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### Open Access Journals

DCDS

We consider the mass-critical generalized Korteweg--de Vries equation
$$(\partial_t + \partial_{xxx})u=\pm \partial_x(u^5)$$
for real-valued functions $u(t,x)$. We prove that if the global well-posedness and scattering conjecture for this equation failed,
then, conditional on a positive answer to the global well-posedness and scattering conjecture for the mass-critical nonlinear
Schrödinger equation $(-i\partial_t + \partial_{xx})u=\pm (|u|^4u)$, there exists a minimal-mass blowup solution to the
mass-critical generalized KdV equation which is almost periodic modulo the symmetries of the equation. Moreover, we can
guarantee that this minimal-mass blowup solution is either a self-similar solution, a soliton-like solution, or a double high-to-low
frequency cascade solution.

CPAA

We consider the Cauchy problem for a family of semilinear defocusing Schrödinger
equations with monomial nonlinearities in one space dimension. We establish global
well-posedness and scattering. Our analysis is based on a four-particle interaction Morawetz
estimate giving

*a priori*$L_{t,x}^8$ spacetime control on solutions.
DCDS

We consider the defocusing energy-critical nonlinear Schrödinger equation with inverse-square potential $iu_t = -Δ u + a|x|^{-2}u + |u|^4u$ in three space dimensions. We prove global well-posedness and scattering for $a > - \frac{1}{4} + \frac{1}{{25}}$. We also carry out the variational analysis needed to treat the focusing case.

DCDS

We consider the mass-subcritical NLS in dimensions $d≥ 3$ with radial initial data. In the defocusing case, we prove that any solution that remains bounded in the critical Sobolev space throughout its lifespan must be global and scatter. In the focusing case, we prove the existence of a threshold solution that has a compact flow.

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