On projectors to subspaces of vector valued functions subject to conditions of the divergence free type

We study operators projecting a vector valued function $v\in W^{1,2}(\Omega,\mathbb Rd)$ to subspaces formed by the condition that the divergence is orthogonal to a certain amount (finite or infinite) of test functions. The condition that divergence is equal to zero almost everywhere presents the first (narrowest) limit case while the integral condition of zero mean divergence generates the other (widest) case. Estimates of the distance between $v$ and the respective projection $\mathsf P_\mathbb Sv$ on such a subspace are important for analysis of various mathematical models related to incompressible media problems (especially in the context of a posteriori error estimates, see [15–17]. We establish different forms of such estimates, which contain only local constants associated with the stability (LBB) inequalities for subdomains. The approach developed in the paper also yields two sided bounds of the inf–sup (LBB) constant.