Local energy bounds and $$\epsilon $$ ϵ -regularity criteria for the 3D Navier–Stokes system

Partial regularity and the Minkowski dimension of singular points for suitable weak solutions to the 3D simplified Ericksen–Leslie system

“…The following two lemmas, which can be found in [7] and [8], respectively, are needed in the proofs of our main results.…”

“…In the following Lemma, by using the modified argument in Guevara-Phuc [8] and the idea of viewing the total Pressure 1 2 (|u| 2 + |∇d| 2 ) + P as a signed distribution in H −σ , gives the bound of A(u, ∇d, z 0 , r) and B(u, ∇d, z 0 , r). Lemma 3.1.…”

“…In 2007, some different type εregularity criteria were obtained by Gustafson-Kang-Tsai [9]. Recently, by viewing the total pressure P + |u| 2 2 as a signed distribution belonging to a certain negative order Sobolev space in local energy inequality, Guevara-Phuc [8] proved that there exists a ε > 0 such that if u is a suitable weak solution in Q r (z) and satisfies…”

“…The singular points Σ of the simplified Ericksen-Leslie system (1) is the set of points z = (x, t) such that in any neighborhood of each z, |u| + |∇d| is unbounded. To this end, we shall first derive a version of local energy estimates using certain type of test function and give an new ε-regularity criterion for suitable weak solutions by modifying the arguments in [8,9]. Since the ε-regularity criteria are local, we suppose our reference balls are in the domain of the equations.…”

“…There exists a ε > 0 such that if (u, d, P ) is a suitable weak solution to the simplified Ericksen-Leslie system (1) in Q r (z 0 ) with 0 < r ≤ 1, and satisfies (8) together with the Hölder's inequality implies that r −2 Qr(z0)…”