  In this paper both we establish the best constants for the Nash inequalities
on the standard unit sphere $\mathbb{S}^n$ of $\mathbb{R}^{n+1}$ and we give
answers on the existence of extremal functions on the corresponding problems.
Also we study the problem of the best constants in the case, where the data are
invariant under the action of the group $G=O(k)\times O(m)$, and we find the
best constants.
