Semi-kinetic and kinetic models are powerful tools for attacking a range of flow conditions involving collisionless plasmas and they have been used for more than two decades to model the polar wind. With this approach, a Maxwellian ion velocity distribution is generally adopted at the lower boundary of the collisionless regime and the resulting distribution is then calculated at higher altitudes by using Liouville's theorem and conservation of energy and angular momentum. However, the resulting velocity distribution and associated moments (density, drift velocity, temperature, etc.) display sharp discontinuities at the boundary, which quantitatively affects the solution at higher altitudes. Analytical and numerical analyses for the polar wind indicate that the adoption of a Maxwellian for an expanding collisionless plasma in the diverging geomagnetic field is inconsistent and causes the discontinuity in the kinetic solutions. A non-Maxwellian velocity distribution with a specific feature must be adopted at the lower boundary in order to get a continuous and consistent kinetic solution. ¿ American Geophysical Union 1991