  Understanding the non-local pressure contributions and viscous effects on the
small-scale statistics remains one of the central challenges in the study of
homogeneous isotropic turbulence. Here we address this issue by studying the
impact of the pressure Hessian as well as viscous diffusion on the statistics
of the velocity gradient tensor in the framework of an exact statistical
evolution equation. This evolution equation shares similarities with earlier
phenomenological models for the Lagrangian velocity gradient tensor evolution,
yet constitutes the starting point for a systematic study of the unclosed
pressure Hessian and viscous diffusion terms. Based on the assumption of
incompressible Gaussian velocity fields, closed expressions are obtained as the
results of an evaluation of the characteristic functionals. The benefits and
shortcomings of this Gaussian closure are discussed, and a generalization is
proposed based on results from direct numerical simulations. This enhanced
Gaussian closure yields, for example, insights on how the pressure Hessian
prevents the finite-time singularity induced by the local self-amplification
and how its interaction with viscous effects leads to the characteristic strain
skewness phenomenon.
