Growth of Sobolev norms for 2D cubic nonlinear Schrödinger equation with partial harmonic potential

In this paper, we study the $2$D cubic nonlinear Schrödinger equation (NLS) with the partial harmonic potential. First, we prove the local well-posedness in Bourgain spaces by establishing a key bilinear estimate associated with the partial harmonic oscillator. Then, we give the polynomial bound of the Sobolev norms for the solutions using the method of the Planchon, Tzvetkov, and Visciglia.