Navier Stokes and smoothness

Basically, a function is differentiable iff it is continuous. Now if there exists a $k$-th derivative for a function, it means that the function is continuous at least $k-1$ times. They are using this to group functions into differentiability classes. A function of class $C^k$ has derivatives $f',f^{\prime\prime},\dots,f^{(k)}.$ Now, a smooth function is in the differentiability class $C^{\infty}$ which means that you can differentiate it as many times as you want. Which means it is always continuous.

Now that we know this, we should first try and understand this operator in Terence Tao's paper $$\partial_t u=\Delta u+B(u,u)$$ and he says this $B$ is equivalent to the energy equation. As far as I understand, Tao doesn't assume the equation to be smooth, instead of $B(u,u)$ he assumed an averaged $\tilde{B}(u,u).$ And then he constructs a smooth solution and demonstrates finite-time blowup.

References: Navier-Stokes existence and smoothness on Wikipedia and Tao's paper: Finite time blowup for an averaged three-dimensional Navier-Stokes equation