On optimal initial value conditions for local strong solutions of the Navier–Stokes equations

Abstract

Consider a smooth bounded domain Ω ⊆ R 3 , and the Navier-Stokes system in [0, ∞) × Ω with initial value u 0 ∈ L 2 σ (Ω) and external force f = div F, F ∈ L 2 (0, ∞; L 2 (Ω))∩L s /2 (0, ∞; L q /2 (Ω)) where 2 < s < ∞, 3 < q < ∞, 2 s + 3 q = 1, are so-called Serrin exponents. It is an important question what is the optimal (weakest possible) initial value condition in order to obtain a unique strong solution u ∈ L s (0, T ; L q (Ω)) in some initial interval [0, T ), 0 < T ≤ ∞. Up to now several sufficient conditions on u 0 are known which need not be necessary. Our main result, see Theorem 1.1, shows that the condition ∞ 0 ||e −t A u 0 || s q dt < ∞, A denotes the Stokes operator, is sufficient and necessary for the existence of such a strong solution u. In particular, if ∞ 0 ||e −t A u 0 || s q dt = ∞, u 0 ∈ L 2 σ (Ω), then any weak solution u in the usual sense does not satisfy Serrin's condition u ∈ L s (0, T ; L q (Ω)) for each 0 < T ≤ ∞.

Regularity of weak solutions to 3D incompressible Navier–Stokes equations

“…Here, the authors for the first time arrived the inequality 2 s + 3 q ≤ 1 2 in connection with one component of the velocity. Motivated by the results in [7,10,11], we find some new local and global regularity properties for weak solutions of (1.1), which can be viewed as a complement of Serrin's condition.…”

Global Weak Solutions of the Navier-Stokes System with Nonzero Boundary Conditions

“…The idea is to reduce the problem to the case of vanishing boundary values and a Navier-Stokes system containing additionally linear perturbation terms. We note that the concept of perturbed Navier-Stokes equations as in (1.10), (1.12) is also used in [6] to find the optimal initial value condition for the existence of local strong solutions.…”

Initial Values for the Navier-Stokes Equations in Spaces with Weights in Time

“…We use a modification of the proof described in [9]. Since for the moment we have no differentiability property for the mild solution u, we apply the Yosida operator J n = (I +…”

“…Since then many results on sufficient initial value conditions for the existence of local strong solutions have been developed, see [2,10,13,14,18,20,22,25,26,27]. Recent results in [8,9] yield sufficient and necessary conditions, also written in terms of (solenoidal) Besov spaces B See Section 4 for a definition of solenoidal Besov spaces; for a review of these results we refer to [5].…”

“…By this means the theory of [8,9] is extended to the scale of Besov q,∞ (Ω). There are also some results using weighted Serrin's conditions related to Kato's approach of construction of mild and strong solutions, see [17,23].…”