On the \(C^{1,\gamma}(\overline \varOmega)\cap W^{2,2}(\varOmega) \) regularity for a class of electro-rheological fluids. (English) Zbl 1178.35300
The stationary boundary value problem for a class of electro-rheological fluids is studied in the paper. The velocity \(v(x)\) and the pressure \(\pi(x)\) satisfy to the equations in a bounded smooth domain \(\Omega\subset \mathbb R^n\), \(n\geq 2\):
\[
\begin{cases} &-\nabla\cdot S(Dv)+(v\cdot\nabla)v+\nabla\pi=f,\quad \text{div}\,v=0, \quad x\in\Omega,\\ &v|_{\partial\Omega}=0. \end{cases}
\]
Here \(S(Dv)\) is the extra-stress tensor of the form
\[
S(Dv)=\big(1+|Dv|\big)^{p\,(x)-2}Dv,\quad Dv=\frac{1}{2}\big(\nabla v+\nabla v^T\big),
\]
where \(p\,(x)\) is a given continuous function such that \(1<p_\infty\leq p\,(x)\leq p_0\), \(p_0>2\).
The author prove the existence and uniqueness of a smooth solution \(v\in C^{1,\gamma}(\bar{\Omega})\cap W^{2,2}(\Omega)\), \(\pi\in C^{0,\gamma}(\bar{\Omega})\cap W^{1,2}(\Omega)\) to the small \(f\) without further restrictions on the constants \(p_\infty,p_0\).
The author prove the existence and uniqueness of a smooth solution \(v\in C^{1,\gamma}(\bar{\Omega})\cap W^{2,2}(\Omega)\), \(\pi\in C^{0,\gamma}(\bar{\Omega})\cap W^{1,2}(\Omega)\) to the small \(f\) without further restrictions on the constants \(p_\infty,p_0\).
Reviewer: Il’ya Sh. Mogilevskij (Tver’)
MSC:
| 35Q35 | PDEs in connection with fluid mechanics |
| 76A05 | Non-Newtonian fluids |
| 76W05 | Magnetohydrodynamics and electrohydrodynamics |
References:
| [1] | Acerbi, E.; Mingione, G., Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164, 213-259 (2002) · Zbl 1038.76058 |
| [2] | Apushkinskaya, D.; Bildhauer, M.; Fuchs, M., Steady states of anisotropic generalized Newtonian fluids, J. Math. Fluid Mech., 7, 261-297 (2005) · Zbl 1162.76375 |
| [3] | Beirão da Veiga, H., On the regularity of flows with Ladyzhenskaya shear dependent viscosity and slip or non-slip boundary conditions, Comm. Pure Appl. Math., 58, 552-577 (2005) · Zbl 1075.35045 |
| [4] | Beirão da Veiga, H., Navier-Stokes equations with shear thinning viscosity. Regularity up to the boundary, J. Math. Fluid Mech. (2008) · Zbl 1190.35174 |
| [5] | Beirão da Veiga, H., Navier-Stokes equations with shear thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech. (2008) · Zbl 1213.76047 |
| [6] | Beirão da Veiga, H., On the Ladyzhenskaya-Smagorinsky turbulence model of the Navier-Stokes equations in smooth domains. The regularity problem, J. Eur. Math. Soc. (JEMS), 11, 127-167 (2009) · Zbl 1155.35073 |
| [7] | Beirão da Veiga, H., Turbulence models, \(p\)-fluid flows and \(W^{2, l}\)-regularity of solutions, Commun. Pure Appl. Anal., 8, 769-783 (2009) · Zbl 1168.35031 |
| [8] | Beirão da Veiga, H., On non-Newtonian \(p\)-fluids. The pseudo-plastic case, J. Math. Anal. Appl., 344, 175-185 (2008) · Zbl 1168.76001 |
| [9] | Beirão da Veiga, H., On the global regularity of shear-thinning flows in smooth domains, J. Math. Anal. Appl., 349, 335-360 (2009) · Zbl 1229.35200 |
| [10] | H. Beirão da Veiga, P. Kaplický, M. Růžička, Boundary regularity of shear thickening flows, Preprint Series of the Charles University, Prague, MATH-KMA-2008/277; H. Beirão da Veiga, P. Kaplický, M. Růžička, Boundary regularity of shear thickening flows, Preprint Series of the Charles University, Prague, MATH-KMA-2008/277 · Zbl 1270.35360 |
| [11] | Berselli, L. C., On the \(W^{2, q}\)-regularity of incompressible fluids with shear-dependent viscosities: The shear thinning case, J. Math. Fluid Mech. (2008) |
| [12] | Bildhauer, M.; Fuchs, M.; Zhong, X., On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids, Algebra i Analiz. Algebra i Analiz, St. Petersburg Math. J., 18, 183-199 (2007), translation in: · Zbl 1129.35061 |
| [13] | Bulíček, M.; Feireisl, E.; Málek, J., A Navier-Stokes-Fourier system for incompressible fluids with temperature dependent material coefficients, Nonlinear Anal. Real World Appl., 10, 992-1015 (2009) · Zbl 1167.76316 |
| [14] | Bulíček, M.; Málek, J.; Rajagopal, K. R., Navier’s slip and evolutionary Navier-Stokes-like systems with pressure and shear-rate dependent viscosity, Indiana Univ. Math. J., 56, 51-86 (2007) · Zbl 1129.35055 |
| [15] | Consiglieri, L., Weak solutions for a class of non-Newtonian fluids with energy transfer, J. Math. Fluid Mech., 2, 267-293 (2000) · Zbl 0974.35090 |
| [16] | Consiglieri, L., Friction boundary conditions on thermal incompressible viscous flows, Ann. Mat. Pura Appl., 187, 647-665 (2008) · Zbl 1223.35255 |
| [17] | Crispo, F., Shear thinning viscous fluids in cylindrical domains. Regularity up to the boundary, J. Math. Fluid Mech., 10, 311-325 (2008) · Zbl 1162.76303 |
| [18] | Crispo, F., Global regularity of a class of \(p\)-fluid flows in cylinders, J. Math. Anal. Appl., 341, 559-574 (2008) · Zbl 1135.35061 |
| [19] | F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids, Port. Math. (2009), in press; F. Crispo, A note on the global regularity of steady flows of generalized Newtonian fluids, Port. Math. (2009), in press · Zbl 1170.35491 |
| [20] | F. Crispo, On the regularity of shear-thickening viscous fluids, Chinese Ann. Math. Ser. B, in press; F. Crispo, On the regularity of shear-thickening viscous fluids, Chinese Ann. Math. Ser. B, in press · Zbl 1179.35237 |
| [21] | Crispo, F.; Grisanti, C. R., On the existence, uniqueness and \(C^{1, \gamma}(\overline{\Omega}) \cap W^{2, 2}(\Omega)\) regularity for a class of shear-thinning fluids, J. Math. Fluid Mech., 10, 455-487 (2008) · Zbl 1189.35221 |
| [22] | Diening, L.; Růžička, M., Non-Newtonian fluids and function spaces, Nonlinear Anal. Funct. Spaces Appl., 8, 95-143 (2007) · Zbl 1289.35104 |
| [23] | 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, 2, 457-472 (2007) · Zbl 1140.65060 |
| [24] | Diening, L.; Ettwein, F.; Růžička, M., \(C^{1, \alpha}\)-regularity for electrorheological fluids in two dimensions, NoDEA Nonlinear Differential Equations Appl., 14, 207-217 (2007) · Zbl 1132.76301 |
| [25] | Franta, M.; Málek, J.; Rajagopal, K., On steady flows of fluids with pressure- and shear-dependent viscosities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 461, 651-670 (2005), (English summary) · Zbl 1145.76311 |
| [26] | Galdi, G. P., An Introduction to the Mathematical Theory of the Navier-Stokes Equations, vol. I, Springer Tracts in Nat. Philos., vol. 38 (1994), Springer-Verlag · Zbl 0949.35004 |
| [27] | Giaquinta, M.; Modica, G., Non linear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330, 173-214 (1982) · Zbl 0492.35018 |
| [28] | Hron, J.; Málek, J.; Rajagopal, K. R., Simple flows of fluids with pressure dependent viscosities, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 477, 277-302 (2001) · Zbl 1052.76017 |
| [29] | Kaplický, P., Regularity of flow of anisotropic fluid, J. Math. Fluid Mech., 10, 71-88 (2008) · Zbl 1162.76304 |
| [30] | Kaplický, P.; Málek, J.; Stará, J., \(C^{1, \alpha}\)-solutions to a class of nonlinear fluids in 2D stationary Dirichlet problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 259, 89-121 (1999) · Zbl 0978.35046 |
| [31] | Kufner, A.; John, O.; Fučík, S., Function Spaces, Monographs and Textbooks on Mechanics of Solids and Fluids. Mechanics: Analysis (1977), Noordhoff International Publishing, Academia: Noordhoff International Publishing, Academia Leyden, Prague · Zbl 0364.46022 |
| [32] | Málek, J.; Necas, J.; Rajagopal, K. R., Global analysis of the flows of fluids with pressure-dependent viscosities, Arch. Ration. Mech. Anal., 165, 243-269 (2002) · Zbl 1022.76011 |
| [33] | Málek, J.; Necas, J.; Růžička, M., On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: The case \(p \geqslant 2\), Adv. Differential Equations, 6, 257-302 (2001) · Zbl 1021.35085 |
| [34] | Málek, J.; Rajagopal, K. R., Mathematical properties of the equations governing the flow of fluids with pressure and shear rate dependent viscosities, (Handbook of Mathematical Fluid Dynamics, vol. 4 (2007), Elsevier B.V.) · Zbl 1112.76005 |
| [35] | Málek, J.; Růžička, M.; Shelukhin, V. V., Herschel-Bulkley fluids: Existence and regularity of steady flows, Math. Models Methods Appl. Sci., 15, 1845-1861 (2005) · Zbl 1098.76008 |
| [36] | Nečas, J., Équations aux dérivées partielles (1965), Presses de l’Université de Montréal: Presses de l’Université de Montréal Montréal · Zbl 0143.14003 |
| [37] | Parés, C., Existence, uniqueness and regularity of solution of the equations of a turbulence model for incompressible fluids, Appl. Anal., 43, 245-296 (1992) · Zbl 0739.35075 |
| [38] | Rajagopal, K. R.; Růžička, M., Mathematical modeling of electrorheological materials, Contin. Mech. Thermodyn., 13, 59-78 (2001) · Zbl 0971.76100 |
| [39] | Růžička, M., Electrorheological Fluids: Modeling and Mathematical Theory, Lecture Notes in Math., vol. 1748 (2000), Springer-Verlag: Springer-Verlag Berlin, Heidelberg, New York · Zbl 0968.76531 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
