Regularity results for parabolic systems related to a class of non-newtonian fluids
Annales de l'I.H.P. Analyse non linéaire, Volume 21 (2004) no. 1, p. 25-60
@article{AIHPC_2004__21_1_25_0,
     author = {Acerbi, Emilio and Mingione, G and Seregin, G. A.},
     title = {Regularity results for parabolic systems related to a class of non-newtonian fluids},
     journal = {Annales de l'I.H.P. Analyse non lin\'eaire},
     publisher = {Elsevier},
     volume = {21},
     number = {1},
     year = {2004},
     pages = {25-60},
     doi = {10.1016/j.anihpc.2002.11.002},
     zbl = {1052.76004},
     mrnumber = {2037246},
     language = {en},
     url = {http://www.numdam.org/item/AIHPC_2004__21_1_25_0}
}
Acerbi, E; Mingione, G; Seregin, G. A. Regularity results for parabolic systems related to a class of non-newtonian fluids. Annales de l'I.H.P. Analyse non linéaire, Volume 21 (2004) no. 1, pp. 25-60. doi : 10.1016/j.anihpc.2002.11.002. http://www.numdam.org/item/AIHPC_2004__21_1_25_0/

[1] Acerbi E., Mingione G., Regularity results for a class of functionals with nonstandard growth, Arch. Rational Mech. Anal. 156 (2001) 121-140. | MR 1814973 | Zbl 0984.49020

[2] Acerbi E., Mingione G., Regularity results for electrorheological fluids: the stationary case, C. R. Acad. Sci. Paris Ser. I 334 (2002) 817-822. | MR 1905047 | Zbl 1017.76098

[3] Acerbi E., Mingione G., Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal. 164 (2002) 213-259. | MR 1930392 | Zbl 1038.76058

[4] Bildhauer M., Fuchs M., Partial regularity for variational integrals with (s,μ,q)-growth, Calc. Var. Partial Differential Equations 13 (2001) 537-560. | Zbl 1018.49026

[5] M. Bildhauer, M. Fuchs, Variants of the Stokes problem: the case of anisotropic potentials J. Math. Fluid Mechanics, submitted for publication. | MR 2004292 | Zbl 1072.76019

[6] Caffarelli L., Kohn R.V., Nirenberg L., Partial regularity of suitable weak solutions of the Navier-Stokes equations, Comm. Pure Appl. Math. 35 (1982) 771-831. | MR 673830 | Zbl 0509.35067

[7] Campanato S., On the nonlinear parabolic systems in divergence form. Hölder continuity and partial Hölder continuity of the solutions, Ann. Mat. Pura Appl. (4) 137 (1984) 83-122. | MR 772253 | Zbl 0704.35024

[8] Coscia A., Mingione G., Hölder continuity of the gradient of p(x)-harmonic mappings, C. R. Acad. Sci. Paris Ser. I 328 (1999) 363-368. | MR 1675954 | Zbl 0920.49020

[9] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis, University of Freiburg, 2002. | Zbl 1022.76001

[10] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p,q) growth, J. Differential Equations, in press. | MR 2076158 | Zbl 1072.49024

[11] Frehse J., Seregin G.A., Full regularity for a class of degenerated parabolic systems in two spatial variables, Manuscripta Math. 99 (1999) 517-539. | MR 1713807 | Zbl 0931.35029

[12] Frehse J., Seregin G.A., Regularity of solutions to variational problems of the deformation theory of plasticity with logarithmic hardening, Amer. Math. Soc. Transl. Ser. 2 (1999) 193. | MR 1736908 | Zbl 0973.74033

[13] Fuchs M., Seregin G.A., Variational Methods for Problems from Plasticity Theory and for Generalized Newtonian Fluids, Lecture Notes in Math., vol. 1749, Springer, Berlin, 2000. | MR 1810507 | Zbl 0964.76003

[14] Fuchs M., Seregin G.A., Variational methods for fluids of Prandtl-Eyring type and plastic materials with logarithmic hardening, Math. Methods Appl. Sci. 22 (1999) 317-351. | MR 1671448 | Zbl 0928.76087

[15] Giaquinta M., Growth conditions and regularity, a counterexample, Manuscripta Math. 59 (1987) 245-248. | MR 905200 | Zbl 0638.49005

[16] Giusti E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. | MR 1962933 | Zbl 1028.49001

[17] Ladyzhenskaya O.A., On nonlinear problems of continuum mechanics, in: Proc. Internat. Congr. Math. (Moscow 1966), Nauka, Moscow, 1968, pp. 560-573, English translation in: , Amer. Math. Soc. Translation (2) 70 (1968). | Zbl 0194.41701

[18] Ladyzhenskaya O.A., New equations for description of motion of viscous incompressible fluids and global solvability of boundary value problems for them, Proc. Steklov Inst. Math. 102 (1967). | Zbl 0202.37802

[19] Ladyzhenskaya O.A., On some modifications of the Navier-Stokes equations for large gradient of velocity, Zap. Nauchn. Sem. Leningrad Odtel. Mat. Inst. Steklov (LOMI) 7 (1968) 126-154, English translation in: , Sem. Math. V.A. Steklov Math. Inst. Leningrad 7 (1968). | MR 241832 | Zbl 0202.37301 | Zbl 0195.10602

[20] Ladyzhenskaya O.A., Seregin G.A., On partial regularity of suitable weak solutions to the three-dimensional Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999) 356-387. | MR 1738171 | Zbl 0954.35129

[21] Ladyzhenskaya O.A., Solonnikov V.A., Uraltseva N.N., Linear and Quasilinear Equations of Parabolic Type, Translations of Mathematical Monographs, vol. 23, American Mathematical Society, 1967. | MR 241822 | Zbl 0164.12302

[22] Lieberman G.M., Gradient estimates for a new class of degenerate elliptic and parabolic equations, Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 21 (1994) 497-522. | Numdam | MR 1318770 | Zbl 0839.35018

[23] Lions J.-L., Quelques methodes de resolution des problemes aux limites non lineaires, Gauthier-Villars, Paris, 1969. | MR 259693 | Zbl 0189.40603

[24] P. Marcellini, Un exemple de solution discontinue d' un probléme variationnel dans le cas scalaire, Preprint Dip. Mat. “U. Dini”, Univ. Firenze, 1987.

[25] Marcellini P., Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations 90 (1991) 1-30. | MR 1094446 | Zbl 0724.35043

[26] Mingione G., The singular set of solutions to non-differentiable elliptic systems, Arch. Rational Mech. Anal. 166 (2003) 287-301. | MR 1961442 | Zbl 1142.35391 | Zbl 01922408

[27] Malek J., Necas J., Růžička M., On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p≥2, Adv. Differential Equations 6 (2001) 257-302. | Zbl 1021.35085

[28] Malek J., Necas J., Rokyta M., Růžička M., Weak and Measure-Valued Solutions to Evolutionary PDEs, Appl. Math. Math. Comp., vol. 13, Chapman-Hall, London, 1996. | MR 1409366 | Zbl 0851.35002

[29] Rajagopal K.R., Wineman A.S., Flow of electrorheological materials, Acta Mech. 91 (1992) 57-75. | MR 1140999 | Zbl 0746.76095

[30] Rajagopal K.R., Růžička M., Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn. 13 (2001) 59-78. | Zbl 0971.76100

[31] Růžička M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer, Berlin, 2000. | MR 1810360 | Zbl 0968.76531 | Zbl 0962.76001

[32] Růžička M., Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Ser. I Math. 329 (1999) 393-398. | MR 1710119 | Zbl 0954.76097

[33] Seregin G.A., Interior regularity for solutions to the modified Navier-Stokes equations, J. Math. Fluid Mech. 1 (1999) 235-281. | MR 1738752 | Zbl 0961.35106

[34] Seregin G.A., On the number of singular points of weak solutions to the Navier-Stokes equations, Comm. Pure Appl. Math. 54 (2001) 1019-1028. | MR 1829531 | Zbl 1030.35133

[35] Seregin G.A., Sverak V., Navier-Stokes equations with lower bounds on the pressure, Arch. Rational Mech. Anal. 163 (2002) 65-86. | MR 1905137 | Zbl 1002.35094

[36] Temam R., Navier-Stokes equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, 1984. | MR 609732 | Zbl 0568.35002

[37] Zhikov V.V., Meyers type estimates for solving the nonlinear Stokes system, Differential Equations 33 (1997) 107-114. | MR 1607245 | Zbl 0911.35089