On robustness of a strong solution to the Navier–Stokes equations with Navier’s boundary conditions in theL³-norm

Abstract

We recall or prove a series of results on solutions to the Navier-Stokes equation with Navier's slip boundary conditions. The main theorem says that a strong solution u on any time interval (0,T ) (where < ∞ T 0 ⩽ ) is robust in the sense that small perturbations of the initial value in the norm of Ω σ L 3 ( ) and the acting body force in the norm of Ω L T L 0, ; 2 32 ( ( )) / cause only a small perturbation of solution u in the norm of Ω L 3 ( ). This result particularly implies that the maximum length of the time interval, on which the solution starting from the initial value ∈ Ω σ u L 0 3 ( ) is regular, is a lower semi-continuous functional on Ω σ L 3 ( ).

Navier–Stokes equations: regularity criteria in terms of the derivatives of several fundamental quantities along the streamlines—the case of a bounded domain

“…Since u(0) ∈ L 3 σ (Ω), χ > 0 (see lemma 12). We will prove that χ = T. Proceeding by contradiction we suppose that χ < T. (22) Due to (13) there exists an arbitrarily small T 1 ∈ (0, χ) such that…”

Regularity criteria for the Navier–Stokes equations in terms of the velocity direction and the flow of energy

“…The following result is proved in Amrouche et al 23, Theorem 1.1 or in Kučera and Neustupa. 28,Lemma 3 Lemma 11. Let u 0 ∈ L 3 𝜎 (Ω).…”

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

“…This is why a series of important results, well known from the theory of equations (1), (2) with the noslip boundary condition, have not been explicitly proven in literature for equations with boundary conditions (3), (4), although many of them can be obtained in a similar or almost the same way. This concerns except others the local in time existence of a strong solution (here, however, one can cite the papers [20,22], where the local in time existence of a strong solution is proven in the case when K = I, ≥ 0), the uniqueness of the weak solution, and the socalled theorem on structure. This theorem states that if the specific volume force f is at least in 2 (0, ; L 2 (Ω)) and k is a weak solution of the Navier-Stokes problem with the no-slip boundary condition, satisfying the strong energy inequality, then (0, ) = ⋃ ∈Γ ( , ) ∪ , where set Γ is at most countable, the intervals ( , ) are pairwise disjoint, the 1D Lebesgue measure of set is zero, and solution k coincides with a strong solution in the interior of each of the time intervals ( , ).…”