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

Abstract

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. We illustrate the technique, using it to 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 regularizations 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.