Anisotropic and Geometric Criteria for Interior Regularity of Weak Solutions to the 3D Navier—Stokes Equations

On Anisotropic Regularity Criteria for the Solutions to 3D Navier–Stokes Equations

“…Thus a natural question appears, whether all these results can be connected via certain anisotropic criteria. The only result in this direction can be found in [12], where the authors showed the regularity provided…”

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

“…on Ω such that K(x) ⋅ a is tangential to Ω at point x ∈ Ω if vector a is tangential to Ω at point x. Condition (4) generalizes the "classical" Navier boundary condition [2]D(k) ⋅ n] + k = 0, where ≥ 0 is the coefficient of friction between the fluid and the boundary. The replacement of k by K ⋅ k reflects the fact that the microscopic structure of Ω can vary from point to point, it need not produce the same resistance in all tangential directions, and it may therefore divert the flow to the side.…”

“…System (1) and (2). Existence of a global regular solution and uniqueness of a weak solution are still the fundamental open questions in the theory of the Navier-Stokes equation in 3D.…”

“…Most of the known regularity criteria can be applied in the case when either Ω = R 3 (like those from [1,3,5]) or they exclude singularities in the interior of Ω, but not the singularities on the boundary. (This concerns, e.g., the criteria from [2,12].) As to criteria, valid up to the boundary, we can cite, for example, the papers [13] (where the socalled suitable weak solution is shown to be bounded locally near the boundary if it satisfies Serrin's conditions near the boundary and the trace of the pressure is bounded on the boundary), [14] (where an analogy of the well-known Caffarelli-Kohn-Nirenberg criterion for the regularity of a suitable weak solution at the point (x 0 , 0 ) ∈ Ω × (0, ), e.g., [15], is also proven for points on a flat part of the boundary), and [16,17] (for some generalizations of the criterion from [14], however, also valid only on a flat part of the boundary).…”

“…(They are usually called the "criteria of regularity.") The assumptions concern various quantities, like the velocity or some of its components (see, e.g., [1][2][3][4]), the gradient of velocity or some of its components (see, e.g., [3,5]), the vorticity or only two of its components (see, e.g., [1,6]), the direction of vorticity (see [7,8]), and the pressure (see, e.g., [9][10][11]). The absence of a blow-up (i.e., the nonexistence of singularities) in a weak solution has also been proven under certain assumptions on the integrability of the positive part of the middle eigenvalue of the rate of deformation tensor D(k) in [12].…”