Remarks on some dispersive estimates

“…Theorem 2.1 (Strichartz estimates [12]). Let d ≥ 1, σ ∈ (0, ∞)\{1}, γ ∈ R and a (weak) solution to the linear fractional Schrödinger equation, namely…”

Section: Strichartz Estimatesmentioning

confidence: 99%

Well-Posedness of Nonlinear Fractional Schr\"odinger and Wave Equations in Sobolev Spaces

Dinh¹

2018

Int. J. of Appl. Math.

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We prove the well-posed results in sub-critical and critical cases for the pure power-type nonlinear fractional Schrödinger equations on R d . These results extend the previous ones in [22] for σ ≥ 2. This covers the well-known result for the Schrödinger equation σ = 2 given in [4]. In the case σ ∈ (0, 2)\{1}, we give the local well-posedness in sub-critical case for all exponent ν > 1 in contrast of ones in [22]. This also generalizes the ones of [11] when d = 1 and of [17] when d ≥ 2 where the authors considered the cubic fractional Schrödinger equation with σ ∈ (1, 2). We also give the global existence in energy space under some assumptions. We finally prove the local well-posedness in sub-critical and critical cases for the pure power-type nonlinear fractional wave equations.

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“…Indeed, these bounds play a crucial role in the poof of the main theorem (see (6.1)). for admissible (q, r) (see [14] for α = 1, and [4] for 0 < α < 1 with α = 1 2 ). By local well-posedness, the solution u(t) satisfies u(t) As an application, we obtain the following desired time-averaged L ∞ bound.…”

Section: Proof Of the Main Theoremmentioning

confidence: 99%

Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

Hong¹,

Yang²

2019

SIAM J. Math. Anal.

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We consider discrete nonlinear Schrödinger equations (DNLS) on the lattice hZ d whose linear part is determined by the discrete Laplacian which accounts only for nearest neighbor interactions, or by its fractional power. We show that in the continuum limit h → 0, solutions to DNLS converge strongly in L 2 to those to the corresponding continuum equations, but a precise rate of convergence is also calculated. In particular cases, this result improves weak convergence in Kirkpatrick, Lenzmann and Staffilani [17]. Our proof is based on a suitable adjustment of dispersive PDE techniques to a discrete setting. Notably, we employ uniform-in-h Strichartz estimates for discrete linear Schrödinger equations in [10], which quantitatively measure dispersive phenomena on the lattice. Our approach could be adapted to a more general setting like [17] as long as the desired Strichartz estimates are obtained.

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Remarks on some dispersive estimates

Cited by 83 publications

References 3 publications

On Hartree equations with derivatives

On Hartree equations with derivatives

Well-Posedness of Nonlinear Fractional Schr\"odinger and Wave Equations in Sobolev Spaces

Strong Convergence for Discrete Nonlinear Schrödinger equations in the Continuum Limit

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