Algebraic decay rates for 3D Navier–Stokes and Navier–Stokes–Coriolis equations in \( \dot{H}^{\frac{1}{2}}\)

An algebraic upper bound for the decay rate of solutions to the Navier–Stokes and Navier–Stokes–Coriolis equations in the critical space \(\dot{H} ^{\frac{1}{2}} (\mathbb {R}^3)\) ${\dot{H}}^{\frac{1}{2}}({\mathbb{R}}^3)$ is derived using the Fourier splitting method. Estimates are framed in terms of the decay character of initial data, leading to solutions with algebraic decay and showing in detail the roles played by the linear and nonlinear parts. The proof is carried on purely in the critical space, as no \(L^2 (\mathbb {R}^3)\) $L^2({\mathbb{R}}^3)$ estimates are available for the solution. This is the first instance in which such a method is used for obtaining decay bounds in a critical space for a nonlinear equation.