L_3,∞-solutions of the Navier-Stokes equations and backward uniqueness

On smoothness of L3,?-solutions to the Navier?Stokes equations up to boundary

“…Moreover, using (4.10) and interpolation, we can show that 15) see details in the proof of (3.23) in [4]. Letting d = x 03 /2 for an arbitrary point x 0 ∈ R 3 + and using (4.2) and (4.15), we find…”

“…Let z 0 ∈ Q + (1/2) be a singular point. As it was shown in [4], Theorem 1.4, z 0 must belong to Γ(1/2). Without loss of generality (just by translation and by scaling), we may assume that z 0 = 0.…”

“…It is known (see, for instance, [4], Lemma 2.2) that, for any k = 0, 1, ..., the function ∇ k u is bounded on the set (R 3 + + 6hi 3 ) \ B(R 1 ) × [−50, 0]. Smoothness (and…”

A new proof of partial regularity of solutions to Navier-Stokes equations

“…This result was later extend in [23] and [3] to the case of equality for p < ∞. Notice that the case of L ∞ (0, ∞; L 3 (Ω)) was proven only very recently by Escauriaza, Seregin and Sverak [10]. In a series of papers [16,17,18,19], Scheffer began to develop the analysis about the possible singular points set, and established various partial regularity results for a class weak solutions named "suitable weak solutions".…”

Logarithmic improvement of regularity criteria for the Navier–Stokes equations in terms of pressure

“…The borderline case u ∈ L ∞ (0, T ; L 3 ) is much more complicated and requires a totally different approach. It was settled much later by Escauriaza, Seregin, and Sverak in [7]. Many generalizations and refinements of (5) have been proved, see e.g.…”