There are solitary wave solutions for the Euler equations, but they do not have the "soliton" property of passing through each other without changing shape. Friedrichs and Hyers proved existence of such solutions in the 1950s for the case of zero surface tension. The problem with surface tension was solved in the 1980s and 1990s.
Here is one reference, which will lead you to the earlier ones:
S.M. Sun, Proc. Roy. Soc. London A 455 (1999), 2191-2228.
To complete Michael's answer, the only situation that is under control is that of the Cauchy problem: the spatial domain is ${\mathbb R}^d$ or ${\mathbb T}^d$ (case of periodic solutions). This means that there is no boundary condition.
If $d=2$, both systems are globally well-posed for $t>0$, with uniformly bounded (in $L^2$) solutions, and $u_\nu$ converges strongly to the solution of the Euler equation. Notice that it is not a trivial fact: the reasons why both Navier-Stokes and Euler Cauchy problems are globally well-posed have nothing in common; for Navier-Stokes, it comes from the Ladyzhenskaia inequality (say, $\|w\|_{L^4}^2\le c\|w\|_{L^2}\|\nabla w\|_{L^2}$), while for Euler, it is the transport of the vorticity.
If $d=3$, both Cauchy problems are locally-in-time well-posed for smooth enough initial data. One has a convergence as $\nu\rightarrow0+$ on some time interval $(0,\tau)$, but $\tau$ might be strictly smaller than both the time of existence of Euler and the $\lim\inf$ of the times of existence for Navier-Stokes.
To my knowledge, the initial-boundary value problem is a nightmare. The only result of convergence is in the case of analytic data (Caflisch & Sammartino, 1998). From time to time, a paper or a preprint appears with a "proof" of convergence, but so far, such papers have all be wrong.
By the way, your question is incorrectly stated, when you say boundary conditions are equal. The boundary condition for NS is $u=0$, whereas that for Euler is $u\cdot\vec n=0$, where $\vec n$ is the normal to the boundary. This discrepancy is the cause of the boundary layer. One may say that the difficulty lies in the fact that this boundary layer is characteristic. Non-characteristic singular limits are easier to handle.
Another remark is that some other boundary condition for NS are better understood. For instance, there is a convergenece result (Bardos) when $u=0$ is replaced by
$$u\cdot\vec n=0,\qquad {\rm curl}u\cdot\vec n=0.$$
Best Answer
There is probably no universally accepted mathematical definition of turbulence. (By the way, is there a physical one?) Moreover, the prevailing definitions seem to be highly volatile and time-dependent themselves.
A few notable examples.
In the Ptolemaic Landau–Hopf theory turbulence is understood as a cascade of bifurcations from unstable equilibriums via periodic solutions (the Hopf bifurcation) to quasiperiodic solutions with arbitrarily large frequency basis.
According to Arnold and Khesin, in the 1960's most specialists in PDEs regarded the lack of global existence and uniqueness theorems for solutions of the 3D Navier–Stokes equation as the explanation of turbulence.
Kolmogorov suggested to study minimal attractors of the Navier-Stokes equations and formulated several conjectures as plausible explanations of turbulence. The weakest one says that the maximum of the dimensions of minimal attractors of the Navier–Stokes equations grows along with the Reynolds number Re.
In 1970 Ruelle and Takens formulated the conjecture that turbulence is the appearance of global attractors with sensitive dependence of motion on the initial conditions in the phase space of the Navier–Stokes equations (link). In spite of the vast popularity of their paper, even the existence of such attractors is still unknown.
Edit 1. Concerning explicit solutions to the Navier-Stokes equations, I don't think any of them really exhibit turbulence features. The thing is that the nonlinear term $v\cdot \nabla v$ is equal to $0$ for most known classical explicit solutions. In other words, these solutions actually solve the linear Stokes equation and don't "see" the nonlinearity of the full Navier-Stokes system. This is probably not what one would expect from a truly turbulent flow.
Edit 2. As for the quick reference, you may find helpful the short survey on turbulence theories by Ricardo Rosa. It appears as an article in the Encyclopedia of Mathematical Physics.