While bright, blue, compact galaxies are common at $rm z sim 1$, they are relatively rare in the local universe, and their evolutionary paths are uncertain. We have obtained resolved H I observations of nine $rm z sim 0$ luminous compact blue galaxies (LCBGs) using the Giant Metrewave Radio Telescope and Very Large Array in order to measure their kinematic and dynamical properties and better constrain their evolutionary possibilities. We find that the LCBGs in our sample are rotating galaxies that tend to have nearby companions, relatively high central velocity dispersions, and can have disturbed velocity fields. We calculate rotation velocities for each galaxy by measuring half of the velocity gradient along their major axes and correcting for inclination using axis ratios derived from SDSS images of each galaxy. We compare our measurements to those previously made with single dishes and find that single dish measurements tend to overestimate LCBGs rotation velocities and H I masses. We also compare the ratio of LCBGs rotation velocities and velocity dispersions to those of other types of galaxies and find that LCBGs are strongly rotationally supported at large radii, similar to other disk galaxies, though within their half-light radii the $rm V_{rot}/ sigma$ values of their H I are comparable to stellar $rm V_{rot}/ sigma$ values of dwarf elliptical galaxies. We find that LCBGs disks on average are gravitationally stable, though conditions may be conducive to local gravitational instabilities at the largest radii. Such instabilities could lead to the formation of star-forming gas clumps in the disk, resulting eventually in a small central bulge or bar.
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