Consider a (Newtonian) incompressible viscous fluid in three spatial dimensions, whose velocity field $\mathbb{v}=\mathbb{v}(x,y,z,t)$ moves according to the Navier-Stokes equations
$$\tag{1}\label{e1}\frac{\partial\mathbb{v}}{\partial t}+(\mathbb{v}\cdot\nabla)\mathbb{v}-\nu\Delta\mathbb{v}=-\nabla p\ , \\ \nabla\cdot\mathbb{v}=0$$
where $\nu$ is the kinematic viscosity and $p$ is the pressure (scalar) field acting on the fluid. Assume that the pressure gradient is always zero: $\nabla p=0$ everywhere for all times $t\geq 0$. Suppose the fluid is in free space (i.e. no boundaries) and we have as an initial condition a smooth velocity field $\mathbb{v}(x,y,z,0)=\mathbb{v}_0(x,y,z)\not\equiv 0$ which vanishes outside a bounded region of $\mathbb{R}^3$.
Question: Is it possible for the flow to develop turbulence in this case?
Edit: as discused in the comments to sammy gerbil's answer below (to whom I thank for helping me make my doubt more precise), my expectation in the absence of a pressure gradient and of boundary drag forces (unlike e.g. in Couette flow between a stationary plate and a moving, parallel one) is that the dissipation term $-\nu\Delta\mathbb{v}$ dominates the convection term $(\mathbb{v}\cdot\nabla)\mathbb{v}$ and the fluid flow should behave like a kind of "heat" flow, dissipating along time until the fluid stops moving (perhaps after an infinite amount of time) - in particular, I expect the flow to remain laminar at all times $t>0$ (hence the tone of the question's title). Put differently, the above question reduces to:
Question (rephrased): Does the linear part of the left hand side of \eqref{e1} (which is essentially a heat operator acting on $\mathbb{v}$) dominate under the above hypotheses?
If that is really true, I would like to see a mathematically precise argument for this, based on the Navier-Stokes equations \eqref{e1}.
