2010/07/31 by Carlo Morosi, Livio Pizzocchero · 6 citations
Mathematics · Physics and Astronomy · #math.AP #math-ph #math.FA #math.MP #msc:76D05 #msc:26D10 #msc:46E35
paper · pdf · doi:10.1016/j.aml.2012.09.007
published as an abridged version published in Applied Mathematics Letters 26 (2013), 277-284 · 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
arxiv created 2012/09/05 · arxiv updated 2013/03/26
We consider the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus Td; the quadratic term in these equations arises from the bilinear map sending two velocity fields v, w : Td -> Rd 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 Kn d = Kn in the basic inequality || L(v . D w)||n <= Kn || v ||n || w ||n+1, where n in (d/2, + infinity) and v, w are in the Sobolev spaces Hn, Hn+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 Kn.