Data di Pubblicazione:
2010
Citazione:
On the constants in a Kato inequality for
the Euler and Navier-Stokes equations / C. Morosi, L. Pizzocchero. - [s.l] : arXiv, 2010 Sep.
Abstract:
We continue an analysis, started in (C. Morosi, L. Pizzocchero, arXiv:1007.4412v2 [math.AP] (2010)), 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 . Dw, 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 . Dw | w>_n| <= G_n || v ||_n || w ||^2_n, where n ∈ (d/2 + 1,+∞) 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 (C. Morosi, L. Pizzocchero, arXiv:1007.4412v2 [math.AP] (2010)) 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.
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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