Navier–Stokes energy class

On $\mathbb R^n\times(0,T)$ , the Navier–Stokes energy class is

u\in L^\infty(0,T;L^2(\mathbb R^n)),\qquad \nabla u\in L^2(\mathbb R^n\times(0,T)),

where $\nabla u$ is the weak spatial gradient and the time-space norms are mixed Lebesgue norms. For finite $T$ this equals $L^\infty_tL^2_x\cap L^2_tH^1_x$ , with $H^1=W^{1,2}$ the Sobolev space. It bounds kinetic energy and integrated dissipation.

Here spatial $L^2$ is the usual Lebesgue space of square-integrable velocity components.

What membership supplies

Membership is a function-space condition. It does not by itself assert the equation, a representative at every time, an initial trace, or an energy inequality. On an infinite interval the analogous local-in-time definition is imposed on each finite interval; global integrated dissipation is an additional requirement.