I'm trying to understand the Navier-Stokes Problem from the Clay Institute text here: http://www.claymath.org/sites/default/files/navierstokes.pdf . I've done an example problem to see what the problem is, but I don't run into any.
We are trying to satisfy:
\begin{equation} \frac{\partial \textbf{u}}{\partial t} + (\textbf{u}\cdot\nabla)\textbf{u}=-\frac{\nabla P}{\rho} + \nu\nabla^{2}\textbf{u}+\textbf{f}, \end{equation}
\begin{equation}\label{incompr} \nabla\cdot\textbf{u}=0. \end{equation}
We get to pick a velocity (u) and pressure (P) function to satisfy these equations. We take $n=3$ by putting our velocity and pressure vectors in the 3D Cartesian plane with $x,y,z$. We let $\textbf{f}$ be 0 according to the problem description, and assume kinematic viscosity (nu) to be greater than 0. Let
\begin{equation} \textbf{u}(x,t)=\begin{bmatrix} x+t \\ -2y+t \\ z+t \end{bmatrix} \end{equation}
Then $\textbf{u}(x,t)$ satisfies the divergence free condition because
\begin{equation} \nabla \cdot \textbf{u} = \frac{\partial u_1}{\partial x}+\frac{\partial u_2}{\partial y}+\frac{\partial u_3}{\partial z} \end{equation}
which is
\begin{equation} \nabla \cdot \textbf{u} = 1-2+1=0 \end{equation}
Then
\begin{equation}\frac{\partial \textbf{u}}{\partial t}=\begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix} \end{equation}
And
\begin{equation} \textbf{u} \cdot \nabla = (x+t)\frac{\partial}{\partial x} + (-2y+t)\frac{\partial}{\partial y}+(z+t)\frac{\partial}{\partial z} \end{equation}
And
\begin{equation} \begin{split} (\textbf{u} \cdot \nabla)\textbf{u} &= (x+t)\frac{\partial \textbf{u}}{\partial x} + (-2y+t)\frac{\partial \textbf{u}}{\partial y}+(z+t)\frac{\partial \textbf{u}}{\partial z} \\ &=(x+t)\begin{bmatrix} 1 \\ 0 \\ 0 \end{bmatrix}+(-2y+t)\begin{bmatrix} 0 \\ -2 \\ 0 \end{bmatrix}+(z+t)\begin{bmatrix} 0 \\ 0 \\ 1 \end{bmatrix} \\ &= \begin{bmatrix} x+t \\ 4y-2t \\ z+t \end{bmatrix} \end{split} \end{equation}
Since laplacian of $\textbf{u}$ is 0 the whole kinematic velocity term goes to 0. And the final Navier-Stokes expression is:
\begin{equation} \begin{bmatrix} 1 \\ 1 \\ 1 \end{bmatrix}+\begin{bmatrix} x+t \\ 4y-2t \\ z+t \end{bmatrix}=-\frac{\nabla P}{\rho} \end{equation}
So we set $\rho$ equal to 1 for easy calculation. Bring the negative to the left hand side and combine the left hand side.
\begin{equation} \begin{bmatrix} -x-t-1 \\ -4y+2t-1 \\ -z-t-1 \end{bmatrix}=\nabla P \end{equation}
and we solve for a solution for P
\begin{equation} P=\frac{-x^2}{2}-tx-x-2y^2+2ty-y-\frac{z^2}{2}-tz-z \end{equation}
So we now have infinitely differentiable functions for u and P and they are smooth since they are polynomial. Where did I make a mistake/misunderstand the problem?
In the context of the Clay Institute's presentation of the Navier-Stokes equations (as subject for the Millennium Problem Prizes), your attempt most closely resembles the challenge to give a proof of statement (A):
The notation of the linked PDF conforms closely to the notation of the Question above. The component version of (1) there has been recast here as a vector equation above, $\textbf{u}=(u_1,u_2,u_3)$, and pressure $p$ there has been labelled $P$ here. Also the mentions of $x$ in the PDF were intended to mean points in $\mathbb{R}^n$ with $n=2,3$. For consistency we will adopt the notation of the Question, asking Readers to supply any necessary rewriting for comparison with the PDF. In particular, except where quoting the PDF, our $x$ is only the first coordinate of $\mathbb{R}^3$, supplemented by variable $y,z$ as used in the Question.
As already noted by Commenters RRL and Winther, the N-S solution provided in the Question satisfies (1) and (2) but only for a specific initial condition (3):
$$ \textbf{u}^\circ(x,y,z) = (x,-2y,z) $$
and this linear initial condition is not "physically reasonable" as defined in the PDF's "bounded energy" restriction (7):
$$ \int_{\mathbb{R}^3} |u(x,t)|^2 \textrm{d}x \lt C \;\;\text{ for all }\; t \ge 0 $$
NB: This is one of the cases where, in quoting the PDF, $x$ has a multidimensional meaning.
Indeed a fair reading of the linked PDF would require the proof of statement (A) to apply to all initial conditions satisfying (4) for some choice of $\alpha,K \gt 0$:
$$ | \partial_x^\alpha u^\circ(x) | \le C_{\alpha K} (1+|x|)^{-K} \;\;\text{ on }\mathbb{R}^n, \text{ for some positive } \alpha \text{ and } K $$
The PDF's bound (4) involves both smoothness and growth of the initial condition and leaves something to the Reader's imagination to interpret how $\alpha$ and $K$ are quantified. However the "bounded energy" inequality (7) must also be satisfied by the initial condition by taking $t=0$, so we may confidently excluded the solution offered above from what would satisfy the Millennium Problem requirements.
For some background and perspective on the incompressible Navier-Stokes in the context of more general fluid flow problems, the Wikipedia article Navier-Stokes equations is a reasonable jumping off point.
In general it should be noted that the two-dimensional problems corresponding to (A) and (B) (periodic solution) were settled by Ladyzhenskaya.
Much literature on the incompressible N-S equations is concerned with numerical approximation schemes, esp. finite elements. In this context the analysis of discrete versions of the N-S equations can be understood as parallels to the continuous problem. In that vein I would recommend my adviser's book, Finite Element Methods for Viscous Incompressible Flows: A Guide to Theory, Practice, and Algorithms by Max Gunzburger (2012).