Data di Pubblicazione:
2010
Citazione:
On the constants in a basic inequality for
the Euler and Navier-Stokes equations / C. Morosi, L. Pizzocchero. - [s.l] : arXiv, 2010 Sep.
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 . Dw, 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 . Dw)||_n <= K_n || v ||_n || w ||_{n+1}, where n ∈ (d/2,+∞) 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.
Tipologia IRIS:
08 - Relazione interna o rapporto di ricerca
Keywords:
Navier-Stokes equations ; inequalities ; Sobolev spaces
Elenco autori:
C. Morosi, L. Pizzocchero
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