The Navier–Stokes equations are approximated by means of a fractional step, Chorin–Temam projection method; the time derivative is approximated by a three-level backward finite difference, whereas the approximation in space is performed by a Galerkin technique. It is shown that the proposed scheme yields an error of $𝒪(\delta t^2+h^{l+1})$ for the velocity in the norm of l2(L2(Ω)d), where l ≥ 1 is the polynomial degree of the velocity approximation. It is also shown that the splitting error of projection schemes based...

The main result of this paper is the proof of uniqueness for mild solutions of the Navier-Stokes equations in L3(R3). This result is extended as well to some Morrey-Campanato spaces.

Consider the Navier-Stokes equation with the initial data $a\in L_{\sigma }^2\left({ℝ}^d\right)$. Let $u$ and $v$ be two weak solutions with the same initial value $a$. If $u$ satisfies the usual energy inequality and if $\nabla v\in L^2((0,T);{\dot{X}}_1{\left({ℝ}^d\right)}^d)$ where ${\dot{X}}_1\left({ℝ}^d\right)$ is the multiplier space, then we have $u=v$.

We prove a uniqueness result of weak solutions to the $nD$$(n\ge 3)$ Cauchy problem of a Keller-Segel-Navier-Stokes system with a logistic term.