  We present a technique for derivation of a priori bounds for Gevrey-Sobolev
norms of space-periodic three-dimensional solutions to evolutionary partial
differential equations of hydrodynamic type. It involves a transformation of
the flow velocity in the Fourier space, which introduces a feedback between the
index of the norm and the norm of the transformed solution, and results in
emergence of a mildly dissipative term. To illustrate the technique, we derive
finite-time bounds for Gevrey-Sobolev norms of solutions to the Euler and
inviscid Burgers equations, and global in time bounds for the Voigt-type
regularisations of the Euler and Navier-Stokes equation (assuming that the
respective norm of the initial condition is bounded). The boundedness of the
norms implies analyticity of the solutions in space.
