A note on unique continuation for the Schrödinger equation

Lee, Sanghyuk; Seo, Ihyeok

doi:10.1016/j.jmaa.2011.11.067

Cited by 14 publications

(21 citation statements)

References 11 publications

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“…In the classical case α = 2, the Strichartz estimates originated by Strichartz [23] have been extensively studied by many authors ( [9,14,2,12,8,24,15,17,18,5,6,20]). Over the past several years, considerable attention has been paid to the fractional order where 1 < α < 2 in the radial case (see [21,11,13] and references therein).…”

Section: Introductionmentioning

confidence: 99%

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

Cho¹,

Koh²,

Seo³

2015

DCDS-A

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In this paper we obtain some new inhomogeneous Strichartz estimates for the fractional Schrödinger equation in the radial case. Then we apply them to the well-posedness theory for the equation i∂tu + |∇| α u = V (x, t)u, 1 < α < 2, with radialḢ γ initial data below L 2 and radial potentials V ∈ L r t L w x under the scaling-critical range α/r + n/w = α.2010 Mathematics Subject Classification. Primary: 35B45, 35A01; Secondary: 35Q40.

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Section: Introductionmentioning

confidence: 99%

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

Cho¹,

Koh²,

Seo³

2015

DCDS-A

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“…Observe that by taking H " L 2 , pB 0 , B 1 q " pL 2 , L 1 q, and U ptq " e it∆ , along with the Lorentz space embedding L r 1 Ă B θ " L r 1 ,2 (r " 2 1´θ ), then we recover the result in [14] with the addition of the case d " 2. Then for all q andq verifying dˆ1 2´1 r˙"…”

Section: Introductionmentioning

confidence: 55%

“…For (1.21), we note that L q,8 cannot be improved to a smaller Lorentz space L q,p with p ă 8. We prove this assertion in Section 3 (the corresponding claim for the Schrödinger equation and (1.10) was explained in [14]). For further remarks on Theorem 1.1 and its connection to other results in the literature, we refer the reader to Section 3.…”

Section: Introductionmentioning

confidence: 56%

Inhomogeneous Strichartz estimatesin some critical cases

Bez

Cunanan

Lee

2019

Proc. Amer. Math. Soc.

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Strong-type inhomogeneous Strichartz estimates are shown to be false for the wave equation outside the so-called acceptable region. On a critical line where the acceptability condition marginally fails, we prove substitute estimates with a weak-type norm in the temporal variable. We achieve this by establishing such weak-type inhomogeneous Strichartz estimates in an abstract setting. The application to the wave equation rests on a slightly stronger form of the standard dispersive estimate in terms of certain Besov spaces.

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“…(We refer the reader to [3,13,8] for other necessary conditions.) Compared with (3.1), we give here the following new necessary condition:…”

Section: Necessary Conditionsmentioning

confidence: 99%

Inhomogeneous Strichartz estimates for Schrödinger's equation

Koh

Seo

2016

Journal of Mathematical Analysis and Applications

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Abstract. Foschi and Vilela in their independent works ([3], [13]) showed that the range of (1/r, 1/ r) for which the inhomogeneous Strichartz estimateholds for some q, q is contained in the closed pentagon with vertices A, B, B ′ , P, P ′ except the points P, P ′ (see Figure 1). We obtain the estimate for the corner points P, P ′ . IntroductionIn this paper we consider the following Cauchy problem for the Schrödinger equation:where (x, t) ∈ R n × R, n ≥ 1. By Duhamel's principle, we have the solutionwhere e it∆ is the free Schrödinger propagator defined byThe Strichartz estimates for the solution play important roles in the study of well-posedness for nonlinear Schrödinger equations (cf. [1,12]). They actually consist of two parts, homogeneous (F = 0) and inhomogeneous (f = 0) part. The homogeneous Strichartz estimateholds if and only if (r, q) is admissible pair, that is, r, q ≥ 2, (n, r, q) = (2, ∞, 2) and n/r + 2/q = n/2 (see [11,4,9,6] and references therein). But determining the optimal range of (r, q) and ( r, q) for which the inhomogeneous Strichartz estimateholds is not completed yet when n ≥ 3. It was observed that this estimate is valid on a wider range than what is given by admissible pairs (r, q), ( r, q) (see [2], [5]).2010 Mathematics Subject Classification. Primary 35B45, 35Q40.

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A note on unique continuation for the Schrödinger equation

Cited by 14 publications

References 11 publications

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

On inhomogeneous Strichartz estimates for fractional Schrödinger equations and their applications

Inhomogeneous Strichartz estimatesin some critical cases

Inhomogeneous Strichartz estimates for Schrödinger's equation

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