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March  2012, 11(2): 557-586. doi: 10.3934/cpaa.2012.11.557

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

 1 Dipartimento di Matematica, Politecnico di Milano, P.za L. da Vinci 32, I-20133 Milano, Italy 2 Dipartimento di Matematica, Università di Milano, Via C. Saldini 50, I-20133 Milano, Italy

Received  September 2010 Revised  March 2011 Published  October 2011

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.
Citation: Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure & Applied Analysis, 2012, 11 (2) : 557-586. doi: 10.3934/cpaa.2012.11.557
##### References:
 [1] A. Abdelrazec and D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems,, Numer. Methods Partial Differential Equations \textbf{27} (2011), 27 (2011), 749.  doi: 10.1002/num.20549.  Google Scholar [2] R. A. Adams, "Sobolev Spaces,'', Academic Press, (1978).   Google Scholar [3] M. V. Bartuccelli, K. B. Blyuss and Y. N. Kyrychko, Length scales and positivity of solutions of a class of reaction-diffusion equations,, Comm. Pure Appl. Anal., 3 (2004), 25.  doi: 10.3934/cpaa.2004.3.25.  Google Scholar [4] S. I. Chernyshenko, P. Constantin, J. C. Robinson and E. S. Titi, A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2372512.  Google Scholar [5] P. Constantin and C. Foias, "Navier Stokes Equations,'', Chicago University Press, (1988).   Google Scholar [6] T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$,, J. Funct. Anal., 9 (1972), 296.   Google Scholar [7] Y. N. Kyrychko and M. V. Bartuccelli, Length scales for the Navier-Stokes equations on a rotating sphere,, Phys. Lett. A, 324 (2004), 179.  doi: 10.1016/j.physleta.2004.03.008.  Google Scholar [8] C. Morosi and L. Pizzocchero, On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations,, Rev. Math. Phys., 20 (2008), 625.  doi: 10.1142/S0129055X08003407.  Google Scholar [9] C. Morosi and L. Pizzocchero, An $H^1$ setting for the Navier-Stokes equations: quantitative estimates,, Nonlinear Analysis, 74 (2011), 2398.  doi: 10.1016/j.na.2010.11.043.  Google Scholar [10] C. Morosi and L. Pizzocchero, On the constants in a basic inequality for the Euler and Navier-Stokes equations,, preprint, ().   Google Scholar [11] C. Morosi and L. Pizzocchero, On approximate solutions of the incompressible Euler and Navier-Stokes equations,, preprint, ().   Google Scholar [12] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Partial Differential Equations, 35 (2010), 613.  doi: 10.1080/03605300903509104.  Google Scholar [13] P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation,, Found. Comput. Math., 1 (2001), 255.  doi: 10.1007/s102080010010.  Google Scholar [14] P. Zgliczyński, Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE - A computer-assisted proof,, Found. Comput. Math., 4 (2004), 157.  doi: 10.1007/s10208-002-0080-8.  Google Scholar

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##### References:
 [1] A. Abdelrazec and D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems,, Numer. Methods Partial Differential Equations \textbf{27} (2011), 27 (2011), 749.  doi: 10.1002/num.20549.  Google Scholar [2] R. A. Adams, "Sobolev Spaces,'', Academic Press, (1978).   Google Scholar [3] M. V. Bartuccelli, K. B. Blyuss and Y. N. Kyrychko, Length scales and positivity of solutions of a class of reaction-diffusion equations,, Comm. Pure Appl. Anal., 3 (2004), 25.  doi: 10.3934/cpaa.2004.3.25.  Google Scholar [4] S. I. Chernyshenko, P. Constantin, J. C. Robinson and E. S. Titi, A posteriori regularity of the three-dimensional Navier-Stokes equations from numerical computations,, J. Math. Phys., 48 (2007).  doi: 10.1063/1.2372512.  Google Scholar [5] P. Constantin and C. Foias, "Navier Stokes Equations,'', Chicago University Press, (1988).   Google Scholar [6] T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$,, J. Funct. Anal., 9 (1972), 296.   Google Scholar [7] Y. N. Kyrychko and M. V. Bartuccelli, Length scales for the Navier-Stokes equations on a rotating sphere,, Phys. Lett. A, 324 (2004), 179.  doi: 10.1016/j.physleta.2004.03.008.  Google Scholar [8] C. Morosi and L. Pizzocchero, On approximate solutions of semilinear evolution equations II. Generalizations, and applications to Navier-Stokes equations,, Rev. Math. Phys., 20 (2008), 625.  doi: 10.1142/S0129055X08003407.  Google Scholar [9] C. Morosi and L. Pizzocchero, An $H^1$ setting for the Navier-Stokes equations: quantitative estimates,, Nonlinear Analysis, 74 (2011), 2398.  doi: 10.1016/j.na.2010.11.043.  Google Scholar [10] C. Morosi and L. Pizzocchero, On the constants in a basic inequality for the Euler and Navier-Stokes equations,, preprint, ().   Google Scholar [11] C. Morosi and L. Pizzocchero, On approximate solutions of the incompressible Euler and Navier-Stokes equations,, preprint, ().   Google Scholar [12] D. Pelinovsky and A. Sakovich, Global well-posedness of the short-pulse and sine-Gordon equations in energy space,, Comm. Partial Differential Equations, 35 (2010), 613.  doi: 10.1080/03605300903509104.  Google Scholar [13] P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation,, Found. Comput. Math., 1 (2001), 255.  doi: 10.1007/s102080010010.  Google Scholar [14] P. Zgliczyński, Rigorous numerics for dissipative partial differential equations II. Periodic orbit for the Kuramoto-Sivashinsky PDE - A computer-assisted proof,, Found. Comput. Math., 4 (2004), 157.  doi: 10.1007/s10208-002-0080-8.  Google Scholar
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