  The present work elucidates the boundary behaviors of the velocity gradient
tensor ($\bm{A}\equiv\bm{\nabla}\bm{u}$) and its principal invariants ($P,Q,R$)
for compressible flow interacting with a stationary rigid wall. Firstly, it is
found that the well-known Caswell formula exhibits an inherent physical
structure being compatible with the normal-nilpotent decomposition, where both
the strain-rate and rotation-rate tensors contain the physical effects from the
spin component of the vorticity. Secondly, we derive the kinematic and dynamic
forms of the boundary $\bm{A}$-flux from which the known boundary fluxes can be
recovered by applying the symmetric-antisymmetric decomposition. Then, we
obtain the explicit expression of the boundary $Q$ flux as a result of the
competition among the boundary fluxes of squared dilatation, enstrophy and
squared strain-rate. Importantly, we emphasize that both the coupling between
the spin and surface pressure gradient, and the spin-curvature quadratic
interaction, are \textit{not} responsible for the generation of the boundary
$Q$ flux, although they contribute to both the boundary fluxes of enstrophy and
squared strain-rate. Moreover, we prove that the boundary $R$ flux must vanish
on a stationary rigid wall. Finally, the boundary fluxes of the invariants of
the strain-rate and rotation-rate tensors are also discussed. It is revealed
that the boundary flux of the third invariant of the strain-rate tensor is
proportional to the wall-normal derivative of the vortex stretching term, which
serves as a source term accounting for the the spatiotemporal evolution rate of
the wall-normal enstrophy flux. These theoretical results provide a unified
description of boundary vorticity and vortex dynamics, which could be valuable
in understanding the formation mechanisms of complex near-wall coherent
structures and the boundary sources of flow noise.
