September 2018, 13(3): 479-491. doi: 10.3934/nhm.2018021

Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth

1. 

Department of Financial Engineering, Ajou University, Suwon, Korea

2. 

Manufacturing Technology Center, Samsung Electronics, Giheung, Korea

3. 

Department of Mathematics, Seoul National University, Seoul 08826, Republic of Korea

* Corresponding author: Yeonghun Youn

Received  November 2017 Revised  March 2018 Published  July 2018

We consider weak solutions to the equations of stationary motion of a class of non-Newtonian fluids which includes the power law model. The power depends on the spatial variable, which is motivated by electrorheological fluids. We prove the existence of second order derivatives of weak solutions in the shear thinning cases.

Citation: Hyeong-Ohk Bae, Hyoungsuk So, Yeonghun Youn. Interior regularity to the steady incompressible shear thinning fluids with non-Standard growth. Networks & Heterogeneous Media, 2018, 13 (3) : 479-491. doi: 10.3934/nhm.2018021
References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. doi: 10.1007/s00205-002-0208-7.

[2]

R. A. Adams, Sobolev Spaces, Academic Press, New York-London, 1975.

[3]

D. ApushkinskayaM. Bildhauer and M. Fuchs, Steady states of anisotropic generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 261-297. doi: 10.1007/s00021-004-0118-6.

[4]

H. Beirão da Veiga, Navier-Stokes Equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2.

[5]

M. BildhauerM. Fuchs and X. Zhong, On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids, Algebra i Analiz, 18 (2006), 1-23. doi: 10.1090/S1061-0022-07-00948-X.

[6]

S.-S. Byun and J. Ok, On $W^{1, q(·)}$-estimates for elliptic equations of $p(x)$-Laplacian type, J. Math. Pures Appl., 106 (2016), 512-545. doi: 10.1016/j.matpur.2016.03.002.

[7]

S. Challal and A. Lyaghfouri, Second order regularity for the $p(x)$-Laplace operator, Math. Nachr., 284 (2011), 1270-1279. doi: 10.1002/mana.200810285.

[8]

F. Crispo and C. R. Grisanti, On the $C^{1, γ}(\bar{Ω}) \cap W^{2, 2} (Ω)$ regularity for a class of electro-rheological fluids, J. Math. Anal. Appl., 356 (2009), 119-132. doi: 10.1016/j.jmaa.2009.02.013.

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

L. Diening and 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.

[11]

F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Comput. Math. Appl., 53 (2007), 595-604. doi: 10.1016/j.camwa.2006.02.032.

[12]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[13]

J. FrehseJ. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal., 34 (2003), 1064-1083. doi: 10.1137/S0036141002410988.

[14]

P. KaplickýJ. Málek and J. Stará, $C^{1, α}$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 259 (1999), 89-121. doi: 10.1023/A:1014440207817.

[15]

J. Naumann and J. Wolf, Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 298-313. doi: 10.1007/s00021-004-0120-z.

[16]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.

show all references

References:
[1]

E. Acerbi and G. Mingione, Regularity results for stationary electro-rheological fluids, Arch. Ration. Mech. Anal., 164 (2002), 213-259. doi: 10.1007/s00205-002-0208-7.

[3]

D. ApushkinskayaM. Bildhauer and M. Fuchs, Steady states of anisotropic generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 261-297. doi: 10.1007/s00021-004-0118-6.

[4]

H. Beirão da Veiga, Navier-Stokes Equations with shear-thickening viscosity. Regularity up to the boundary, J. Math. Fluid Mech., 11 (2009), 233-257. doi: 10.1007/s00021-008-0257-2.

[5]

M. BildhauerM. Fuchs and X. Zhong, On strong solutions of the differential equations modeling the steady flow of certain incompressible generalized Newtonian fluids, Algebra i Analiz, 18 (2006), 1-23. doi: 10.1090/S1061-0022-07-00948-X.

[6]

S.-S. Byun and J. Ok, On $W^{1, q(·)}$-estimates for elliptic equations of $p(x)$-Laplacian type, J. Math. Pures Appl., 106 (2016), 512-545. doi: 10.1016/j.matpur.2016.03.002.

[7]

S. Challal and A. Lyaghfouri, Second order regularity for the $p(x)$-Laplace operator, Math. Nachr., 284 (2011), 1270-1279. doi: 10.1002/mana.200810285.

[8]

F. Crispo and C. R. Grisanti, On the $C^{1, γ}(\bar{Ω}) \cap W^{2, 2} (Ω)$ regularity for a class of electro-rheological fluids, J. Math. Anal. Appl., 356 (2009), 119-132. doi: 10.1016/j.jmaa.2009.02.013.

[9]

L. Diening, P. Harjulehto, P. Hästö and M. Růžička, Lebesgue and Sobolev Spaces with Variable Exponents, Springer, Heidelberg, 2011. doi: 10.1007/978-3-642-18363-8.

[10]

L. Diening and 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.

[11]

F. Ettwein and M. Růžička, Existence of local strong solutions for motions of electrorheological fluids in three dimensions, Comput. Math. Appl., 53 (2007), 595-604. doi: 10.1016/j.camwa.2006.02.032.

[12]

L. C. Evans, Partial Differential Equations, 2$^{nd}$ edition, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI, 2010. doi: 10.1090/gsm/019.

[13]

J. FrehseJ. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the Lipschitz truncation method, SIAM J. Math. Anal., 34 (2003), 1064-1083. doi: 10.1137/S0036141002410988.

[14]

P. KaplickýJ. Málek and J. Stará, $C^{1, α}$-solutions to a class of nonlinear fluids in two dimensions-stationary Dirichlet problem, Zap. Nauchn. Sem. S.-Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 259 (1999), 89-121. doi: 10.1023/A:1014440207817.

[15]

J. Naumann and J. Wolf, Interior differentiability of weak solutions to the equations of stationary motion of a class of non-Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 298-313. doi: 10.1007/s00021-004-0120-z.

[16]

M. Růžička, Electrorheological Fluids: Modeling and Mathematical Theory, Springer-Verlag, Berlin, 2000.

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