The purpose of this paper is to study the nonstationary Navier-Stokes equations in \(\Omega\times(0,T)\): \[ {{\partial u}\over{\partial t}}=\Delta u-(u\cdot\nabla)u-\nabla n+f,\qquad \text{div u}=0 \] with the initial and Dirichlet condition \[ u(X,t)=0\quad \text{for }(X,t)\in\partial\Omega \times(0,T),\qquad u(X,0)=u_0(X),\quad X\in\Omega. \] The authors describe \(D(A)\), the domain of the Stokes operator \(A\), in terms of Sobolev spaces, and prove that the domain of \(A\) is contained in \(W_0^{1,p}(\Omega)\cap W^{3/2,2}(\Omega)\) for some \(p>3\). Certain \(L^\infty\)-estimates are also given. Moreover, a simple proof of area integral estimates for solutions of Stokes equations is provided.