Entire solutions of hydrodynamical equations with exponential dissipation

TL;DR

This paper studies modified hydrodynamical equations with exponential dissipation, showing solutions become entire and decay rapidly in Fourier space, with implications for turbulence and spectral simulations.

Contribution

It introduces a novel exponential dissipation operator in hydrodynamical equations and analyzes the resulting solutions' regularity and decay properties, extending to Navier-Stokes and Burgers equations.

Findings

Solutions become immediately entire in space variables.

Fourier coefficients decay faster than exponential with a universal constant.

The behavior suggests reduced intermittency in turbulence models.

Abstract

We consider a modification of the three-dimensional Navier--Stokes equations and other hydrodynamical evolution equations with space-periodic initial conditions in which the usual Laplacian of the dissipation operator is replaced by an operator whose Fourier symbol grows exponentially as $\ue ^{|k|/\kd}$ at high wavenumbers $|k|$. Using estimates in suitable classes of analytic functions, we show that the solutions with initially finite energy become immediately entire in the space variables and that the Fourier coefficients decay faster than $\ue ^{-C(k/\kd) \ln (|k|/\kd)}$ for any $C<1/(2\ln 2)$. The same result holds for the one-dimensional Burgers equation with exponential dissipation but can be improved: heuristic arguments and very precise simulations, analyzed by the method of asymptotic extrapolation of van der Hoeven, indicate that the leading-order asymptotics is precisely of the above form with $C= C_\star =1/\ln2$. The same behavior with a universal constant $C_\star$ is conjectured for the Navier--Stokes equations with exponential dissipation in any space dimension. This universality prevents the strong growth of intermittency in the far dissipation range which is obtained for ordinary Navier--Stokes turbulence. Possible applications to improved spectral simulations are briefly discussed.