On the constants in a basic inequality for the Euler and Navier-Stokes equations
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.
Keywords
Cite
@article{arxiv.1007.4412,
title = {On the constants in a basic inequality for the Euler and Navier-Stokes equations},
author = {Carlo Morosi and Livio Pizzocchero},
journal= {arXiv preprint arXiv:1007.4412},
year = {2013}
}
Comments
LaTeX, 36 pages. The numerical values of the upper bounds K^{+}_{5} and K^{+}_{10} for d=3 have been corrected. Some references have been updated. arXiv admin note: text overlap with arXiv:1009.2051 by the same authors, not concerning the main results