(...Ah, the Navier-Stokes equations, always a treat

Not!

I hate 'em )

Well... the relation

is equivalent to

,

which will ofcourse hold true if-f does not have zero limit as . Call this requirement .

So, the whole idea is to get (theBochnernorms, I think) bounded as .

Remember that spatially so by the Poincare inequality, these norms are bounded above and below by the norms of the respective gradient times a non-singular function of . So, we need just bound the norms of the gradients.

Back to the equations. Multiply the first by and integrate. Using some calculus and the fact that is divergence-free, we will obtain

.

Now, the left hand side can be made less than

for some function (calculations up to you ), while the right hand side will be less that for some non singular function .

Let us demand . This gives , and by making an even grander demand we take the right hand side to have a nonzero limes inferior as .

In this way, the half of is met.

Proceed in the same manner with the other equation.

Ps. Ofcourse, this proof is a bit rough around the edges,

but I bet you' re a smart kid and will straighten things out.