On regularity of weak solutions for the Navier–Stokes equations in general domains

• Published in: Mathematische Nachrichten 2021-12, Vol.294 (12), p.2302-2316
• Main Authors: Duong, V. T. T., Khai, D. Q., Tri, N. M.
• Format: Article
• Language: English
• Subjects: Domains; energy criteria; Fluid flow; Hölder continuity; Kinetic energy; Mathematical analysis; Navier-Stokes equations; regularity; weak solutions
• ISSN: 0025-584X; 1522-2616
• DOI: 10.1002/mana.201900407

Let u be a weak solution of the instationary Navier–Stokes equations in a completely general domain Ω⊆R3$\Omega \subseteq \mathbb {R}^3$ which additionally satisfies the strong energy inequality. Firstly, we prove that u is regular if the kinetic energy 12∥u(t)∥22$\frac{1}{2}\big \Vert u(t)\big \Vert _2^2$ is left‐side Hölder continuous with Hölder exponent 12$\frac{1}{2}$ and with a sufficiently small Hölder seminorm. This result extends the previous ones by several authors [5, 6, 7, 8] in which the domain Ω is additionally supposed to be bounded or have the uniform C2‐boundary ∂Ω$\partial \Omega$. Secondly, we show that if u(t)∈D(A14)$u(t) \in \mathbb {D}\Big(A^\frac{1}{4}\Big)$ and limδ→0+∥A14(u(t−δ)−u(t))∥2