Lq-Estimates of the First-Order Derivatives of Solutions to the Nonstationary Stokes Problem

For a solution to the nonstationary Stokes problem in the half-space $$ \mathbb{R}_{ + }^3 $$ with the external force $$ f = \nabla \; \cdot \;F,\;F\; \in \;{L_q}\left( {\mathbb{R}_{ + }^3\; \times \;\left( {0,\;T} \right)} \right) $$, we establish the L _q -estimates for the first-order derivatives of the vector field of velocities and prove that the pressure does not belong to the space $$ {L_q}\left( {\mathbb{R}_{ + }^3\; \times \;\left( {0,\;T} \right)} \right) $$.