On well-posedness results for the cubic-quintic NLS on ${\mathbb{T}}^3$

We consider the periodic cubic-quintic nonlinear Schrödinger equation \begin{align}\label{cqnls_abstract} (i\partial_t +Δ)u=μ_1 |u|^2 u+μ_2 |u|^4 u\tag{CQNLS} \end{align} on the three-dimensional torus ${\mathbb{T}}^3$ with ${\mu }_1,{\mu }_2\in \mathbb{R}\setminus \{0\}$. As a first result, we establish the small data well-posedness of \eqref{cqnls_abstract} for arbitrarily given ${\mu }_1$ and ${\mu }_2$. By adapting the crucial perturbation arguments in \cite{zhang2006cauchy} to the periodic setting, we also prove that \eqref{cqnls_abstract} is always globally well-posed in $H^1({\mathbb{T}}^3)$ in the case ${\mu }_2>0$.