On Regularity of a Weak Solution to the Navier–Stokes Equations with the Generalized Navier Slip Boundary Conditions

Abstract

The paper shows that the regularity up to the boundary of a weak solution k of the Navier-Stokes equation with generalized Navier's slip boundary conditions follows from certain rate of integrability of at least one of the functions 1 , ( 2 ) + (the positive part of 2 ), and 3 , where 1 ≤ 2 ≤ 3 are the eigenvalues of the rate of deformation tensor D(k). A regularity criterion in terms of the principal invariants of tensor D(k) is also formulated.

Conditional regularity for the 3D Navier-Stokes equations in terms of the middle eigenvalue of the strain tensor

“…The author of [7] wrote in paper: "It is natural, however, to ask whether the main theorem in this paper (i.e., condition (8))can be extended to the critical Besov spaces, so in that sense the result may be pushed further". Our regularity condition (10) gives this problem an affirmative answer. But, unfortunately, our criteria (10) not cover the case p = ∞, it is not clear to us if regularity criteria…”

“…In particular, Neustupa-Penel [8] first formulated sufficient conditions for regularity of a so called suitable weak solution (u, p) in a sub-domain D of the time-space cylinder Q T = (Ω × (0, T )) by means of requirements on one of the eigenvalues of strain tensor. Later, for the generalized impermeability boundary conditions and the Navier-Slip boundary condition, in [9,10], they showed that the similar condition holds, then the solution u is regular on [0, T]. As explained in [8], such deformations, when the "infinitely small" volumes of the fluid are compressed in two dimensions and stretched in one dimension support regularity, while the cases when the "infinitely small" volumes of the fluid are compressed in one dimension and stretched in two dimensions support the hypothetical blow up.…”

On the Deformation Tensor Regularity for the Navier–Stokes Equations in Lorentz Spaces

“…However, full regularity of Leary-Hopf weak solutions to the 3D Navier-Stokes system is still a fundamental open question. Starting from Serrin's famous work, regularity criteria of Leray-Hopf weak solutions are extensively studied (see [1][2][3]5,7,[9][10][11]13,[16][17][18][19][20][21][23][24][25][26]28,29] and references therein). The so-called Serrin type regularity criteria is that a weak solution u is regular on…”

“…Denote λ 1 , λ 2 , λ 3 by the eigenvalue of the deformation tensor D(u). In a series of works [17][18][19][20][21], Neustupa-Penel obtained regularity criteria via only the middle eigenvalue λ 2 of the deformation tensor below…”

On the Motion of a Fluid-Filled Rigid Body with Navier Boundary Conditions

“…From the mathematical point of view, so far there have been only few contributions aimed at furnishing a rigorous treatment of the problem at hand. Major results concerning the study of viscous incompressible fluids subject to Navier boundary conditions either consider the case where the fluid is confined to a fixed immovable domain ( [1,3,25,29,31]), or the case where the fluid is flowing around moving (rigid or elastic) structures ( [23,10,27,20,2]).…”