Existence of weak solutions for steady flows of electrorheological fluid with Navier-slip type boundary conditions

Sin, Cholmin ; Ri, Sin-Il

Mathematica Bohemica, Tome 147 (2022) no. 4, pp. 567-585

Abstract (VO)

We prove the existence of weak solutions for steady flows of electrorheological fluids with homogeneous Navier-slip type boundary conditions provided $p(x)>2n/(n+2)$. To prove this, we show Poincaré- and Korn-type inequalities, and then construct Lipschitz truncation functions preserving the zero normal component in variable exponent Sobolev spaces.

Détail

DOI : 10.21136/MB.2022.0200-20

Classification : 35A23, 35D30, 46E30, 46E35, 76A05, 76D03
Keywords: existence of weak solutions; electrorheological fluid; Lipschitz truncation; variable exponent

Bibliographie

[1] Abbatiello, A., Crispo, F., Maremonti, P.: Electrorheological fluids: Ill posedness of uniqueness backward in time. Nonlinear Anal., Theory Methods Appl., Ser. A 170 (2018), 47-69. | DOI | MR | JFM

[2] Bauer, S., Pauly, D.: On Korn's first inequality for mixed tangential and normal boundary conditions on bounded Lipschitz domains in $\mathbb{R}^N$. Ann. Univ. Ferrara, Sez. VII, Sci. Mat. 62 (2016), 173-188. | DOI | MR | JFM

[3] Veiga, H. Beirão da: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9 (2004), 1079-1114. | MR | JFM

[4] Veiga, H. Beirão da: On the regularity of flows with Ladyzhenskaya shear-dependent viscosity and slip or non-slip boundary conditions. Commun. Pure Appl. Math. 58 (2005), 552-577. | DOI | MR | JFM

[5] Bögelein, V., Duzaar, F., Habermann, J., Scheven, C.: Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients. Adv. Calc. Var. 5 (2012), 1-57. | DOI | MR | JFM

[6] Breit, D., Diening, L., Fuchs, M.: Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Equations 253 (2012), 1910-1942. | DOI | MR | JFM

[7] Breit, D., Diening, L., Schwarzacher, S.: Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013), 2671-2700. | DOI | MR | JFM

[8] Bulíček, M., Gwiazda, P., Málek, J., Świerczewska-Gwiazda, A.: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012), 2756-2801. | DOI | MR | JFM

[9] 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 (2007), 51-85. | DOI | MR | JFM

[10] Chen, P., Xiao, Y., Zhang, H.: Vanishing viscosity limit for the 3D nonhomogeneous incompressible Navier-Stokes equations with a slip boundary condition. Math. Methods Appl. Sci. 40 (2017), 5925-5932. | DOI | MR | JFM

[11] Crispo, F.: A note on the existence and uniqueness of time-periodic electro-rheological flows. Acta Appl. Math. 132 (2014), 237-250. | DOI | MR | JFM

[12] Desvillettes, L., Villani, C.: On a variant of Korn's inequality arising in statistical mechanics. ESAIM, Control Optim. Calc. Var. 8 (2002), 603-619. | DOI | MR | JFM

[13] Desvillettes, L., Villani, C.: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159 (2005), 245-316. | DOI | MR | JFM

[14] Diening, L., Harjulehto, P., Hästö, P., Růžička, M.: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). | DOI | MR | JFM

[15] Diening, L., Málek, J., Steinhauer, M.: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM, Control Optim. Calc. Var. 14 (2008), 211-232. | DOI | MR | JFM

[16] Diening, L., Růžička, M.: An existence result for non-Newtonian fluids in non-regular domains. Discrete Contin. Dyn. Syst., Ser. S 3 (2010), 255-268. | DOI | MR | JFM

[17] Diening, L., Růžička, M., Schumacher, K.: A decomposition technique for John domains. Ann. Acad. Sci. Fenn., Math. 35 (2010), 87-114. | DOI | MR | JFM

[18] Diening, L., Schwarzacher, S., Stroffolini, B., Verde, A.: Parabolic Lipschitz truncation and caloric approximation. Calc. Var. Partial Differ. Equ. 56 (2017), Article ID 120, 27 pages. | DOI | MR | JFM

[19] Ebmeyer, C.: Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity. Math. Methods Appl. Sci. 29 (2006), 1687-1707. | DOI | MR | JFM

[20] Fan, X.: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339 (2008), 1395-1412. | DOI | MR | JFM

[21] Frehse, J., Málek, J., Steinhauer, M.: An existence result for fluids with shear dependent viscosity---steady flows. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3041-3049. | DOI | MR | JFM

[22] Frehse, J., Málek, J., Steinhauer, M.: On analysis of steady flows of fluids with sheardependent viscosity based on the Lipschitz truncation method. SIAM J. Math. Anal. 34 (2003), 1064-1083. | DOI | MR | JFM

[23] Galdi, G. P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems. Springer Monographs in Mathematics. Springer, New York (2011). | DOI | MR | JFM

[24] Jiang, R., Kauranen, A.: Korn's inequality and John domains. Calc. Var. Partial Differ. Equ. 56 (2017), Article ID 109, 18 pages. | DOI | MR | JFM

[25] Kaplický, P., Tichý, J.: Boundary regularity of flows under perfect slip boundary conditions. Cent. Eur. J. Math. 11 (2013), 1243-1263. | DOI | MR | JFM

[26] Kučera, P., Neustupa, J.: On robustness of a strong solution to the Navier-Stokes equations with Navier's boundary conditions in the $L^3$-norm. Nonlinearity 30 (2017), 1564-1583. | DOI | MR | JFM

[27] Ladyzhenskaya, O. A.: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969). | MR | JFM

[28] Li, Y., Li, K.: Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions. J. Math. Anal. Appl. 381 (2011), 1-9. | DOI | MR | JFM

[29] Lions, J. L.: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969), French. | MR | JFM

[30] Mácha, V., Tichý, J.: Higher integrability of solutions to generalized Stokes system under perfect slip boundary conditions. J. Math. Fluid Mech. 16 (2014), 823-845. | DOI | MR | JFM

[31] Malý, J., Ziemer, W. P.: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs 51. American Mathematical Society, Providence (1997). | DOI | MR | JFM

[32] Neustupa, J., Penel, P.: On regularity of a weak solution to the Navier-Stokes equations with the generalized Navier slip boundary conditions. Adv. Math. Phys. 2018 (2018), Article ID 4617020, 7 pages. | DOI | MR | JFM

[33] Rădulescu, D. V., Repovš, D. D.: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015). | DOI | MR | JFM

[34] Růžička, M.: A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3029-3039. | DOI | MR | JFM

[35] Růžička, M.: Electrorheological Fluid: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). | DOI | MR | JFM

[36] Sin, C.: The existence of strong solutions to steady motion of electrorheological fluids in 3D cubic domain. J. Math. Anal. Appl. 445 (2017), 1025-1046. | DOI | MR | JFM

[37] Sin, C.: The existence of weak solutions for steady flow of electrorheological fluids with nonhomogeneous Dirichlet boundary condition. Nonlinear Anal., Theory Methods Appl., Ser. A 163 (2017), 146-162. | DOI | MR | JFM

[38] Sin, C.: Global regularity of weak solutions for steady motions of electrorheological fluids in 3D smooth domain. J. Math. Anal. Appl. 461 (2018), 752-776. | DOI | MR | JFM

[39] Sin, C.: Boundary partial regularity for steady flows of electrorheological fluids in 3D bounded domains. Nonlinear Anal., Theory Methods Appl., Ser. A 179 (2019), 309-343. | DOI | MR | JFM

[40] Solonnikov, V. A., Scadilov, V. E.: On a boundary value problem for a stationary system of Navier-Stokes equations. Proc. Steklov Inst. Math. 125 (1973), 186-199 translation from Trudy Mat. Inst. Steklov 125 1973 196-210. | MR | JFM