Mathematics – Analysis of PDEs
Scientific paper
2010-09-10
Commun. Pure Appl. Anal. 11(2)(2012), 557-586
Mathematics
Analysis of PDEs
LaTeX, 39 pages
Scientific paper
10.3934/cpaa.2012.11.557
We continue an analysis, started in [10], 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 . D w, 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 . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + 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. When combined with the results of [10] 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.
Morosi Carlo
Pizzocchero Livio
No associations
LandOfFree
On the constants in a Kato inequality for the Euler and Navier-Stokes equations does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On the constants in a Kato inequality for the Euler and Navier-Stokes equations, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On the constants in a Kato inequality for the Euler and Navier-Stokes equations will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-672650