Clay Institute Navier Stokes Part 2

I think you're misunderstanding the problem. The problem is not to show that functions $\{\mathbf{u}(\mathbf{x},t), P(\mathbf{x},t)\}$ exist which satisfy the Navier-Stokes equations; that much is relatively easy. The problem is to show that given any function $\mathbf{u}^\circ(\mathbf{x})$, there exists functions $\{\mathbf{u}(\mathbf{x},t), P(\mathbf{x},t)\}$ that satisfy the Navier-Stokes equations and satisfy $$ \mathbf{u}(\mathbf{x},0) = \mathbf{u}^\circ(\mathbf{x}). $$ This function $\mathbf{u}^\circ(\mathbf{x})$ is called the initial data; it basically tells you what the system is doing at the moment you start your stopwatch.

For your function, you have $\mathbf{u}(\mathbf{x},0) = (1,1,1)$. I haven't gone carefully through your math, but even if this were a solution of the Navier-Stokes equations, it would only be a drop in the bucket. You'd then have to find a solution with $\mathbf{u}(\mathbf{x},0) = (e^{-(x^2+y^2 + z^2)}, 0, 0)$. And then $\mathbf{u}(\mathbf{x},0) = e^{-(x^2+y^2 + z^2)}(\sin(z^2+x), x^2, \tanh(y))$. And then every other conceivable function that I could give you for the initial data.

Obviously, it's impossible to write down a solution for every conceivable function, for the simple reason that there are infinitely many such functions and you don't have an infinite amount of time. Rather, when mathematicians are trying to prove existence of solutions to PDEs, they try to find a general solution in the form of integrals involving the initial data, and then prove that these integrals are well-behaved no matter what the initial data is. Sometimes, they have to impose certain conditions on the initial data to make sure that their integrals are well-behaved; these are the conditions that you ran into in your last post.

As with many hard math problems, the trick is to prove something always exists. This was a good exercise, though - you found a certain function that satisfies a nonlinear PDE, and that's not always easy to do explicitly. Now can you find a solution that satisfies something like $$ {\bf u}(x,0) = {\bf u}^0(x) = \exp(-\|x\|^2)[y,-x,0]^T? $$