Abstract:
2D equations, describing the motion of generalized Newtonian fluids, are considered. It is shown that for smooth data of corresponding initial-boundary value problems any spatial derivative of the velocity field is a locally Lipschitz function.
Citation:
O. A. Ladyzhenskaya, G. A. Seregin, “On reqularity of solutions to two-dimensional equations of the dynamics of fluids with nonlinear viscosity”, Boundary-value problems of mathematical physics and related problems of function theory. Part 30, Zap. Nauchn. Sem. POMI, 259, POMI, St. Petersburg, 1999, 145–166; J. Math. Sci. (New York), 109:5 (2002), 1911–1928
\Bibitem{LadSer99}
\by O.~A.~Ladyzhenskaya, G.~A.~Seregin
\paper On reqularity of solutions to two-dimensional equations of the dynamics of fluids with nonlinear viscosity
\inbook Boundary-value problems of mathematical physics and related problems of function theory. Part~30
\serial Zap. Nauchn. Sem. POMI
\yr 1999
\vol 259
\pages 145--166
\publ POMI
\publaddr St.~Petersburg
\mathnet{http://mi.mathnet.ru/znsl1054}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=1754361}
\zmath{https://zbmath.org/?q=an:1060.76005}
\transl
\jour J. Math. Sci. (New York)
\yr 2002
\vol 109
\issue 5
\pages 1911--1928
\crossref{https://doi.org/10.1023/A:1014444308725}
Linking options:
https://www.mathnet.ru/eng/znsl1054
https://www.mathnet.ru/eng/znsl/v259/p145
This publication is cited in the following 17 articles:
Xin Liu, Edriss S. Titi, “Global Existence of Weak Solutions to the Compressible Primitive Equations of Atmospheric Dynamics with Degenerate Viscosities”, SIAM J. Math. Anal., 51:3 (2019), 1913
Macha V., Tichy J., “Holder Continuity of Velocity Gradients For Shear-Thinning Fluids Under Perfect Slip Boundary Conditions”, NoDea-Nonlinear Differ. Equ. Appl., 24:3 (2017), 24
da Veiga H.B., “On the global regularity of shear thinning flows in smooth domains”, Journal of Mathematical Analysis and Applications, 349:2 (2009), 335–360
A. E. Mamontov, “Globalnaya razreshimost mnogomernykh uravnenii szhimaemoi nenyutonovskoi zhidkosti, transportnoe uravnenie i prostranstva Orlicha”, Sib. elektron. matem. izv., 6 (2009), 120–165
da Veiga H.B., “Navier–Stokes Equations with Shear-Thickening Viscosity. Regularity up to the Boundary”, J Math Fluid Mech, 11:2 (2009), 233–257
da Veiga H.B., “On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier–Stokes equations in smooth domains. The regularity problem”, J Eur Math Soc (JEMS), 11:1 (2009), 127–167
Lei Zh., Liu Ch., Zhou Y., “Global solutions for incompressible viscoelastic fluids”, Arch Ration Mech Anal, 188:3 (2008), 371–398
Lei Zh., Liu Ch., Zhou Y., “Global existence for a 2D incompressible viscoelastic model with small strain”, Communications in Mathematical Sciences, 5:3 (2007), 595–616
Kaplicky P., Prazak D., “Differentiability of the solution operator and the dimension of the attractor for certain power-law fluids”, J Math Anal Appl, 326:1 (2007), 75–87
Lei Z., Zhou Y., “Global existence of classical solutions for the two-dimensional Oldroyd model via the incompressible limit”, SIAM J Math Anal, 37:3 (2005), 797–814
Lin F.H., Liu C., Zhang P., “On hydrodynamics of viscoelastic fluids”, Comm Pure Appl Math, 58:11 (2005), 1437–1471
Kaplicky P., “Regularity of flows of a non-Newtonian fluid subject to Dirichlet boundary conditions”, Z Anal Anwend, 24:3 (2005), 467–486
da Veiga H.B., “On some boundary value problems for flows with shear dependent viscosity”, Variational Analysis and Applications, Nonconvex Optimization and its Applications, 79, 2005, 161–172
H. Beirão da Veiga, “On the regularity of flows with Ladyzhenskaya Shear‐dependent viscosity and slip or non‐slip boundary conditions”, Comm Pure Appl Math, 58:4 (2005), 552
G. A. Seregin, N. N. Ural'tseva, “Ol'ga Aleksandrovna Ladyzhenskaya (on her 80th birthday)”, Russian Math. Surveys, 58:2 (2003), 395–425
Kaplicky P., Malek J., Stara J., “Global-in-time Holder continuity of the velocity gradients for fluids with shear-dependent viscosities”, NoDEA Nonlinear Differential Equations Appl, 9:2 (2002), 175–195
Malek J., “Global analysis for the fluids of a power-law type”, Differential Equations and Nonlinear Mechanics, Mathematics and its Applications, 528, 2001, 213–233
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