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

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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

Carlo Morosi ^1, and Livio Pizzocchero ^2,

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

Abstract
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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.

Keywords: Navier-Stokes equations, inequalities, Sobolev spaces..

Mathematics Subject Classification: Primary: 76D05, 26D10; Secondary: 46E3.

Citation: Carlo Morosi, Livio Pizzocchero. On the constants in a Kato inequality for the Euler and Navier-Stokes equations. Communications on Pure and 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 27 (2011), 749-766. doi: 10.1002/num.20549.
[2]	R. A. Adams, "Sobolev Spaces,'' Academic Press, Boston, 1978.
[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), 25-40. doi: 10.3934/cpaa.2004.3.25.
[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), 065204/10. doi: 10.1063/1.2372512.
[5]	P. Constantin and C. Foias, "Navier Stokes Equations,'' Chicago University Press, 1988.
[6]	T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305.
[7]	Y. N. Kyrychko and M. V. Bartuccelli, Length scales for the Navier-Stokes equations on a rotating sphere, Phys. Lett. A, 324 (2004), 179-184. doi: 10.1016/j.physleta.2004.03.008.
[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), 625-706. doi: 10.1142/S0129055X08003407.
[9]	C. Morosi and L. Pizzocchero, An $H^1$ setting for the Navier-Stokes equations: quantitative estimates, Nonlinear Analysis, 74 (2011), 2398-2414. doi: 10.1016/j.na.2010.11.043.
[10]	C. Morosi and L. Pizzocchero, On the constants in a basic inequality for the Euler and Navier-Stokes equations,, preprint, ().
[11]	C. Morosi and L. Pizzocchero, On approximate solutions of the incompressible Euler and Navier-Stokes equations,, preprint, ().
[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), 613-629. doi: 10.1080/03605300903509104.
[13]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s102080010010.
[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), 157-185. doi: 10.1007/s10208-002-0080-8.

show all references

References:

[1]	A. Abdelrazec and D. Pelinovsky, Convergence of the Adomian decomposition method for initial-value problems, Numer. Methods Partial Differential Equations 27 (2011), 749-766. doi: 10.1002/num.20549.
[2]	R. A. Adams, "Sobolev Spaces,'' Academic Press, Boston, 1978.
[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), 25-40. doi: 10.3934/cpaa.2004.3.25.
[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), 065204/10. doi: 10.1063/1.2372512.
[5]	P. Constantin and C. Foias, "Navier Stokes Equations,'' Chicago University Press, 1988.
[6]	T. Kato, Nonstationary flows of viscous and ideal fluids in $R^3$, J. Funct. Anal., 9 (1972), 296-305.
[7]	Y. N. Kyrychko and M. V. Bartuccelli, Length scales for the Navier-Stokes equations on a rotating sphere, Phys. Lett. A, 324 (2004), 179-184. doi: 10.1016/j.physleta.2004.03.008.
[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), 625-706. doi: 10.1142/S0129055X08003407.
[9]	C. Morosi and L. Pizzocchero, An $H^1$ setting for the Navier-Stokes equations: quantitative estimates, Nonlinear Analysis, 74 (2011), 2398-2414. doi: 10.1016/j.na.2010.11.043.
[10]	C. Morosi and L. Pizzocchero, On the constants in a basic inequality for the Euler and Navier-Stokes equations,, preprint, ().
[11]	C. Morosi and L. Pizzocchero, On approximate solutions of the incompressible Euler and Navier-Stokes equations,, preprint, ().
[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), 613-629. doi: 10.1080/03605300903509104.
[13]	P. Zgliczyński and K. Mischaikow, Rigorous numerics for partial differential equations: the Kuramoto-Sivashinsky equation, Found. Comput. Math., 1 (2001), 255-288. doi: 10.1007/s102080010010.
[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), 157-185. doi: 10.1007/s10208-002-0080-8.

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