Sharp dispersive estimates for an anisotropic linear operator group

Applied Mathematics Letters 103 (2020) 106212

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Applied Mathematics Letters www.elsevier.com/locate/aml

Sharp dispersive estimates for an anisotropic linear operator group Jinyi Sun a,b ,∗, Boling Guo a , Minghua Yang c a

Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China College of Mathematics and Statistics, Northwest Normal University, Lanzhou 730070, PR China Department of Mathematics, Jiangxi University of Finance and Economics, Nanchang 330032, PR China b c

article

info

abstract By exploiting the stationary phase methods, we obtain the sharp dispersive estimates of linear operators group {eR1 t }t∈R , which is related to the linearize model of the dispersive surface quasi-geostrophic equations. Here R1 is the Riesz ξ1 transform with symbol −i |ξ| in R2 . Our main results improve the related ones obtained in Elgindi and Widmayer (2015), Wan and Chen (2016), Wan and Chen (2017). © 2020 Elsevier Ltd. All rights reserved.

Article history: Received 13 November 2019 Received in revised form 28 December 2019 Accepted 28 December 2019 Available online 7 January 2020 Keywords: Linear operator group Dispersive estimate

1. Introduction In this paper, we aim to establish the sharp dispersive estimates of the linear operators group {eR1 t }t∈R related to the following linear equation ∂t θ − R1 θ = 0,

for (x, t) ∈ R2 × R,

(1.1)

ξ1 , and the operator eR1 t is defined via Fourier transform where R1 is the Riesz transform with symbol −i |ξ|

e

R1 t

ξ

∫ f :=

e

1 ix·ξ−it |ξ|

R2

fˆ(ξ)dξ,

for x ∈ R2 and t ∈ R.

(1.2)

Such estimates are the key to study the dispersive surface quasi-geostrophic equations, which have been proposed as a simple model describing the evolution of a surface buoyancy involved with investigating waveturbulence interactions, cf. [1]. More precisely, from the view point of the hyperbolic systems, the dispersive and the inviscid surface quasi-geostrophic equations have the same energy estimates, thus solutions exist a priori only for a time span inversely proportional to the size of the initial data. A finer understanding of the dispersive properties allows us to improve this time span. ∗ Corresponding author at: Institute of Applied Physics and Computational Mathematics, Beijing 100088, PR China. E-mail addresses: [email protected] (J. Sun), [email protected] (B. Guo), [email protected] (M. Yang).

https://doi.org/10.1016/j.aml.2020.106212 0893-9659/© 2020 Elsevier Ltd. All rights reserved.

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J. Sun, B. Guo and M. Yang / Applied Mathematics Letters 103 (2020) 106212

By the stationary phase methods, Elgindi and Widmayer [1] proved the following dispersive estimate of semigroup {eR1 t }t≥0 : 1 ∥eR1 t f ∥L∞ ≤ Ct− 2 ∥f ∥B˙ 2 . (1.3) 1,1

1 2

They also pointed out that the decay rate cannot be improved to a larger one. Wan and Chen [2,3] further verified that 2 3 4 ∥eR1 t f ∥L∞ ≤ C(1 + t)−( p − 2 ) ∥f ∥ 2 for 1 ≤ p < (1.4) p 3 ˙ B p,1

and

1

1

∥eR1 t f ∥Lp′ ≤ Ct−( p − 2 ) ∥f ∥

for 1 < p ≤ 2,

4 −2

p B˙ p,2

(1.5)

respectively. However, we notice that the decay rate p2 − 23 in (1.4) is not optimal for 1 < p < 43 and the estimations (1.3) and (1.5) do not perform a careful control when t → 0. Inspired by above mentioned results, we are going to establish the sharp dispersive estimate of group ξ1 {eR1 t }t∈R in more general cases. Denote that p(ξ) := |ξ| . Since p(ξ) is homogeneous of degree 0, we note that the matter is reduced to the frequency localized case following from the Littlewood–Paley decomposition and scaling. Now, let us just assume that suppfˆ ⊂ {2−1 ≤ |ξ| ≤ 2}. It is well known that the decay estimate of eR1 t is determined by the number of non-vanishing principle curvature of the surface Σ := {(ξ, τ ) ∈ R2 × R | τ = p(ξ), 2−1 ≤ |ξ| ≤ 2}. More specifically, from the Littman theorem [4] (see also Greenleaf [5]), the decay rate is determined by the rank of Hessian matrix of p(ξ) on {2−1 ≤ |ξ| ≤ 2}. Our first result reads as follows. Theorem 1.1. that

Let f ∈ S (R2 ) with suppfˆ ⊂ {2−1 ≤ |ξ| ≤ 2}. Then there exists a constant C > 0 such 1

∥eR1 t f ∥L∞ ≤ C(1 + |t|)− 2 ∥f ∥L1 for all t ∈ R. Also, the decay rate

1 2

(1.6)

is sharp.

By applying the Riesz–Thorin interpolation theorem between (1.6) with the L2 − L2 estimate and using the scaling argument, we obtain the following dispersive estimate associated to the frequency-dyadic decomposition. Corollary 1.2.

Let 1 ≤ p ≤ 2. There exists a constant C = C(p) > 0 such that 1

1

2

∥eR1 t ∆j f ∥Lp′ ≤ C(1 + |t|)−( p − 2 ) 22j( p −1) ∥∆j f ∥Lp

(1.7)

for all t ∈ R, j ∈ Z and f ∈ S ′ (R2 ), where p1 + p1′ = 1 and ∆j is frequency localization operator defined in (2.1). Moreover, 1 1 ∥eR1 t f ∥B˙ s ≤ C(1 + |t|)−( p − 2 ) ∥f ∥ s+2( 2 −1) (1.8) p p′ ,r ˙ Bp,r

2 −1) s+2( p

for all t ∈ R and f ∈ B˙ p,r

(R2 ) with s ∈ R and 1 ≤ r ≤ ∞, where

1 p′

+

1 p

= 1.

0 Remark 1.3. Taking s = 0, p = 1 and r = 1, and combining with the fact that B˙ ∞,1 (R2 ) ↪→ L∞ (R2 ), (1.8) yields that 1 ∥eR1 t f ∥L∞ ≤ ∥eR1 t f ∥B˙ 0 ≤ C(1 + |t|)− 2 ∥f ∥B˙ 2 . ∞,1

1,1

J. Sun, B. Guo and M. Yang / Applied Mathematics Letters 103 (2020) 106212

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′ On the other hand, taking s = 0 and r = 2, and combining with the fact that B˙ p0′ ,2 (R2 ) ↪→ Lp (R2 ) for 2 ≤ p′ < ∞, (1.8) implies that 1

∥eR1 t f ∥Lp′ ≤ ∥eR1 t f ∥B˙ 0

p′ ,2

1

≤ C(1 + |t|)−( p − 2 ) ∥f ∥

4 −2

p B˙ p,2

,

for 1 < p ≤ 2,

where p1 + p1′ = 1. Therefore, Corollary 1.2 improves (1.3) given by Elgindi and Widmayer [1] and (1.5) given by Wan and Chen [3]. Particularly, the estimate (1.8) has a more careful control when t → 0. 2

′ Remark 1.4. Taking s = p2′ and r = 1, and combining with the fact that B˙ pp′ ,1 (R2 ) ↪→ L∞ (R2 ), (1.8) gives that 1 1 ∥eR1 t f ∥L∞ ≤ ∥eR1 t f ∥ 2 ≤ C(1 + |t|)−( p − 2 ) ∥f ∥ 2 , for 1 ≤ p ≤ 2, ′

p B˙ p,1

p B˙ ′

p ,1

where p1 + p1′ = 1. It is easy to check that p1 − 12 > p2 − 32 for p > 1. Therefore, Corollary 1.2 improves (1.4) given by Wan and Chen [2] in the sense that it provides a larger decay rate p1 − 12 and an enlarged index range 1 ≤ p ≤ 2. The rest of this paper is organized as follows. In Section 2, we collect some basic facts on Littlewood– Paley theory and Besov spaces. Finally, by exploiting the stationary phase methods, we investigate the sharp dispersive estimate of linear operators group {eR1 t }t∈R . 2. Preliminaries Let S (R2 ) be the Schwartz space of smooth functions over R2 , and let S ′ (R2 ) be the space of tempered distributions. First, we recall the homogeneous Littlewood–Paley decomposition. Let φ, ψ ∈ S (R2 ) be two radial functions such that their Fourier transforms φˆ and ψˆ satisfy the following properties: 4 supp φˆ ⊂ B := {ξ ∈ R2 : |ξ| ≤ }, 3 8 3 supp ψˆ ⊂ C := {ξ ∈ R2 : ≤ |ξ| ≤ } 4 3 and ∑ ˆ −j ξ) = 1 for all ξ ∈ R2 \ {0}. ψ(2 j∈Z

Let φj (x) := 2 φ(2 x) and ψj (x) := 22j ψ(2j x) for j ∈ Z. We define by ∆j the following frequency localization operator in S ′ (R2 ): 2j

j

∆j f := ψj ∗ f

for j ∈ Z and f ∈ S ′ (R2 ).

(2.1)

Define Sh′ (R2 ) := S ′ (R2 )/P[R2 ], where P[R2 ] denotes the linear space of polynomials on R2 . s Now, we introduce the definitions of homogeneous Besov space B˙ p,r (R2 ). Definition 2.1. r = ∞)

Let s ∈ R and 1 ≤ p, r ≤ ∞, and let u ∈ Sh′ (R2 ), we set (with the usual convection if ∥u∥B˙ p,r := s

(∑

2jsr ∥∆j u∥rLp

) r1

.

j∈Z

• For s <

2 p

(or s = p2 , if r = 1), we define s B˙ p,r (R2 ) := {u ∈ Sh′ (R2 )|∥u∥B˙ p,r < ∞}; s

s • If k ∈ N, p2 + k ≤ s < p2 + k + 1 (or s = p2 + k + 1, if r = 1), then B˙ p,r (R2 ) is defined as the subset of ′ 2 δ s−k distributions u ∈ Sh (R ) such that ∂ u ∈ B˙ p,r whenever |δ| = k.

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J. Sun, B. Guo and M. Yang / Applied Mathematics Letters 103 (2020) 106212

3. Proofs of Theorem 1.1 and Corollary 1.2 In this section, we give the proof of Theorem 1.1 and Corollary 1.2. Let Ψ ∈ S (R2 ) satisfy suppΨ ⊂ {2−2 ≤ |ξ| ≤ 22 } and Ψ = 1 on {2−1 ≤ |ξ| ≤ 2}. Theorem 1.1 follows from the following results. Proposition 3.1.

There exists a constant C > 0 such that ⏐∫ ⏐ ξ1 1 ix·ξ−it |ξ| ⏐ ⏐ e Ψ (ξ)dξ ⏐ ≤ C(1 + |t|)− 2 ⏐

(3.1)

R2

for all (x, t) ∈ R2 × R. Also, the decay rate

1 2

is optimal.

In order to prove Proposition 3.1, we recall the following two classical results. The first one, as a consequence of stationary phase method, is due to Littman [4] (see also Lemma 3.1 in [6]), and the second one is due to Keel and Tao [7], which are useful for showing the sharpness of the decay rate 12 . Lemma 3.2 ([4,6]). Let Ψ ∈ C0∞ (Rd ), and let p be a real-value C ∞ function on the support of Ψ . Suppose that the rank of Hp (ξ) is at least ρ > 0 on the support of Ψ . Then, there exists a positive constant C = C(d, Ψ , p) such that ⏐ ⏐∫ ρ ⏐ ⏐ eix·ξ−itp(ξ) Ψ (ξ)dξ ⏐ ≤ C(1 + |t|)− 2 (3.2) ⏐ R2

holds for all (x, t) ∈ R2 × R. Lemma 3.3 (Keel and Tao [7]). Let {U (t)}t∈R be a family of operators. For all t, s ∈ R, if ∥U (s)(U (t))∗ f ∥L∞ ≲ (1 + |t − s|)−σ ∥f ∥L1 , and ∥U (s)(U (t))∗ f ∥L2 ≲ ∥f ∥L2 with some σ > 0, then ∥U (t)f ∥Lq Lrx ≲ ∥f ∥L2 t

holds for all 2 ≤ q, r ≤ ∞ with (q, r, σ) ̸= (2, ∞, 1) satisfying

1 q

≤ σ( 12 − 1r ).

Proof of Proposition 3.1. A direct computation gives the Hessian matrix of p(ξ) as ) ( ξ2 (2ξ12 − ξ22 ) −3ξ1 ξ22 −5 Hp (ξ) = |ξ| ξ2 (2ξ12 − ξ22 ) −ξ1 (2ξ12 − ξ22 )

(3.3)

ξ2

2 with determinant det Hp (ξ) = − |ξ|−6 . Obviously, for the stationary points, the dispersive relation p(ξ) has degeneracy along {ξ2 = 0}. Moreover, it is easy to see that the rank of Hessian matrix Hp (ξ) is 2 on {2−2 ≤ |ξ| ≤ 22 }\{ξ2 = 0} and is 1 on {2−2 ≤ |ξ| ≤ 22 } ∩ {ξ2 = 0}. Therefore, applying Lemma 3.2 gives ⏐∫ ⏐ ξ1 1 ix·ξ−it |ξ| ⏐ ⏐ e Ψ (ξ)dξ ⏐ ≤ C(1 + |t|)− 2 . ⏐

R2

Next, we verify the decay rate cannot be improved by any number larger than 12 . To this end, we claim that the following necessary range result holds, and its proof will be shown later.

J. Sun, B. Guo and M. Yang / Applied Mathematics Letters 103 (2020) 106212

Proposition 3.4.

5

Let 2 ≤ q, r ≤ ∞ with (q, r) ̸= (2, ∞). Then the necessary condition such that ∥eR1 t f ∥Lq Lrx ≲ ∥f ∥L2 t

is

1 1 1 1 ≤ ( − ). q 2 2 r

(3.4)

However, on the other hand, applying Plancherel’s Theorem gives that ∥eR1 t f ∥L2 = ∥f ∥L2 , Thus, if (1.6) holds for a decay rate σ > ∥eR1 t f ∥Lq Lrx ≲ ∥f ∥L2 , if t

1 2,

for all t ∈ R.

by the standard argument and Lemma 3.3, we have

1 1 1 ≤ σ( − ), q 2 r

which leads to a contradiction with Proposition 3.4. This thus gives the sharpness of decay rate completes the proof of Proposition 3.1. □ We now give the proof of Proposition 3.4.

1 2,

and

Proof of Proposition 3.4. For 0 < a ≪ 1 and N ≫ 1, we define 1 ≤ ξ1 ≤ 1, a ≤ ξ2 ≤ 2a} 2

Ea := {ξ = (ξ1 , ξ2 ) ∈ R2 : and A := {(x, t) ∈ R2 × R : |x1 | ≤

1 1 1 , |x2 | ≤ , |t| ≤ }. N Na N a2

It is easy to check that ξ1 ξ1 − |ξ| ξ22 −1= =− = O(a2 ), |ξ| |ξ| |ξ|(ξ1 + |ξ|) and further that

⏐ ⏐ ξ1 1 ⏐ ⏐ − 1)⏐ ≲ , ⏐x · ξ − t( |ξ| N

for ξ ∈ Ea ,

for ξ ∈ Ea and (x, t) ∈ A.

Define fa by fˆa (ξ) := χEa . Since Ea ⊂ {2−1 ≤ |ξ| ≤ 2}, we have ∫ ⏐ ⏐ ξ1 ix·ξ−it( |ξ| −1) ⏐ ⏐ |eR1 t fa | = ⏐e−it e dξ ⏐ Ea ⏐∫ ( )) ⏐⏐ ( ξ1 ⏐ − 1 dξ ⏐ ≳⏐ cos x · ξ − t |ξ| Ea ≳ |Ea | ∼ a for (x, t) ∈ A with a sufficiently large N . The estimate ∥eR1 t fa ∥Lq Lrx ≲ ∥fa ∥L2 t

implies that 2

1

a1− q − r 1+2 q r

1

≤ ∥eR1 t fa ∥Lq Lrx (A) ≤ ∥eR1 t fa ∥Lq Lrx ≲ ∥fa ∥L2 ∼ a 2 . t

t

N For a sufficiently large N , and then letting a → 0 gives the inequality (3.4). This completes the proof. □

J. Sun, B. Guo and M. Yang / Applied Mathematics Letters 103 (2020) 106212

6

Proof of Theorem 1.1. Using Proposition 3.1 completes the proof of Theorem 1.1 easily. We omit the detail here. □ Finally, we shall show the proof of Corollary 1.2. Proof of Corollary 1.2. From the Plancherel’s Theorem, it is easy to see that ∥eR1 t ∆0 f ∥L2 ≲ ∥∆0 f ∥L2 ,

(3.5)

for all t ∈ R, where ∆0 is defined as in (2.1). Due to suppψ ⊂ {2−1 ≤ |ξ| ≤ 2}, from Theorem 1.1, we have 1

∥eR1 t ∆0 f ∥L∞ ≲ (1 + |t|)− 2 ∥∆0 f ∥L1 ,

(3.6)

for all t ∈ R. Interpolating (3.5) with (3.6) gives 1

1

∥eR1 t ∆0 f ∥Lp′ ≲ (1 + |t|)−( p − 2 ) ∥∆0 f ∥Lp ,

(3.7)

ξ1 is homogeneous with degree 0, scaling gives for all t ∈ R and p ∈ [1, 2], where p1 + p1′ = 1. Moreover, since |ξ| that ξ1 [ −it ξ1 ] [ ] (3.8) ˆ −j ξ)fˆ(ξ) (x) = F −1 e−it |ξ| ψ(ξ) ˆ fˆj (ξ) (2j x), F −1 e |ξ| ψ(2

where fj (x) := f (2−j x). From the definition of ∆j in (2.1), we see that 2j

− ′ p

∥eR1 t ∆j f ∥Lp′ = 2

1

1

2

∥eR1 t ∆0 fj ∥Lp′ ≤ C(1 + |t|)−( p − 2 ) 22j( p −1) ∥∆j f ∥Lp .

(3.9)

Multiplying both sides of (3.9) by 2sj , and then taking the ℓq (Z)-norm, we complete the proof of Corollary 1.2. □ CRediT authorship contribution statement Jinyi Sun: Conceptualization, Methodology, Investigation, Writing - original draft, Writing - review & editing, Software, Funding acquisition, Visualization. Boling Guo: Supervision, Conceptualization, Methodology, Validation, Project administration, Funding acquisition. Minghua Yang: Formal analysis, Writing - review & editing, Data curation, Software. Acknowledgment This work is partially supported by the National Natural Science Foundation of China (Grant No. 11571381), the China Postdoctoral Science Foundation (Grant No. 2019M660555), the Natural Science Foundation of Gansu Province, China for Young Scholars (Grant No. 18JR3RA102), the Innovation capacity improvement project for colleges and universities of Gansu Province, China (Grant No. 2019A-011). References [1] T.M. Elgindi, K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684. [2] R. Wan, J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (4) (2016) 22, Art. 104, 22 pp. [3] R. Wan, J. Chen, Global well-posedness of smooth solution to the supercritical SQG equation with large dispersive forcing and small viscosity, Nonlinear Anal. 164 (2017) 54–66. [4] W. Littman, Fourier transforms of surface-carried measures and differentiability of surface averages, Bull. Amer. Math. Soc. 69 (1963) 766–770. [5] A. Greenleaf, Principal curvature and harmonic analysis, Indiana Univ. Math. J. 30 (1981) 519–537. [6] B. Wang, Z. Huo, C. Hao, Z. Guo, Harmonic Analysis Method for Nonlinear Evolution Equations, Vol. I, World Scientific Publishing Co. Pte. Ltd., Hackensack, NJ, 2011. [7] M. Keel, T. Tao, Endpoint Strichartz estimates, Amer. J. Math. 120 (1998) 955–980.