On Scattering for the Defocusing Quintic Nonlinear Schrödinger Equation on the Two-Dimensional Cylinder

Cheng, Xing; Guo, Zihua; Zhao, Zehua

doi:10.1137/19m1270586

Cited by 32 publications

(38 citation statements)

References 49 publications

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“…In Section 2, we will generalize (1.7) and create the 2-particle or interaction Morawetz estimate (Theorem 2.1) for the evolution of mesoscopic interactions. This estimate has analogies to the interaction Morawetz estimate for non linear Schrodinger's equation [4,5,6,18] and has physical interpretations. This estimate implies that, along the evolution of solutions to the equation (1.1) and as the time increases, only particles with identical velocities can remain within an arbitrary fixed distance of one another.…”

Section: Introductionmentioning

confidence: 73%

Kinetic interaction Morawetz and concentration estimates

Moini¹

2021

Preprint

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In this paper we will develop the kinetic interaction Morawetz estimate for mesoscopic evolutions. Furthermore, we will find a new a-priori bound and introduce the notion of interaction uncertainty. We will demonstrate that interaction uncertainty goes to infinity with time and that the interactions, averaged over time, concentrate within a specific collection of blind cones with an arbitrarily small apex angle. These results are solely based on conservation laws and thereby are all true for the Boltzmann equation and its different variants. Contents 1 Introduction 2 Interaction Morawetz estimate 3 New a priori bound and the interaction uncertainty 4 Concentration of interactions

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Section: Introductionmentioning

confidence: 73%

Kinetic interaction Morawetz and concentration estimates

Moini¹

2021

Preprint

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“…For dispersive models rather than 4NLS on Euclidean spaces (with scattering behavior), similar method may be applied to obtain the nonlinear decay property (i.e. the nonlinear solutions enjoy the same pointwise decay property as the linear solutions), such as, higher order (more than four) NLS, 4NLS on waveguide manifolds (see [31] for a recent result), NLS on waveguide manifolds (see [11,15,32] for example), NLS with partial harmonic potentials (see [1,3,12]), resonant system (see [4,30]), nonlinear wave equations (see [29]), Klein-Gordon equation (see [29]). We did not list them explicitly.…”

Section: Further Remarksmentioning

confidence: 99%

On the decay property of the cubic fourth-order Schrödinger equation

Yu¹,

Yue²,

Zhao³

2022

Preprint

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In this short paper, we prove that the solution of the cubic fourth-order Schrödinger equation (4NLS) on R d (5 ≤ d ≤ 8) enjoys the same (pointwise) decay property as its linear solution does. This result is proved via a bootstrap argument based on the corresponding global result Pausader [24]. This result can be extended to more general dispersive equations (including some more 4NLS models) with scattering asymptotics.

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“…New techniques are needed including function spaces, profile decomposition, profile approximations and even resonant systems. See [4,18,62] for the NLS case.…”

Section: Further Remarksmentioning

confidence: 99%

“…We refer to [8,10,30] for some important Euclidean results. Moreover, we refer to [5,4,18,22,23,25,26,32,61,62,63,64] with regard to the torus and waveguide settings. We may roughly think of the waveguide case as the "intermediate point" between the Euclidean case and the torus case since the waveguide manifold is a product of Euclidean spaces and the tori.…”

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confidence: 99%

Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

Yu¹,

Yue²,

Zhao³

2021

Preprint

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In this paper, we study the well-posedness theory and the scattering asymptotics for fourth-order Schrödinger equations (4NLS) on waveguide manifolds (semiperiodic spaces) R d × T n , d ≥ 5, n = 1, 2, 3. The tori component T n can be generalized to n-dimensional compact manifolds M n . First, we modify Strichartz estimates for 4NLS on waveguide manifolds, with which we establish the well-posedness theory in proper function spaces via the standard contraction mapping method. Moreover, we prove the scattering asymptotics based on an interaction Morawetz-type estimate established for 4NLS on waveguides. At last, we discuss the higher dimensional analogue, the focusing scenario and give some further remarks on this research line. This result can be regarded as the waveguide analogue of Pausader [46,47,48] and the 4NLS analogue of Tzvetkov-Visciglia [58].

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On Scattering for the Defocusing Quintic Nonlinear Schrödinger Equation on the Two-Dimensional Cylinder

Cited by 32 publications

References 49 publications

Kinetic interaction Morawetz and concentration estimates

Kinetic interaction Morawetz and concentration estimates

On the decay property of the cubic fourth-order Schrödinger equation

Global Well-posedness and scattering for fourth-order Schrödinger equations on waveguide manifolds

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