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<C$\lim _{\delta \rightarrow 0^+}\Big \Vert A^\frac{1}{4}\big (u(t-\delta )-u(t)\big )\Big \Vert _2<C$ for all t∈0,T)$t\in 0, T)$ with a sufficiently small positive constant C then u is regular in 0, T). Our proofs use the theory about the existence of local strong solutions and uniqueness arguments in the general domain.