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Keywords:
Electrorheological fluid; generalized Newtonian fluids; existence theory; function spaces with variable exponents; harmonic analysis; Orlicz spaces; shifted $N$-function
Electrorheological fluid; generalized Newtonian fluids; existence theory; function spaces with variable exponents; harmonic analysis; Orlicz spaces; shifted $N$-function
Summary:
In this note we give an overview of recent results in the theory of electrorheological fluids and the theory of function spaces with variable exponents. Moreover, we present a detailed and self-contained exposition of shifted $N$-functions that are used in the studies of generalized Newtonian fluids and problems with $p$-structure.
References:
[1] Jdayil B. Abu,- Brunn P. O.: Effects of nonuniform electric field on slit flow of an electrorheological fluid. J. Rheol. 39 (1995), 1327–1341. DOI 10.1122/1.550639
[2] Jdayil B. Abu,- Brunn P. O.: Effects of electrode morphology on the slit flow of an electrorheological fluid. J. Non-New. Fluid Mech. 63 (1996), 45–61. DOI 10.1016/0377-0257(95)01416-0
[3] Jdayil B. Abu,- Brunn P. O.: Study of the flow behaviour of electrorheological fluids at shear- and flow- mode. Chem. Eng. and Proc. 36 (1997), 281–289.
[4] Agmon S., Douglis A., Nirenberg L.: Estimates near the boundary for solutions of elliptic partial differential equations satisfying general boundary conditions. I. Comm. Pure Appl. Math. 12 (1959), 623–727. Zbl 0093.10401, MR 23 #A2610. DOI 10.1002/cpa.3160120405 | MR 0125307 | Zbl 0093.10401
[5] Bogovskii M. E.: Solution of the first boundary value problem for the equation of continuity of an incompressible medium. Dokl. Akad. Nauk SSSR 248 (1979), 1037–1040. English transl. in Soviet Math. Dokl. 20 (1979), 1094–1098. Zbl 0499.35022, MR 82b:35135. MR 0553920
[6] Bogovskii M. E.: Solution of some problemsof vector analysis related to the operators $\operatorname{div}$ and $\operatorname{grad}$. Trudy Sem. S. L. Soboleva, no. 1, Akad. Nauk SSSR, Sibirsk. Otdel., Inst. Mat., Novosibirsk (1980), 5–40. MR 0631691
[7] Calderón A. P., Zygmund A.: On the existence of certain singular integrals. Acta Math. 88 (1952), 85–139. Zbl 0047.10201, MR 14,637f. DOI 10.1007/BF02392130 | MR 0052553
[8] Cianchi A.: Symmetrization in anisotropic elliptic problems. Preprint, 2006. MR 2334829 | Zbl 1219.35028
[9] Uribe D. Cruz,- Fiorenza A., Martell J. M., Pérez C.: The boundedeness of classical operators on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math. 31 (2004), no. 1, 239–264. Zbl 1100.42012, MR 2006m:42029.
[10] Uribe D. Cruz,- Fiorenza A., Neugebauer C. J.: The maximal function on variable $L^p$ spaces. Ann. Acad. Sci. Fenn. Math. 28 (2003), no. 1, 223–238. Zbl 1037.42023, MR 2004c:42039. MR 1976842
[11] Diening L.: Maximal function on generalized Lebesgue spaces $L^{p(\cdot )}$. Math. Inequal. Appl. 7 (2004), 245–253. Zbl 1071.42014, MR 2005k:42048. MR 2057643
[12] Diening L.: Riesz potential and Sobolev embeddings on generalized Lebesgue and Sobolev spaces $L^{p(\cdot )}$ and $W^{k,p(\cdot )}$. Math. Nachr. 268 (2004), 31–43. Zbl 1065.46024, MR 2005d:46071. MR 2054530
[13] Diening L.: Maximal function on Musielak-Orlicz spaces and generalized Lebesgue spaces. Bull. Sci. Math. 129 (2005), no. 8, 657–700. MR 2006e:46032. DOI 10.1016/j.bulsci.2003.10.003 | MR 2166733
[14] Diening L., Ebmeyer C., Růžička M.: Optimal convergence for the implicit space-time discretization of parabolic systems with $p$-structure. SIAM J. Numer. Anal. 45 (2007), no. 2, 457–472. MR2300281. DOI 10.1137/05064120X | MR 2300281 | Zbl 1140.65060
[15] Diening L., Ettwein F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math. (2006), accepted. MR 2418205 | Zbl 1188.35069
[16] Diening L., Kreuzer C.: Linear convergence of an adaptive finite element method for the $p$-Laplacian equation. SIAM J. Numer. Anal. (2007), submitted. MR 2383205 | Zbl 1168.65060
[17] Diening L., Málek J., Steinhauer M.: On Lipschitz truncations of sobolev functions (with variable exponent) and their selected applications. ESAIM, Control, Optim. Calc. Var. (2006), submitted. MR 2394508 | Zbl 1143.35037
[18] Diening L., Růžička M.: Error estimates for interpolation operators in Orlicz-Sobolev spaces and quasi norms. Num. Math. (2007), DOI 10.1007/s00211-007-0079-9.
[19] Diening L., Růžička M.: Calderón–Zygmund operators on generalized Lebesgue spaces $L^{p(\cdot )}$ and problems related to fluid dynamics. J. Reine Angew. Math. 563 (2003), 197–220. Zbl 1072.76071, MR 2005g:42054. MR 2009242
[20] Diening L., Růžička M.: Integral operators on the halfspace in generalized Lebesgue spaces $L^{p(\cdot )}$, Part I. J. Math. Anal. Appl. 298 (2004), 559–571. Zbl pre02107284, MR 2005k:47098a. DOI 10.1016/j.jmaa.2004.05.048 | MR 2086975 | Zbl 1128.47044
[21] Diening L., Růžička M.: Integral operators on the halfspace in generalized Lebesgue spaces $L^{p(\cdot )}$, Part II. J. Math. Anal. Appl. 298 (2004), 572–588. Zbl pre02107285, MR 2005k:47098b. DOI 10.1016/j.jmaa.2004.05.049 | MR 2086976 | Zbl 1128.47044
[22] Donaldson T.: Nonlinear elliptic boundary value problems in Orlicz-Sobolev spaces. J. Differential Equations 10 (1971), 507–528. Zbl 0218.35028, MR 45 #7524. DOI 10.1016/0022-0396(71)90009-X | MR 0298472 | Zbl 0218.35028
[23] Eckart W.: Theoretische Untersuchungen von elektrorheologischen Flüssigkeiten bei homogenen und inhomogenen elektrischen Feldern. Aachen: Shaker Verlag.Darmstadt: TU Darmstadt, Fachbereich Mathematik, 2000. Zbl 0958.76003. Zbl 0958.76003
[24] Ettwein F., Růžička M.: Existence of local strong solutions for motions of electrorheological fluids in three dimensions. Comput. Appl. Math. 53 (2007), 595–604. DOI 10.1016/j.camwa.2006.02.032 | MR 2323712 | Zbl 1122.76092
[25] Frehse J., Málek J., Steinhauer M.: An existence result for fluids with shear dependent viscosity-steady flows. Nonlinear Anal., Theory Methods Appl. 30 (1997), no. 5, 3041–3049. Zbl 0902.35089, MR 99g:76006. DOI 10.1016/S0362-546X(97)00392-1 | MR 1602949
[26] Frehse J., Málek J., Steinhauer M.: On existence results for fluids with shear dependent viscosity – unsteady flows. Partial differential equations: theory and numerical solution. Proceedings of the ICM’98 satellite conference, Praha, 1998 (W. Jäger, J. Nečas, O. John, K. Najzar and J. Stará, eds.) 121–129, Chapman & Hall/CRC Res. Notes Math. 406, Boca Raton, FL, 2000. MR 1713880
[27] Frehse J., Málek J., Steinhauer M.: On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003), no. 5, 1064–1083 (electronic). Zbl 1050.35080, MR 2005c:76007. DOI 10.1137/S0036141002410988 | MR 2001659 | Zbl 1050.35080
[28] Giusti E.: Metodi diretti nel calcolo delle variazioni. Unione Matematica Italiana, Bologna, 1994. Zbl 0942.49002, MR 2000f:49001. MR 1707291 | Zbl 0942.49002
[29] Gossez J. P.: Nonlinear elliptic boundary value problems for equations with rapidly (or slowly) increasing coefficients. Trans. Amer. Math. Soc. 190 (1974), 163–205. Zbl 0239.35045, MR 49 #7598. DOI 10.1090/S0002-9947-1974-0342854-2 | MR 0342854
[30] Halsey T. C., Martin J. E., Adolf D.: Rheology of electrorheological fluids. Phys. Rev. Letters 68 (1992), 1519–1522. DOI 10.1103/PhysRevLett.68.1519
[31] Huber A.: Die Divergenzgleichung in gewichteten Räumen und Flüssigkeiten mit $p(\cdot )$-Struktur. Diplomarbeit, Universität Freiburg, 2005.
[32] Hudzik H.: The problems of separability, duality, reflexivity and of comparison for generalized orlicz-sobolev spaces $W^k_m(\omega )$. Comment. Math. Prace Mat. 21 (1979), 315–324. Zbl 0429.46017, MR 81k:46031. MR 0577521
[33] Kokilashvili V., Krbec M.: Weighted inequalities in Lorentz and Orlicz spaces. World Scientific Publishing Co., Inc., River Edge, NJ, 1991. Zbl 0751.46021, 93g:42013. MR 1156767 | Zbl 0751.46021
[34] Kováčik O., Rákosník J.: On spaces ${L}^{p(x)}$ and ${W}^{k,p(x)}$. Czechoslovak Math. J. 41(116) (1991), no. 4, 592–618. Zbl 0784.46029, MR 92m:46047.
[35] skii M. A. Krasnosel,’ Rutickii, Ja. B.: Convex functions and Orlicz spaces. Translated from the first Russian edition by Leo F. Boron, P. Noordhoff Ltd., Groningen, 1961. Zbl 0095.09103, MR 23 #A4016. MR 0126722
[36] Málek J., Nečas J., Rokyta M., Růžička M.: Weak and measure-valued solutions to evolutionary PDEs. Applied Mathematics and Mathematical Computations, vol. 13, Chapman & Hall, London, 1996. Zbl 0851.35002, MR 97g:35002. MR 1409366 | Zbl 0851.35002
[37] Marcellini P., Papi G.: Nonlinear elliptic systems with general growth. J. Differential Equations 221 (2006), no. 2, 412–443. Zbl pre05012954, 2007a:35024. DOI 10.1016/j.jde.2004.11.011 | MR 2196484
[38] Milani A., Picard R.: Decomposition theorems and their application to non-linear electro- and magneto-static boundary value problems. Partial differential equations and calculus of variations, Lecture Notes Math. 1357, 317–340, Springer, Berlin, 1988, LNM 1357. Zbl 0684.35084, MR 90b:35218. DOI 10.1007/BFb0082873 | MR 0976242
[39] Muckenhoupt B.: Weighted norm inequalities for the Hardy maximal function. Trans. Amer. Math. Soc. 165 (1972), 207–226. Zbl 0236.26016, MR 45 #2461. DOI 10.1090/S0002-9947-1972-0293384-6 | MR 0293384 | Zbl 0236.26016
[40] Musielak J., Orlicz W.: On modular spaces. Studia Math. 18 (1959), 49–65. Zbl 0086.08901, MR 21 #298. MR 0101487 | Zbl 0099.09202
[41] Musielak J.: Orlicz spaces and modular spaces. Lecture Notes Math. 1034. Springer-Verlag, Berlin, 1983. Zbl 0557.46020, MR 85m:46028. MR 0724434 | Zbl 0557.46020
[42] Nakano H.: Modulared semi-ordered linear spaces. Tokyo Math. Book Series, Vol. 1. Maruzen Co. Ltd., Tokyo, 1950. Zbl 0041.23401, MR 12,420a. MR 0038565 | Zbl 0041.23401
[43] Nekvinda A.: Hardy-Littlewood maximal operator on $L^{p(x)}(\mathbb R)$. Math. Inequal. Appl. 7 (2004), no. 2, 255–265. Zbl 1059.42016, MR 2005f:42045. MR 2057644 | Zbl 1059.42016
[44] Orlicz W.: Über konjugierte Exponentenfolgen. Studia Math. 3 (1931), 200–211. Zbl 0003.25203. Zbl 0003.25203
[45] Parthasarathy M., Klingenberg D. J.: Electrorheology: Mechanism and models. Materials, Sciences and Engineering R 17,2 (1996), 57–103. DOI 10.1016/0927-796X(96)00191-X
[46] Pick L., Růžička M.: An example of a space $L^{p(x)}$ on which the hardy-littlewood maximal operator is not bounded. Expo. Math. 19 (2001), 369–371. Zbl 1003.42013, MR 2002m:42016. DOI 10.1016/S0723-0869(01)80023-2 | MR 1876258
[47] Rajagopal K. R., Růžička M.: On the modeling of electrorheological materials. Mech. Research Comm. 23 (1996), 401–407. Zbl 0890.76007. DOI 10.1016/0093-6413(96)00038-9 | Zbl 0890.76007
[48] Rajagopal K. R., Růžička M.: Mathematical modeling of electrorheological materials. Cont. Mech. Thermodynamics 13 (2001), 59–78. Zbl 0971.76100. DOI 10.1007/s001610100034 | Zbl 0971.76100
[49] Růžička M.: A note on steady flow of fluids with shear dependent viscosity. Proceedings of the Second World Congress of Nonlinear Analysts, Part 5 (Athens, 1996). Nonlinear Anal., Theory Methods Appl. 30 (1997), no. 5, 3029–3039. Zbl 0906.35076, MR 99g:76005. MR 1602945
[50] Růžička M.: Flow of shear dependent electrorheological fluids: unsteady space periodic case. Applied nonlinear analysis (A. Sequeira, ed.), Kluwer/Plenum, New York, 1999, pp. 485–504. Zbl 0954.35138, MR 2001h:76115. MR 1727468 | Zbl 0954.35138
[51] Růžička M.: Electrorheological fluids: Modeling and mathematical theory. Lecture Notes in Math. 1748, Springer-Verlag, Berlin, 2000. MR 2002a:76004. MR 1810360 | Zbl 0968.76531
[52] Růžička M.: Modeling, mathematical and numerical analysis of electrorheological fluids. Appl. Math. 49 (2004), no. 6, 565–609. Zbl 1099.35103, MR 2005i:35223. DOI 10.1007/s10492-004-6432-8 | MR 2099981 | Zbl 1099.35103
[53] Samko S. G.: Density $C_0^\infty (\mathbb R^n)$ in the generalized Sobolev spaces $W^{m,p(x)}(\mathbb R^n)$. Dokl. Akad. Nauk 369 (1999), no. 4, 451–454. Zbl 1052.46028, MR2001a:46036. MR 1748959
[54] Samko S. G.: Convolution and potential type operators in $L^{p(x)}(\mathbb R^n)$. Integral Transform. Spec. Funct. 7 (1998), no. 3-4, 261–284. Zbl 1023.31009, MR 2001d:47076. DOI 10.1080/10652469808819204 | MR 1775832
[55] Sharapudinov I. I.: The basis property of the Haar system in the space $\Cal L^{p(t)}([0,1])$ and the principle of localization in the mean. Mat. Sb. (N.S.) 130(172) (1986), no. 2, 275–283, 286. English transl. in Math. USSR, Sb. 58 (1987), 279–287. Zbl 0639.42026, MR 88b:42034. MR 0854976
[56] Sharapudinov I. I.: On the uniform boundedness in $L^p\ (p=p(x))$ of some families of convolution operators. Mat. Zametki 59 (1996), no. 2, 291–302, 320. English transl. in Math. Notes 59 (1996), no. 1-2, 205–212. Zbl 0873.47023, MR 97c:47035. MR 1391844
[57] Talenti G.: Nonlinear elliptic equations, rearrangements of functions and Orlicz spaces. Ann. Mat. Pura Appl., IV. Ser. 120 (1979), 160–184. Zbl 0419.35041, MR 81i:35068. MR 0551065 | Zbl 0419.35041
[58] Wolf J.: Existence of weak solutions to the equations of nonstationary motion of non-Newtonian fluids with shear-dependent viscosity. J. Math. Fluid Mech. (2006), no. 1, 104–138. MR2305828. MR 2305828
[59] Zhikov V. V.: Averaging of functionals of the calculus of variations and elasticity theory. Izv. Akad. Nauk SSSR Ser. Mat. 50 (1986), no. 4, 675–710, 877. English transl. in Math. USSR Izv. 29 (1987), no. 1, 33–66. Zbl 0599.49031, MR 88a:49026. MR 0864171 | Zbl 0599.49031
[60] Zhikov V. V.: Meyer-type estimates for solving the nonlinear Stokes system. Differ. Uravn. 33 (1997), no. 1, 107–114, 143. English transl. in Differential Equations 33 (1997), no. 1, 108–115. Zbl 0911.35089, MR 99b:35170. MR 1607245
[61] Zhikov V. V.: On some variational problems. Russian J. Math. Phys. 5 (1997), no. 1, 105–116 (1998). Zbl 0917.49006, MR 98k:49004. MR 1486765 | Zbl 0917.49006
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