A Note on Prodi–Serrin Conditions for the Regularity of a Weak Solution to the Navier–Stokes Equations

Abstract

The paper is concerned with the regularity of weak solutions to the Navier-Stokes equations. The aim is to investigate on a relaxed Prodi-Serrin condition in order to obtain regularity for t > 0. The most interesting aspect of the result is that no compatibility condition is required to the initial data v• ∈ J 2 (Ω).

On the energy equality for the 3D Navier–Stokes equations

“…Even if not explicitly stated, it is clear that any of the conditions on the gradient in Theorem 1 implies, directly by the standard theory of traces, an extra condition on the initial datum. To keep the paper self-contained and more understandable we do not elaborate on this technical point, which can be probably improved by using the theory of weighted estimates as in Farwig, Giga, and Hsu [21] or by dealing with conditions valid on any time interval of the type [ε, T ] (with ε > 0), as done in the recent paper by Maremonti [39].…”

Existence and nonlinear stability of convective solutions for almost compressible fluids in Bénard problem

“…Moreover, we know that the norms of ∂ xx , ∂ yy are equivalent to the full set of the second derivatives. In particular 12) and the same for τ . The proof can be found in [3].…”

“…In the subsequent Section 3 we devote our effort to the proof of existence for the nonlinear perturbation problem. We exploit the usual techniques of functional analysis applied to the study of this type of questions [2,7,12,13]. In particular, we derive several a priori "energy" estimates and couple them with the classical Gälerkin method with a special basis, to prove existence of solutions.…”

“…For all Rayleigh numbers and initial data of arbitrary "size", it is shown that a strong solution (in the sense of Ladyzhenskaya; see e.g. [12]) does exist for some time interval [0, T ) where T can be estimated in terms of the initial data. However, if only the L 2 -norm of these data is sufficiently small and the Rayleigh number is below a certain constant, then we can take T = ∞.…”

On the uniqueness of a suitable weak solution to the Navier–Stokes Cauchy problem

“…Actually, the assumptions fit to ensure the uniqueness imply the regularity of a weak solution. However in [13] relaxed Prodi-Serrin conditions are considered, valid on any time interval properly contained in the one of existence, implying the regularity of the solutions for t > 0. Since no equivalence between the Prodi-Serrin conditions and the uniqueness is known, as well as no example of non-uniqueness is known for weak solutions corresponding to an initial datum only in L 2 , it is an open question if a weak solution satisfying a relaxed Prodi-Serrin condition is also unique.…”