Abstract
We formulate su cient conditions for regularity of a so-called suitable weak solution (v; p) in a sub-domain D of the time-space cylinder Q T by means of requirements on one of the eigenvalues or on the eigenvectors of the rate of deformation tensor.
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References
R. Betchow, An inequality concerning the production of vorticity in isotropic turbulence. Physics of Fluids (1956), 497–504.
W. Borchers and H. Sohr, On the equations rot v = g and div u = f with zero boundary conditions. Hokkaido Math. J. 19 (1990), 67–87.
L. Caffarelli, R. Kohn and L. Nirenberg, Partial regularity of suitable weak solutions of the Navier-Stokes equations. Comm. on Pure and Appl. Math. 35 (1982), 771–831.
D. Chae, H.J. Choe, Regularity of solutions to the Navier-Stokes equations. Electronic J. of Diff. Equations 5 (1999), 1–7.
G.P. Galdi, An Introduction to the Mathematical Theory of the Navier-Stokes Equations, Vol. I: Linearized Steady Problems. Springer-Verlag, New-York-Berlin-Heidelberg (1994).
G.P. Galdi, An Introduction to the Navier-Stokes initial-boundary value problem. In Fundamental Directions in Mathematical Fluid Mechanics, editors G.P. Galdi, J. Heywood, R. Rannacher, series “Advances in Mathematical Fluid Mechanics”, Birkhäuser Verlag, Basel (2000), 1–98.
J. Leray, Sur le mouvements d’un liquide visqueux emplissant l’espace. Acta Mathematica 62 (1934), 193–248.
J. Neustupa, Partial regularity of weak solutions to the Navier-Stokes equations in the class L ∞(0, T; L 3(Ω)3). J. of Mathematical Fluid Mechanics 1 (1999), 309–325
J. Neustupa, A. Novotný, P. Penel, An Interior Regularity of a Weak Solution to the Navier-Stokes Equations in Dependence on One Component of Velocity. Quaderni di Matematica, vol. 10, Topics in Mathematical Fluid Mechanics, Eds. G.P. Galdi and R. Ranacher, 2002, 163–183.
J. Neustupa and P. Penel, Anisotropic and geometric criteria for interior regularity of weak solutions to the 3D Navier-Stokes equations. In Mathematical Fluid Mechanics, Recent Results and Open Problems, editors J. Neustupa and P. Penel, series “Advances in Mathematical Fluid Mechanics”, Birkhäuser Verlag, Basel (2001), 237–268.
J. Neustupa and P. Penel, The role of eigenvalues and eigenvectors of the symmetrized gradient of velocity in the theory of the Navier-Stokes equations. Comptes Rendus Acad. Sci. Paris, Sér. I 336(10), 2003, 805–810.
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Neustupa, J., Penel, P. (2005). Regularity of a Weak Solution to the Navier-Stokes Equation in Dependence on Eigenvalues and Eigenvectors of the Rate of Deformation Tensor. In: Rodrigues, J.F., Seregin, G., Urbano, J.M. (eds) Trends in Partial Differential Equations of Mathematical Physics. Progress in Nonlinear Differential Equations and Their Applications, vol 61. Birkhäuser Basel. https://doi.org/10.1007/3-7643-7317-2_15
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DOI: https://doi.org/10.1007/3-7643-7317-2_15
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