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

TL;DR

This paper establishes bounds for constants in key inequalities related to the quadratic term in the Euler and Navier-Stokes equations on a torus, with implications for understanding fluid dynamics stability and regularity.

Contribution

It provides new upper and lower bounds for the sharp constants in fundamental inequalities involving the bilinear form in Euler and Navier-Stokes equations, including numerical estimates for three-dimensional cases.

Findings

Derived bounds for the constants K_n in the basic inequality.

Numerical bounds reported for d=3 and specific n values.

Highlights importance of precise constants for fluid dynamics analysis.

Abstract

We consider the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w : T^d -> R^d into v . D w, and also involves the Leray projection L onto the space of divergence free vector fields. We derive upper and lower bounds for the constants in some inequalities related to the above quadratic term; these bounds hold, in particular, for the sharp constants K_{n d} = K_n in the basic inequality || L(v . D w)||_n <= K_n || v ||_n || w ||_{n+1}, where n in (d/2, + 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. Some practical motivations are indicated for an accurate analysis of the constants K_n.