Regularity results for parabolic systems related to a class of non-newtonian fluids

 Access to full text

 Full (PDF)

1. [1] Acerbi E., Mingione G., Regularity results for a class of functionals with nonstandard growth, Arch. Rational Mech. Anal.156 (2001) 121-140. Zbl0984.49020MR1814973
2. [2] Acerbi E., Mingione G., Regularity results for electrorheological fluids: the stationary case, C. R. Acad. Sci. Paris Ser. I334 (2002) 817-822. Zbl1017.76098MR1905047
3. [3] Acerbi E., Mingione G., Regularity results for stationary electro-rheological fluids, Arch. Rational Mech. Anal.164 (2002) 213-259. Zbl1038.76058MR1930392
4. [4] Bildhauer M., Fuchs M., Partial regularity for variational integrals with (s,μ,q)-growth, Calc. Var. Partial Differential Equations13 (2001) 537-560. Zbl1018.49026
5. [5] M. Bildhauer, M. Fuchs, Variants of the Stokes problem: the case of anisotropic potentials J. Math. Fluid Mechanics, submitted for publication. Zbl1072.76019MR2004292
6. [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. Zbl0509.35067MR673830
7. [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. Zbl0704.35024MR772253
8. [8] Coscia A., Mingione G., Hölder continuity of the gradient of p(x)-harmonic mappings, C. R. Acad. Sci. Paris Ser. I328 (1999) 363-368. Zbl0920.49020MR1675954
9. [9] L. Diening, Theoretical and numerical results for electrorheological fluids, Ph.D. Thesis, University of Freiburg, 2002. Zbl1022.76001
10. [10] L. Esposito, F. Leonetti, G. Mingione, Sharp regularity for functionals with (p,q) growth, J. Differential Equations, in press. Zbl1072.49024MR2076158
11. [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. Zbl0931.35029MR1713807
12. [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. Zbl0973.74033MR1736908
13. [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. Zbl0964.76003MR1810507
14. [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. Zbl0928.76087MR1671448
15. [15] Giaquinta M., Growth conditions and regularity, a counterexample, Manuscripta Math.59 (1987) 245-248. Zbl0638.49005MR905200
16. [16] Giusti E., Direct Methods in the Calculus of Variations, World Scientific Publishing Co., Inc., River Edge, NJ, 2003. Zbl1028.49001MR1962933
17. [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). Zbl0194.41701
18. [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). Zbl0202.37802
19. [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. Leningrad7 (1968). Zbl0195.10602MR241832
20. [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. Zbl0954.35129MR1738171
21. [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. MR241822
22. [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. Zbl0839.35018MR1318770
23. [23] Lions J.-L., Quelques methodes de resolution des problemes aux limites non lineaires, Gauthier-Villars, Paris, 1969. Zbl0189.40603MR259693
24. [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. [25] Marcellini P., Regularity and existence of solutions of elliptic equations with p,q-growth conditions, J. Differential Equations90 (1991) 1-30. Zbl0724.35043MR1094446
26. [26] Mingione G., The singular set of solutions to non-differentiable elliptic systems, Arch. Rational Mech. Anal.166 (2003) 287-301. Zbl1142.35391MR1961442
27. [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 Equations6 (2001) 257-302. Zbl1021.35085
28. [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. Zbl0851.35002MR1409366
29. [29] Rajagopal K.R., Wineman A.S., Flow of electrorheological materials, Acta Mech.91 (1992) 57-75. Zbl0746.76095MR1140999
30. [30] Rajagopal K.R., Růžička M., Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn.13 (2001) 59-78. Zbl0971.76100
31. [31] Růžička M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748, Springer, Berlin, 2000. Zbl0962.76001MR1810360
32. [32] Růžička M., Flow of shear dependent electrorheological fluids, C. R. Acad. Sci. Paris Ser. I Math.329 (1999) 393-398. Zbl0954.76097MR1710119
33. [33] Seregin G.A., Interior regularity for solutions to the modified Navier–Stokes equations, J. Math. Fluid Mech.1 (1999) 235-281. Zbl0961.35106MR1738752
34. [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. Zbl1030.35133MR1829531
35. [35] Seregin G.A., Sverak V., Navier–Stokes equations with lower bounds on the pressure, Arch. Rational Mech. Anal.163 (2002) 65-86. Zbl1002.35094MR1905137
36. [36] Temam R., Navier–Stokes equations. Theory and Numerical Analysis, Studies in Mathematics and its Applications, vol. 2, North-Holland, Amsterdam, 1984. Zbl0568.35002MR609732
37. [37] Zhikov V.V., Meyers type estimates for solving the nonlinear Stokes system, Differential Equations33 (1997) 107-114. Zbl0911.35089MR1607245

1. Frank Duzaar, Giuseppe Mingione, Second order parabolic systems, optimal regularity, and singular sets of solutions
2. Giuseppe Mingione, Regularity of minima: an invitation to the Dark Side of the Calculus of Variations