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

TL;DR

This paper introduces a new technique to derive a priori bounds for Gevrey-Sobolev norms of 3D space-periodic solutions to hydrodynamic PDEs, ensuring their analyticity and stability.

Contribution

It presents a novel Fourier space transformation method that provides finite-time and global bounds for solutions to Euler, Burgers, and regularized Navier-Stokes equations.

Findings

Finite-time bounds for Euler and Burgers equations

Global bounds for Voigt-type regularizations of Euler and Navier-Stokes

Norm boundedness implies spatial analyticity of solutions

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. 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.