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In this paper, we study the $2$D cubic nonlinear Schr\"odinger equation (NLS)with the partial harmonic potential. First, we prove the local well-posednessin Bourgain spaces by establishing a key bilinear estimate associated with thepartial harmonic oscillator. Then, we give the polynomial bound of the Sobolevnorms for the solutions using the method of the Planchon, Tzvetkov, andVisciglia.

Growth of Sobolev norms for 2D cubic nonlinear Schr\"odinger equation with partial harmonic potential