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Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on Torus

  Published:2015-05-05
 Printed: Sep 2015
  • Seckin Demirbas,
    Department of Mathematics, University of Illinois, Urbana, IL 61801, U.S.A.
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Abstract

In a previous paper, we proved that $1$-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed in $H^s$ for $s\gt \frac{1-\alpha}{2}$ and globally well-posed for $s\gt \frac{10\alpha-1}{12}$. In this paper we define an invariant probability measure $\mu$ on $H^s$ for $s\lt \alpha-\frac{1}{2}$, so that for any $\epsilon\gt 0$ there is a set $\Omega\subset H^s$ such that $\mu(\Omega^c)\lt \epsilon$ and the equation is globally well-posed for initial data in $\Omega$. We see that this fills the gap between the local well-posedness and the global well-posedness range in almost sure sense for $\frac{1-\alpha}{2}\lt \alpha-\frac{1}{2}$, i.e. $\alpha\gt \frac{2}{3}$ in almost sure sense.
Keywords: NLS, fractional Schrodinger equation, almost sure global wellposedness NLS, fractional Schrodinger equation, almost sure global wellposedness
MSC Classifications: 35Q55 show english descriptions NLS-like equations (nonlinear Schrodinger) [See also 37K10] 35Q55 - NLS-like equations (nonlinear Schrodinger) [See also 37K10]
 

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