Article
Full entry |
PDF
(0.2 MB)
Feedback
PDF
(0.2 MB)
Feedback
Keywords:
Stokes problem; $L^q$ theory; pressure-dependent viscosity
Stokes problem; $L^q$ theory; pressure-dependent viscosity
Summary:
We study higher local integrability of a weak solution to the steady Stokes problem. We consider the case of a pressure- and shear-rate-dependent viscosity, i.e., the elliptic part of the Stokes problem is assumed to be nonlinear and it depends on $p$ and on the symmetric part of a gradient of $u$, namely, it is represented by a stress tensor $T(Du,p):=\nu (p,|D|^2)D$ which satisfies $r$-growth condition with $r\in (1,2]$. In order to get the main result, we use Calderón-Zygmund theory and the method which was presented for example in the paper Caffarelli, Peral (1998).
References:
[1] Amrouche, C., Girault, V.: Decomposition of vector spaces and application to the Stokes problem in arbitrary dimension. Czech. Math. J. 44 (1994), 109-140. MR 1257940 | Zbl 0823.35140
[2] Bridgman, P. W.: The Physics of High Pressure. MacMillan, New York (1931).
[3] Bulíček, M., Fišerová, V.: Existence theory for steady flows of fluids with pressure and shear rate dependent viscosity, for low values of the power-law index. Z. Anal. Anwend. 28 (2009), 349-371. DOI 10.4171/ZAA/1389 | MR 2506365 | Zbl 1198.35174
[4] Bulíček, M., Kaplický, P.: Incompressible fluids with shear rate and pressure dependent viscosity: regularity of steady planar flows. Discrete Contin. Dyn. Syst. Ser. S 1 (2008), 41-50. MR 2375580 | Zbl 1153.35314
[5] Bulíček, M., Málek, J., Rajagopal, K. R.: Analysis of the flows of incompressible fluids with pressure dependent viscosity fulfilling $\nu(p,\cdot)\rightarrow +\infty$ as $p\rightarrow +\infty$. Czech. Math. J. 59 (2009), 503-528. DOI 10.1007/s10587-009-0034-2 | MR 2532387
[6] Caffarelli, L. A., Peral, I.: On $W^{1,p}$ estimates for elliptic equations in divergence form. Commun. Pure Appl. Math. 51 (1998), 1-21. DOI 10.1002/(SICI)1097-0312(199801)51:1<1::AID-CPA1>3.0.CO;2-G | MR 1486629 | Zbl 0906.35030
[7] Diening, L., Ettwein, F.: Fractional estimates for non-differentiable elliptic systems with general growth. Forum Math. 20 (2008), 523-556. DOI 10.1515/FORUM.2008.027 | MR 2418205 | Zbl 1188.35069
[8] Diening, L., Kaplický, P.: $L^q$ theory for a generalized Stokes system. Manuscr. Math. 141 (2013), 333-361. DOI 10.1007/s00229-012-0574-x | MR 3042692 | Zbl 1263.35175
[9] Diening, L., Růžička, M., Schumacher, K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn. Math. 35 (2010), 87-114. DOI 10.5186/aasfm.2010.3506 | MR 2643399 | Zbl 1194.26022
[10] Franta, M., Málek, J., Rajagopal, K. R.: On steady flows of fluids with pressure- and shear-dependent viscosities. Proc. R. Soc. Lond., Ser. A, Math. Phys. Eng. Sci. (2005), 461 651-670. MR 2121929 | Zbl 1145.76311
[11] Gazzola, F., Secchi, P.: Some results about stationary Navier-Stokes equations with a pressure-dependent viscosity. Navier-Stokes Equations: Theory and Numerical Methods, Varenna, 1997 R. Salvi 31-37 Pitman Res. Notes Math. Ser. 388 Longman, Harlow (1998). MR 1773582 | Zbl 0940.35156
[12] Giusti, E.: Metodi Diretti Nel Calcolo Delle Variazioni. Italian Unione Matematica Italiana, Bologna (1994). MR 1707291 | Zbl 0942.49002
[13] Iwaniec, T.: On $L^p$-integrability in PDE's and quasiregular mappings for large exponents. Ann. Acad. Sci. Fenn. Ser. A I, Math. 7 (1982), 301-322. DOI 10.5186/aasfm.1982.0719 | MR 0686647 | Zbl 0505.30011
[14] Knauf, S., Frei, S., Richter, T., Rannacher, R.: Towards a complete numerical description of lubricant film dynamics in ball bearings. Comput. Mech. 53 (2014), 239-255. DOI 10.1007/s00466-013-0904-1 | MR 3158820
[15] Lanzendörfer, M.: On steady inner flows of an incompressible fluid with the viscosity depending on the pressure and the shear rate. Nonlinear Anal., Real World Appl. 10 (2009), 1943-1954. MR 2508405 | Zbl 1163.76335
[16] Lanzendörfer, M.: Numerical Simulations of the Flow in the Journal Bearing. Master's Thesis. Charles University in Prague, Faculty of Mathematics and Physics (2003).
[17] Lanzendörfer, M., Stebel, J.: On pressure boundary conditions for steady flows of incompressible fluids with pressure and shear rate dependent viscosities. Appl. Math., Praha 56 (2011), 265-285. DOI 10.1007/s10492-011-0016-1 | MR 2800578 | Zbl 1224.35347
[18] Mácha, V.: Partial regularity of solution to generalized Navier-Stokes problem. Cent. Eur. J. Math. 12 (2014), 1460-1483. MR 3224012 | Zbl 1303.35058
[19] Mácha, V., Tichý, J.: Higher integrability of solution to generalized Stokes system under perfect slip boundary conditions. J. Math. Fluid Mech. 16 (2014), 823-845. DOI 10.1007/s00021-014-0190-5 | MR 3267551
[20] Málek, J.: Mathematical properties of flows of incompressible power-law-like fluids that are described by implicit constitutive relations. Electron. Trans. Numer. Anal. 31 (2008), 110-125. MR 2569596 | Zbl 1182.35182
[21] Málek, J., Mingione, G., Stará, J.: Fluids with Pressure Dependent Viscosity, Partial Regularity of Steady Flows. F. Dumortier, et al. Equadiff 2003. Proc. Int. Conf. Differential Equations World Sci. Publ., Hackensack 380-385 (2005). MR 2185057 | Zbl 1101.76021
[22] Málek, J., Nečas, J., Rajagopal, K. R.: Global analysis of the flows of fluids with pressure-dependent viscosities. Arch. Ration. Mech. Anal. 165 (2002), 243-269. DOI 10.1007/s00205-002-0219-4 | MR 1941479 | Zbl 1022.76011
[23] Roubíček, T.: Nonlinear Partial Differential Equations with Applications. International Series of Numerical Mathematics 153 Birkhäuser, Basel (2005). MR 2176645 | Zbl 1087.35002
Search
Partner of
