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

Abstract

We consider the nonstationary Navier-Stokes system in a smooth bounded domain Ω ⊂ R 3 with initial value u 0 ∈ L 2 σ (Ω). It is an important question to determine the optimal initial value condition in order to prove the existence of a unique local strong solution satisfying Serrin's condition. In this paper, we introduce a weighted Serrin condition that yields a necessary and sufficient initial value condition to guarantee the existence of local strong solutions u(·) contained in the weighted Serrin classMoreover, we prove a restricted weak-strong uniqueness theorem in this Serrin class.

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].…”

Analyticity of solutions to the primitive equations

“…These spaces correspond in the situation of the Navier–Stokes equations to the critical function spaces introduced by Cannone for the full space case , and by Prüss and Wilke for bounded domains. Also, for other solution classes for the 3‐d Navier–Stokes equations initial conditions in Besov spaces occur such as in the works by Farwig, Giga and Hsu which investigate solutions which are continuous in time and taking values in the class of Besov spaces for suitable coefficients including the case .…”

On Critical Spaces for the Navier–Stokes Equations

“…(Ω) d . This result has been extended by Farwig et al [8] to a time weighted version of Serrins class. To be more precise, it is shown in [8] that (1.1) subject to Dirichlet boundary conditions has a unique local strong solution with t α u ∈ L p (0, T ; L q (Ω) d ),…”