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

Abstract

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.

On blow-up solutions to the focusing mass-critical nonlinear fractional Schrödinger equation

“…We have the following result due to [21] (see also [11]). Proposition 2.2 (Non-radial local theory [21,11]). Let s ∈ (0, 1)\{1/2} and α > 0 be such that…”

“…• For general data (see e.g. [6] or [11]): the following estimates hold for d ≥ 1 and s ∈ (0, 1)\{1/2},…”

“…In the last decade, the fractional nonlinear Schrödinger equation has attracted a lot of interest in mathematics, numerics and physics (see e.g. [1,2,3,7,8,4,6,5,9,11,14,15,13,20,18,21,25,23,24,40,42,46,41] and references therein). The equation (1.1) enjoys the scaling invariance u λ (t, x) = λ 2s α u(λ 2s t, λx), λ > 0.…”

“…Note that the unitary group e −it(−∆) s enjoys several types of Strichartz estimates (see e.g. [6] or [11] for Strichartz estimates with non-radial data; and [19], [27] or [4] for Strichartz estimates with radially symmetric data; and [12] or [5] for weighted Strichartz estimates). For non-radial data, these Strichartz estimates have a loss of derivatives.…”

“…This makes the study of local well-posedness more difficult and leads to a weak local theory comparing to the standard nonlinear Schrödinger equation (see e.g. [21] or [11]). One can remove the loss of derivatives in Strichartz estimates by considering radially symmetric initial data.…”