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On the constants in a Kato inequality for the Euler and Navier-Stokes equations

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  • 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 $(v, w) \mapsto v \cdot \partial w$, where $v, w : T^d \to R^d$ 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 $G_{n d} \equiv G_n$ in the Kato inequality $| < v \cdot \partial w | w >_n | \leq G_n || v ||_n || w ||^2_n$, where $n \in (d/2 + 1, + \infty)$ and $v, w$ are in the Sobolev spaces $H^n_{\Sigma_0}, H^{n+1}_{\Sigma_0}$ 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.
    Mathematics Subject Classification: Primary: 76D05, 26D10; Secondary: 46E35.


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