A local smoothness criterion for solutions of the 3D Navier-Stokes equations

Robinson, James C.; Sadowski, Witold

Rendiconti del Seminario Matematico della Università di Padova, Tome 131 (2014), pp. 159-178

[1] H. Beirão Da Veiga. Existence and asymptotic behavior for strong solutions of the Navier-Stokes equations in the whole space, Indiana Univ. Math. J., 36 (1987), pp. 149–166. | Zbl | MR

[2] L.C. Berselli and G.P. Galdi. On the space–time regularity of $C(0,T;L^n)$ -very weak solutions to the Navier—Stokes equations, Nonlinear Analysis, 58 (2004), pp. 703–717. | Zbl | MR

[3] M. Cannone. Harmonic analysis tools for solving the incompressible Navier-Stokes equations, in S. Friedlander & D. Serre (Eds.) Handbook of Mathematical Fluid Dynamics, vol. 3, Elsevier, 2003. | Zbl | MR

[4] J.-Y. Chemin, B. Desjardins, I. Gallagher and E. Grenier. Mathematical Geophysics, Oxford University Press, Oxford, 2006. | Zbl | MR

[5] P. Constantin and C. Foias. Navier-Stokes equations. University of Chicago Press, Chicago, 1988. | Zbl | MR

[6] L. Escauriaza, G. Seregin, and V. Šverák. $L_{3,\infty }$ -solutions of Navier-Stokes equations and backward uniqueness, Russian Math. Surveys, 58 (2003), pp. 211–250. | Zbl | MR

[7] L.C. Evans. Partial differential equations. Second edition. Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. | Zbl | MR

[8] R. Farwig, H. Sohr and W. Varnhorn. Necessary and sufficient conditions on local strong solvability of the Navier-Stokes system Applicable Analysis, Vol. 90, No. 1, January (2011), pp. 47–58. | Zbl | MR

[9] H. Fujita and T. Kato. On the Navier-Stokes initial value problem. I. Arch. Rational Mech. Anal., 16 (1964), pp. 269–315. | Zbl | MR

[10] G.P. Galdi. An introduction to the Navier-Stokes initial-boundary value problem, Fundamental directions in Mathematical Fluid Dynamics, Birkhauser, Basel, 1–70, 2011. | Zbl | MR

[11] G.P. Galdi and S. Rionero. The weight function approach to uniqueness of viscous flows in unbounded domains, Arch. Ratl Mech. Anal., 69, 37 (1979). | Zbl | MR

[12] Y. Giga. Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system. J. Differential Equations, 62 (1986), pp. 186–212. | Zbl | MR

[13] P.G. Lemarié-Rieusset. Recent developments in the Navier-Stokes problem, Chapman & Hall/CRC Research Notes in Mathematics, 431. Chapman & Hall/CRC, Boca Raton, FL, 2002. | Zbl | MR

[14] T. Kato, Strong $L^p$ -solutions of the Navier-Stokes equations in $R^m$ with applications to weak solutions, Math. Zeit., 187 (1984), pp. 471–480. | Zbl | MR

[15] J. Leray. Sur le mouvement d’un liquide visqueux emplissant l’espace, Acta Math., 63 (1934), pp. 193–248. | MR | JFM

[16] P. Marín-Rubio, J.C. Robinson and W. Sadowski. Solutions of the 3D Navier-Stokes equations for initial data in $H^{1/2}$ : robustness of regularity and numerical verification of regularity for bounded sets of initial data in $H^1$ , J. Math. Anal. Appl., 400 (2013), pp. 76–85. | Zbl | MR

[17] A. Rodríguez Bernal. Introduction to semigroup theory for partial differential equations. Lectures notes, 2005. Available online at http://www.opencontent.org/openpub/

[18] J.C. Robinson, W. Sadowski and R. Silva. Lower bounds on blow up solutions of the three-dimensional Navier-Stokes equations in homogeneous Sobolev spaces, J. Math. Phys., 53 (2012), 115618. | Zbl | MR

[19] V. Shapiro. Fourier Series in Several Variables with Applications to Partial Differential Equations, Taylor & Francis Group, LLC, Chapman & Hall/CRC Applied Mathematics and Nonlinear Science Series Edt. Goong Chen, 2011. | Zbl | MR

[20] W. Von Wahl. Regularity of weak solutions of the Navier-Stokes equations. Nonlinear functional analysis and its applications, Part 2 (Berkeley, Calif., 1983), 497–503, Proc. Sympos. Pure Math., 45, Part 2, Amer. Math. Soc., Providence, RI, 1986. | Zbl | MR