Well-posedness and Ill-posedness for the cubic fractional Schrödinger equations

Abstract

We study the low regularity well-posedness of the 1-dimensional cubic nonlinear fractional Schrödinger equations with Lévy indices $1<\alpha <2$. We consider both non-periodic and periodic cases, and prove that the Cauchy problems are locally well-posed in $H^s$ for $s\ge \frac{2-\alpha }{4}$. This is shown via a trilinear estimate in Bourgain's $X^{s,b}$ space. We also show that non-periodic equations are ill-posed in $H^s$ for $\frac{2-3\alpha }{4(\alpha +1)}<s<\frac{2-\alpha }{4}$ in the sense that the flow map is not locally uniformly continuous.

References in corpus (4)

• Instability of the periodic nonlinear Schrodinger equation
• Nonlinear fractional Schrödinger equations in one dimension
• Existence and Uniqueness theory for the fractional Schrödinger equation on the torus
• Profile decompositions and Blowup phenomena of mass critical fractional Schrödinger equations

Cited by in corpus (3)

• Existence and Uniqueness theory for the fractional Schrödinger equation on the torus
• On Fractional Schrodinger Equations in sobolev spaces
• A new class of Traveling Solitons for cubic Fractional Nonlinear Schrodinger equations