New results on the constants in some inequalities for the Navier-Stokes quadratic nonlinearity

We give fully explicit upper and lower bounds for the constants in two known inequalities related to the quadratic nonlinearity of the incompressible (Euler or) Navier-Stokes equations on the torus T^d. These inequalities are "tame" generalizations (in the sense of Nash-Moser) of the ones analyzed in the previous works [Morosi and Pizzocchero: CPAA 2012, Appl.Math.Lett. 2013].

[1]  C. Morosi,et al.  On the constants in a Kato inequality for the Euler and Navier-Stokes equations , 2010, 1009.2051.

[2]  C. Morosi,et al.  On approximate solutions of the incompressible Euler and Navier–Stokes equations , 2011, 1104.3832.

[3]  Richard S. Hamilton,et al.  The inverse function theorem of Nash and Moser , 1982 .

[4]  On power series solutions for the Euler equation, and the Behr-Necas-Wu initial datum , 2012, 1203.6865.

[5]  Gerd Baumann,et al.  Navier–Stokes Equations on R3 × [0, T] , 2016 .

[6]  Tosio Kato,et al.  Remarks on the breakdown of smooth solutions for the 3-D Euler equations , 1984 .

[7]  C. Morosi,et al.  Smooth solutions of the Euler and Navier–Stokes equations from the a posteriori analysis of approximate solutions , 2014, 1405.3421.

[8]  Carlo Morosi,et al.  Large order Reynolds expansions for the Navier-Stokes equations , 2014, Appl. Math. Lett..

[9]  Carlo Morosi,et al.  On the constants in a basic inequality for the Euler and Navier-Stokes equations , 2013, Appl. Math. Lett..

[10]  C. Morosi,et al.  An H1 setting for the Navier–Stokes equations: Quantitative estimates , 2009, 0909.3707.

[11]  James C. Robinson,et al.  Lower bounds on blow up solutions of the three-dimensional Navier–Stokes equations in homogeneous Sobolev spaces , 2012 .

[12]  James C. Robinson,et al.  A posteriori regularity of the three-dimensional Navier–Stokes equations from numerical computations , 2006, math/0607181.

[13]  R. Temam.  Navier-Stokes Equations , 1977 .

[14]  Tosio Kato.  Nonstationary flows of viscous and ideal fluids in R3 , 1972 .

[15]  R. Temam,et al.  Local existence of ℓ∞ solutions of the euler equations of incompressible perfect fluids , 1976 .