## Functiones et Approximatio Commentarii Mathematici

### Renormalized estimates for solutions to the Navier-Stokes equation

#### Abstract

For weak solutions to the three-dimensional Navier-Stokes equations the interior regularity problem for the renormalized velocity $u(1+|u|^2)^{-\alpha/2}$ and pressure $p(1+|u|^2)^{-\beta/2}$ is investigated. If a velocity component is locally semibounded and $\nabla u$ slightly more regular than suitable weak solutions the regularity estimates for the renormalized velocity are improved. Furthermore, estimates for the negative part of a renormalized pressure are presented.

#### Article information

Source
Funct. Approx. Comment. Math., Volume 40, Number 1 (2009), 11-32.

Dates
First available in Project Euclid: 30 March 2009

https://projecteuclid.org/euclid.facm/1238418795

Digital Object Identifier
doi:10.7169/facm/1238418795

Mathematical Reviews number (MathSciNet)
MR2527626

Zentralblatt MATH identifier
05620882

#### Citation

Frehse, Jens; Specovius-Neugebauer, Maria. Renormalized estimates for solutions to the Navier-Stokes equation. Funct. Approx. Comment. Math. 40 (2009), no. 1, 11--32. doi:10.7169/facm/1238418795. https://projecteuclid.org/euclid.facm/1238418795

#### References

• R. A. Adams and J. Fournier, Sobolev Spaces, Academic Press, Elsevier, New York, 2003, 2. ed.
• H. Beirão da Veiga, On the regular solutions of the evolution Navier-Stokes equations. Magalhães, L. (ed.) et al., International conference on differential equations. Papers from the conference, EQUADIFF 95, Lisboa, Portugal, July 24-29, 1995. Singapore: World Scientific. 18-23 (1998)., 1998.
• H. Beirão da Veiga, A sufficient condition on the pressure for the regularity of weak solutions to the Navier-Stokes equations., J. Math. Fluid Mech. 2 (2000), 99--106.
• F. Bernis, Integral inequalities with applications to nonlinear degenerate parabolic equations. Angell, T. S. (ed.) et al., Nonlinear problems in applied mathematics. In honor of Ivar Stakgold on his 70th birthday. Philadelphia, PA: SIAM. 57-65 (1996)., 1996.
• L. Caffarelli, R. Kohn, and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982), 771--831.
• P. Constantin, Navier-Stokes equations and area of interfaces, Comm. Math. Phys. 129 (1990), 241--266.
• L. Escauriaza, G. Seregin, and V. Šverák, Backward uniqueness for parabolic equations, Arch. Ration. Mech. Anal. 169 (2003), 147--157.
• J. Frehse and M. Ružička, On the regularity of the stationary Navier-Stokes equations., Ann. Sc. Norm. Super. Pisa, Cl. Sci., IV. Ser. 21 (1994), 63--95.
• O. A. Ladyženskaja, The mathematical theory of viscous incompressible flow, Gordon & Breach, New York, 1966
• F. Lin, A new proof of the Caffarelli-Kohn-Nirenberg theorem, Comm. Pure Appl. Math. 51 (1998), 241--257.
• P.-L. Lions, Mathematical topics in fluid mechanics. Vol. 1, vol. 3 of Oxford Lecture Series in Mathematics and its Applications, The Clarendon Press Oxford University Press, New York, 1996. Incompressible models, Oxford Science Publications.
• M. Ružička, Nonlinear functional analysis. An introduction. (Nichtlineare Funktionalanalysis. Eine Einführung.), Berlin: Springer. xii, 208 S. EUR 29.95; sFr. 51.00, 2004.
• G. Seregin, Estimates of suitable weak solutions to the Navier-Stokes equations in critical Morrey spaces, Zapiski Nauchn. Seminar, POMI 336 (2006), 199--210.
• H. Sohr, The Navier-Stokes Equations. An Elementary Functional Analytic Approach, Birkhäuser V., Basel, 2001.
• R. Temam, Navier-Stokes Equations, North Holland PC, Amsterdam, 1977.
• J. Wolf, The Caffarelli-Kohn-Nirenberg theorem - a direct proof by Campanato's method, Preprint of the Nečas Center for Mathematicak Modelling, Prague, 2006.