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We present a technique for derivation of a priori bounds for Gevrey-Sobolevnorms of space-periodic three-dimensional solutions to evolutionary partialdifferential equations of hydrodynamic type. It involves a transformation ofthe flow velocity in the Fourier space, which introduces a feedback between theindex of the norm and the norm of the transformed solution, and results inemergence of a mildly dissipative term. To illustrate the technique, we derivefinite-time bounds for Gevrey-Sobolev norms of solutions to the Euler andinviscid Burgers equations, and global in time bounds for the Voigt-typeregularisations of the Euler and Navier-Stokes equation (assuming that therespective norm of the initial condition is bounded). The boundedness of thenorms implies analyticity of the solutions in space.

A priori bounds for Gevrey-Sobolev norms of space-periodic three-dimensional solutions to equations of hydrodynamic type