Mathematica Bohemica, Vol. 147, No. 4, pp. 567-585, 2022


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

Cholmin Sin, Sin-Il Ri

Received December 22, 2020.   Published online February 7, 2022.

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


References:
[1] A. Abbatiello, F. Crispo, P. Maremonti: Electrorheological fluids: Ill posedness of uniqueness backward in time. Nonlinear Anal., Theory Methods Appl., Ser. A 170 (2018), 47-69. DOI 10.1016/j.na.2017.12.014 | MR 3765555 | Zbl 1469.35172
[2] S. Bauer, D. Pauly: 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 10.1007/s11565-016-0247-x | MR 3570353 | Zbl 1364.46028
[3] H. Beirão da Veiga: Regularity for Stokes and generalized Stokes systems under nonhomogeneous slip-type boundary conditions. Adv. Differ. Equ. 9 (2004), 1079-1114. MR 2098066 | Zbl 1103.35084
[4] H. Beirão da Veiga: 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 10.1002/cpa.20036 | MR 2119869 | Zbl 1075.35045
[5] V. Bögelein, F. Duzaar, J. Habermann, C. Scheven: Stationary electro-rheological fluids: Low order regularity for systems with discontinuous coefficients. Adv. Calc. Var. 5 (2012), 1-57. DOI 10.1515/acv.2011.009 | MR 2879566 | Zbl 1238.35095
[6] D. Breit, L. Diening, M. Fuchs: Solenoidal Lipschitz truncation and applications in fluid mechanics. J. Differ. Equations 253 (2012), 1910-1942. DOI 10.1016/j.jde.2012.05.010 | MR 2943947 | Zbl 1245.35080
[7] D. Breit, L. Diening, S. Schwarzacher: Solenoidal Lipschitz truncation for parabolic PDEs. Math. Models Methods Appl. Sci. 23 (2013), 2671-2700. DOI 10.1142/S0218202513500437 | MR 3119635 | Zbl 1309.76024
[8] M. Bulíček, P. Gwiazda, J. Málek, A. Świerczewska-Gwiazda: On unsteady flows of implicitly constituted incompressible fluids. SIAM J. Math. Anal. 44 (2012), 2756-2801. DOI 10.1137/110830289 | MR 3023393 | Zbl 1256.35074
[9] M. Bulíček, J. Málek, K. R. Rajagopal: 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 10.1512/iumj.2007.56.2997 | MR 2305930 | Zbl 1129.35055
[10] P. Chen, Y. Xiao, H. Zhang: 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 10.1002/mma.4443 | MR 3713338 | Zbl 1390.35226
[11] F. Crispo: A note on the existence and uniqueness of time-periodic electro-rheological flows. Acta Appl. Math. 132 (2014), 237-250. DOI 10.1007/s10440-014-9897-9 | MR 3255040 | Zbl 1295.76004
[12] L. Desvillettes, C. Villani: On a variant of Korn's inequality arising in statistical mechanics. ESAIM, Control Optim. Calc. Var. 8 (2002), 603-619. DOI 10.1051/cocv:2002036 | MR 1932965 | Zbl 1092.82032
[13] L. Desvillettes, C. Villani: On the trend to global equilibrium for spatially inhomogeneous kinetic systems: The Boltzmann equation. Invent. Math. 159 (2005), 245-316. DOI 10.1007/s00222-004-0389-9 | MR 2116276 | Zbl 1162.82316
[14] L. Diening, P. Harjulehto, P. Hästö, M. Růžička: Lebesgue and Sobolev Spaces with Variable Exponents. Lecture Notes in Mathematics 2017. Springer, Berlin (2011). DOI 10.1007/978-3-642-18363-8 | MR 2790542 | Zbl 1222.46002
[15] L. Diening, J. Málek, M. Steinhauer: On Lipschitz truncations of Sobolev functions (with variable exponent) and their selected applications. ESAIM, Control Optim. Calc. Var. 14 (2008), 211-232. DOI 10.1051/cocv:2007049 | MR 2394508 | Zbl 1143.35037
[16] L. Diening, M. Růžička: An existence result for non-Newtonian fluids in non-regular domains. Discrete Contin. Dyn. Syst., Ser. S 3 (2010), 255-268. DOI 10.3934/dcdss.2010.3.255 | MR 2610563 | Zbl 1193.35150
[17] L. Diening, M. Růžička, K. Schumacher: 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
[18] L. Diening, S. Schwarzacher, B. Stroffolini, A. Verde: Parabolic Lipschitz truncation and caloric approximation. Calc. Var. Partial Differ. Equ. 56 (2017), Article ID 120, 27 pages. DOI 10.1007/s00526-017-1209-6 | MR 3672391 | Zbl 1377.35144
[19] C. Ebmeyer: Regularity in Sobolev spaces of steady flows of fluids with shear-dependent viscosity. Math. Methods Appl. Sci. 29 (2006), 1687-1707. DOI 10.1002/mma.748 | MR 2248563 | Zbl 1124.35053
[20] X. Fan: Boundary trace embedding theorems for variable exponent Sobolev spaces. J. Math. Anal. Appl. 339 (2008), 1395-1412. DOI 10.1016/j.jmaa.2007.08.003 | MR 2377096 | Zbl 1136.46025
[21] J. Frehse, J. Málek, M. Steinhauer: An existence result for fluids with shear dependent viscosity - steady flows. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3041-3049. DOI 10.1016/S0362-546X(97)00392-1 | MR 1602949 | Zbl 0902.35089
[22] J. Frehse, J. Málek, M. Steinhauer: 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 10.1137/S0036141002410988 | MR 2001659 | Zbl 1050.35080
[23] G. P. Galdi: An Introduction to the Mathematical Theory of the Navier-Stokes Equations: Steady-State Problems. Springer Monographs in Mathematics. Springer, New York (2011). DOI 10.1007/978-0-387-09620-9 | MR 2808162 | Zbl 1245.35002
[24] R. Jiang, A. Kauranen: Korn's inequality and John domains. Calc. Var. Partial Differ. Equ. 56 (2017), Article ID 109, 18 pages. DOI 10.1007/s00526-017-1196-7 | MR 3669778 | Zbl 1373.35015
[25] P. Kaplický, J. Tichý: Boundary regularity of flows under perfect slip boundary conditions. Cent. Eur. J. Math. 11 (2013), 1243-1263. DOI 10.2478/s11533-013-0232-x | MR 3047056 | Zbl 1278.35040
[26] P. Kučera, J. Neustupa: 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 10.1088/1361-6544/aa6166 | MR 3636311 | Zbl 1367.35109
[27] O. A. Ladyzhenskaya: The Mathematical Theory of Viscous Incompressible Flow. Gordon and Breach, New York (1969). MR 0254401 | Zbl 0184.52603
[28] Y. Li, K. Li: Existence of the solution to stationary Navier-Stokes equations with nonlinear slip boundary conditions. J. Math. Anal. Appl. 381 (2011), 1-9. DOI 10.1016/j.jmaa.2011.04.020 | MR 2796187 | Zbl 1221.35282
[29] J. L. Lions: Quelques méthodes de résolution des problèmes aux limites non linéaires. Dunod, Paris (1969). (In French.) MR 0259693 | Zbl 0189.40603
[30] V. Mácha, J. Tichý: Higher integrability of solutions 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 | Zbl 1309.35089
[31] J. Malý, W. P. Ziemer: Fine Regularity of Solutions of Elliptic Partial Differential Equations. Mathematical Surveys and Monographs 51. American Mathematical Society, Providence (1997). DOI 10.1090/surv/051 | MR 1461542 | Zbl 0882.35001
[32] J. Neustupa, P. Penel: 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 10.1155/2018/4617020 | MR 3773415 | Zbl 1406.35236
[33] D. V. Rădulescu, D. D. Repovš: Partial Differential Equations with Variable Exponents: Variational Methods and Qualitative Analysis. Monographs and Research Notes in Mathematics. CRC Press, Boca Raton (2015). DOI 10.1201/b18601 | MR 3379920 | Zbl 1343.35003
[34] M. Růžička: A note on steady flow of fluids with shear dependent viscosity. Nonlinear Anal., Theory Methods Appl. 30 (1997), 3029-3039. DOI 10.1016/S0362-546X(97)00391-X | MR 1602945 | Zbl 0906.35076
[35] M. Růžička: Electrorheological Fluid: Modeling and Mathematical Theory. Lecture Notes in Mathematics 1748. Springer, Berlin (2000). DOI 10.1007/BFb0104029 | MR 1810360 | Zbl 0962.76001
[36] C. Sin: The existence of strong solutions to steady motion of electrorheological fluids in 3D cubic domain. J. Math. Anal. Appl. 445 (2017), 1025-1046. DOI 10.1016/j.jmaa.2016.07.019 | MR 3543809 | Zbl 1352.35124
[37] C. Sin: 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 10.1016/j.na.2017.06.014 | MR 3695973 | Zbl 1375.35400
[38] C. Sin: Global regularity of weak solutions for steady motions of electrorheological fluids in 3D smooth domain. J. Math. Anal. Appl. 461 (2018), 752-776. DOI 10.1016/j.jmaa.2017.10.081 | MR 3759566 | Zbl 1387.35082
[39] C. Sin: 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 10.1016/j.na.2018.08.009 | MR 3886635 | Zbl 1404.35079
[40] V. A. Solonnikov, V. E. Scadilov: 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 0172014 | Zbl 0313.35063

Affiliations:   Cholmin Sin, Sin-Il Ri, Institute of Mathematics, State Academy of Sciences, KwaHaK-1Dong, Unjong District, Pyongyang, Democratic People's Republic of Korea, e-mail: sincm1223@star-co.net.kp, si.ri@star-co.net.kp


 
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