen-cies.

Hence,

kPLe^{itψξ0(∇/i)}w0k_L4

t,x(I×Tⁿ).^ε,CϕL^s(n,k)+εkPLw0k_L2, and from carrying out the sum and the relation ofu0and w0, we find

kP≤Ke^{itψξ0(∇/i)}w₀kL⁴(I×Tⁿ).^ε,ϕK^s(n,k)+εkRK1u₀kL². The claim follows from almost orthogonality, i.e.,

X

K₁

kRK₁u0k²_L2

!1/2

.kPNu0k_L2.

This bilinear improvement can also stem from transversality: Write

|∇ϕ(ξ1)± ∇ϕ(ξ2)| ∼N^α, whenever|ξ1| ∼K, |ξ2| ∼N. (T^α) The corresponding short-time estimate from Section 3.1 is sufficient to derive an L²_t,x-estimate for finite times by gluing together the short time intervals:

Proposition 3.2.4. Let α > 0, K N, K, N ∈ 2^N and suppose that ϕ satisfies (T^α). Then, we find the following estimate to hold:

kPNe^{±itϕ(∇/i)}u0PKe^{±itϕ(∇/i)}v0k_L²

t,x(I×T).^ϕ|I|^1/2kPNu0kL²kPKv0kL² (3.31) provided thatI⊆Ris a compact time interval with|I|&N^−α.

Proof. LetI=S

jIj,|Ij| ∼N^−α, where theIj are disjoint. Then, lhs(3.31)².X

Ij

kPNe^{±itϕ(∇/i)}u0PKe^{±itϕ(∇/i)}v0k²_L2 t,x(I_j×T)

.(#Ij)N^−αkPNu0k²_L2kPKv0k²_L2, and the claim follows from #Ij∼ |I|N^α.

Invoking Proposition 3.2.2 together with Propositions 3.2.3 or 3.2.4, the below theorem follows:

Theorem 3.2.5. Suppose that ϕ ∈ C²(Rⁿ,R) satisfies (E^k(Cϕ)). Then, there is s0(n, k)such that (3.28) is locally well-posed fors > s0(n, k).

Let n= 1 and suppose thatϕ satisfies (T^α). Then, there is s0 =s0(ϕ) such that (3.28)is locally well-posed for s > s0(ϕ).

We give examples: In one dimension we treat the fractional Schr¨odinger equation i∂tu+D^au =±|u|²u, (t, x)∈R×T,

u(0) =u0∈H^s(T), (3.32)

whereD= (−∆)^1/2.

Theorem 3.2.5 yields uniform local well-posedness fors > ^2−a₄ , 1< a <2, which is presumably sharp up to endpoints as discussed in [CHKL15], where the endpoint s=^2−a₄ was covered by resonance considerations.

For 0< a <1 varying the above arguments, we can also prove local well-posedness for s > ^2−a₄ , which was previously proved in [Din17] in the context of Strichartz estimates for fractional Schr¨odinger equations on compact manifolds.

Moreover, in Euclidean space fractional Schr¨odinger equations were considered in [HS15]. Key ingredient to well-posedness are linear and bilinear Strichartz esti-mates, which hold globally in time due to dispersive effects. On the circle we can reach the same regularity up to the endpoint like in [HS15].

It might well be the case that the linear Strichartz estimates are sharp in higher dimensions because they match the estimates from Euclidean space. However, satis-factory bilinearL²_t,x-Strichartz estimates appear to be beyond the above arguments and possibly require additional angular decompositions (cf. [CKS⁺08]).

We also discuss hyperbolic Schr¨odinger equations. The well-posedness result from [Wan13, GT12] is recovered for the hyperbolic nonlinear Schr¨odinger equation in two dimensions, which is known to be sharp up to endpoints.

Generalizing the example probing sharpness to higher dimensions indicates that there is only a significant difference between hyperbolic and elliptic Schr¨odinger equations in low dimensions.

For hyperbolic phase functions, Theorem 3.2.5 recovers the results from [Wan13, GT12], where essentially sharp local well-posedness of

i∂_tu+ (∂_x²

1−∂_x²

2)u =±|u|²u, (t, x)∈R×T²,

u(0) =u₀∈H^s(T²), (3.33)

was proved fors > 1/2. Notably, due to subcriticality of the L⁴_t,x-Strichartz esti-mate, already for the hyperbolic equations

i∂tu+ (∂_x²₁−∂_x²₂+∂_x²₃)u =±|u|²u, (t, x)∈R×T³,

u(0) =u₀, (3.34)

and

i∂tu+ (∂_x²₁−∂²_x2+∂_x²₃−∂²_x4)u =±|u|²u, (t, x)∈R×T⁴,

u(0) =u0, (3.35)

the (essentially sharp) Strichartz estimates yield the same well-posedness results as for the elliptic counterparts:

Firstly, recall the counterexample from [Wan13], which showedC³-ill-posedness of (3.33) fors <1/2. As initial data consider

φN(x) =N^−1/2 X

|k|≤N

e^ikx1e^−ikx2,

which satisfieskφ_Nk_Hs ∼N^s andS[φ_N](t) :=e^{it(∂x21−∂x22)}φ_N =φ_N. This implies

Z T 0

|S[φN](s)|²S[φN](s)ds _Hs

=Tk|φN|²φNkH^s&T N^1+s. For details on this estimate, see [Wan13].

The validity of the estimate

Z T 0

|S[φN](s)|²S[φN](s)ds _Hs

.kφNk³_Hs (T .1) requiress≥1/2.

The same counterexample shows thats≥1/2 is required forC³-well-posedness of (3.34). This regularity is reached up to the endpoint in Theorem 3.2.5.

When considering (3.35), we modify the above example to φN(x) =N⁻¹ X

|k1|,|k2|≤N

e^ik1x1e^−ik1x2e^ik2x3e^−ik2x4, which again satisfieskφNkH^s ∼N^s.

Carrying out the estimate for the first Picard iterate with the necessary modifica-tions yields

Z T 0

|S[φN](s)|²S[φN](s)ds _Hs

=Tk|φN|²φNkH^s&T N^2+s,

which impliesC³-ill-posedness, unlesss≥1. This regularity is again obtained up to the endpoint in Theorem 3.2.5.

Apparently, for other hyperbolic Schr¨odinger equations, theL⁴_t,x-Strichartz es-timate also coincides with the ellipticL⁴_t,x-estimate, and modifications of the above counterexample yield lower thresholds than in the elliptic case. This indicates that the difference between elliptic and hyperbolic Schr¨odinger equations is only signifi-cant in lower dimensions.

Im Dokument Short-time Fourier transform restriction phenomena and applications to nonlinear dispersive equations (Seite 35-40)