On the constants in a Kato inequality for the Euler and Navier-Stokes equations

,Carlo Morosi,Livio Pizzocchero

doi:10.3934/cpaa.2012.11.557

Abstract

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.

Highlights

The present paper continues our previous work on some inequalities related to the Euler or Navier-Stokes (NS) equations
Our aim is to analyze quantitatively, in terms of the Sobolev inner products, the quadratic map appearing in (1.1). Some aspects of this map have been already examined in the companion paper [10]; here we have considered the bilinear maps sending two vector fields v, w on Td into v∂w or L(v∂w), and we have discussed some inequalities about them, the basic one being
We use for Sobolev spaces and the Euler/NS bilinear map the same notations proposed in [10]; for the reader’s convenience, these are summarized hereafter

Summary

Introduction

The present paper continues our previous work on some inequalities related to the Euler or Navier-Stokes (NS) equations. In the present work we discuss other inequalities related to the quadratic Euler/NS nonlinearity, discovered by Kato in [6], and establish upper and lower bounds for the unknown sharp constants appearing therein. In the present paper we derive fully computable upper and lower bounds G±n ≡ G±nd such that. In [9] we have considered the NS equations in H1Σ0(T3); here we have derived a fully quantitative upper bound on the vorticity curl u0 L2 of the initial datum, which ensures global existence of the solution.

Some preliminaries

The Kato inequality

A Some tools preparing the analysis of the function Gn

B The function Gn