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### Open Access Journals

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 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.

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.

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