The focus of this work is on the analysis of the Westervelt equation modeling nonlinear propagation of high intensity ultrasound, in the practically relevant setting of a truncated computational domain with absorbing boundary conditions. We especially consider the zero and first order nonlinear absorbing boundary conditions devised in [38] in one and two space dimensions. As a matter of fact, the energy identities and estimates presented here were crucial for designing these absorbing boundary conditions in such a way that the desired energy dissipation through the boundary is guaranteed. Under the hypothesis of small initial data, we establish local well-posedness and provide higher order energy estimates, that we expect to be of additional use in boundary control and stabilization.