Local in Time Regularity Properties of the Navier-Stokes Equations

Let u be a weak solution of the Navier-Stokes equations in a smooth domain Ω ⊆ ℝ3 and a time interval [0,T), 0 < T ≤ ∞, with initial value u0, and vanishing external force. As is well known, global regularity of u for general u0 is an unsolved problem unless we pose additional assumptions on u0 or on the solution u itself such as Serrin's condition ${\Vert \mathrm{u}\Vert }_{{\mathrm{L}}^{\mathrm{s}}(0,\mathrm{T};{\mathrm{L}}^{\mathrm{q}}\left(\mathrm{\Omega }\right))}