We consider the dynamics of a supersolid state that is based on Leggett's idea of local phase flow around the atomic sites of the solid. This leads to a local microscopic superfluid velocity that is not Galilean and is defined only relative to the solid and a $T=0$ superfluid fraction ${\rho }_S/\rho <1$ (with $\rho$ being the macroscopic mass density). Landau's excitation-based argument for superfluids—that at temperature $T=0$ the normal fluid mass density ${\rho }_N$ is zero—should apply to supersolids. Together these two arguments imply that at low $T$ there is a missing mass density. We attribute this to a supersolid density ${\rho }_L$, with ${\rho }_L\equiv \rho -{\rho }_S-{\rho }_N$ and a momentum-bearing supersolid velocity $v_{Li}$. We derive the macroscopic dynamical equations for this system with three momentum-bearing macroscopic velocities $(v_{Li},v_{Ni},v_{Si})$ using Onsager's irreversible thermodynamics. For slow spatial variations, $v_{Si}$ accelerates along the negative gradient of the chemical potential or mass, like for an ordinary superfluid, and $v_{Ni}$ is subject to drag against $v_{Li}$ with a characteristic drag time $\tau$. There are three contributions to the acceleration of $v_{Li}$: drag against $v_{Ni}$, along the negative gradient of the chemical potential or mass, and due to the forces of lattice elasticity. For high-frequency motion relative to the drag time $\tau$, ${\rho }_L$ moves independently of ${\rho }_N$. The normal modes for such a system have a crossover frequency below which the normal fluid velocity $v_{Ni}$ is dragged into motion with $v_{Li}$ (so the theory reduces to the $v_{Si},v_{Ni}$ theory of Andreev and Lifshitz) and above which the normal fluid velocity $v_{Ni}$ is an independent variable. For an isotropic lattice we study the transverse and longitudinal responses. A frequency-dependent rotational inertia experiment is predicted to show, at frequencies that are low relative to $\tau$, that ${\rho }_L$ and ${\rho }_N$ move together but that at high frequencies ${\rho }_L$ moves much more than ${\rho }_N$.

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• Bose gases
• Order parameters
• Superfluidity