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. 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.

21 pp. Version 4 withdrawn due to an error. Submitted to Differential and integral equations

References in corpus (2)

• On the convergence rate of the Euler-$\alpha$, an inviscid second-grade complex fluid, model to the Euler equations
• Blow Ups of Complex Solutions of the 3D-Navier-Stokes System

Cited by in corpus (1)

• Dissipative structures in a nonlinear dynamo