Log improvement of the Prodi-Serrin criteria for Navier-Stokes equations

where u is a vector-valued function representing the velocity of the fluid, and p is the pressure. Note that the pressure depends in a non local way on the velocity u. It can be seen as a Lagrange multiplier associated to the incompressible condition (2). The initial value problem of the above equation is endowed with the condition that u(0, ·) = u0 ∈ L (R). Leray [11] and Hopf [6] had already established the existence of global weak solutions for the Navier-Stokes equation. In particular, Leray introduced a notion of weak solutions for the Navier-Stokes equation, and proved that, for every given initial datum u0 ∈ L (R), there exists a global weak solution u ∈ L(0,∞;L(R)) ∩ L(0,∞; Ḣ(R)) verifying the Navier-Stokes equation in the sense of distribution. From that time on, much effort has been devoted to establish the global existence and uniqueness of smooth solutions to the Navier-Stokes equation. Different Criteria for regularity of the weak solutions have been proposed. The Prodi-Serrin conditions (see Serrin [16], Prodi [14], and [17]) states that any weak Leray-Hopf solution verifying u ∈ L(0,∞;L(R)) with 2/p + 3/q = 1, 2 ≤ p < ∞, is regular on (0,∞)×R. Notice that if p = q, this corresponds to u ∈ L((0,∞)×R). The limit case of L(0,∞;L(R)) has been solved very recently by L. Escauriaza,