September  2017, 37(9): 4815-4834. doi: 10.3934/dcds.2017207

Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure

1. 

Department of Mathematics, Mokpo National University, Mokpo, Republic of Korea

2. 

Department of Mathematics, Yonsei University, Seoul, Republic of Korea

* Corresponding author: Kyungkeun Kang

Received  June 2016 Revised  April 2017 Published  June 2017

Fund Project: Bum Ja Jin's work is supported by NRF-2014R1A1A3A04049515 and Kyungkeun Kang's work was partially supported by NRF-2014R1A2A1A11051161 and NRF20151009350.

We prove a Caccioppoli type inequality for the solution of a parabolic system related to the nonlinear Stokes problem. Using the method of Caccioppoli type inequality, we also establish the existence of weak solutions satisfying a local energy inequality without pressure for the non-Newtonian Navier-Stokes equations.

Citation: Bum Ja Jin, Kyungkeun Kang. Caccioppoli type inequality for non-Newtonian Stokes system and a local energy inequality of non-Newtonian Navier-Stokes equations without pressure. Discrete & Continuous Dynamical Systems - A, 2017, 37 (9) : 4815-4834. doi: 10.3934/dcds.2017207
References:
[1]

H. Amman, Stability of the rest state of viscous incompressible fluid, Arch. Rat. Mech. Anal., 126 (1994), 231-242.  doi: 10.1007/BF00375643.  Google Scholar

[2]

H.-O. Bae and J.-B. Jin, Regularity of Non-Newtonian fluids, J. Math. Fluid Mech., 16 (2014), 225-241.  doi: 10.1007/s00021-013-0149-y.  Google Scholar

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H. Beirão da Veiga, On some boundary value problems for incompressible viscous flows with Shear dependent viscosity, Progress in Nonlinear Differentail Equations, 63 (2005), 23-32.  doi: 10.1007/3-7643-7384-9_3.  Google Scholar

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H. Beirão da VeigaP. Kaplický and M. Ružička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404.  doi: 10.1007/s00021-010-0025-y.  Google Scholar

[6]

H. BelloutF. Bloom and J. Nečas, Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids, Comm. in PDE, 19 (1994), 1763-1803.  doi: 10.1080/03605309408821073.  Google Scholar

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L. C. BerselliL. Diening and M. Ružička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech., 12 (2010), 101-132.  doi: 10.1007/s00021-008-0277-y.  Google Scholar

[8]

D. Bothe and J. Prüss, Lp-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635.  Google Scholar

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M. BuličekF. EttweinP. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 1995-2010.  doi: 10.1002/mma.1314.  Google Scholar

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H. J. Choe and M. Yang, Hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations, Comm. Math. phys., 336 (2015), 171-198.  doi: 10.1007/s00220-015-2307-y.  Google Scholar

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L. Diening and M. Ružička, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 413-450.  doi: 10.1007/s00021-004-0124-8.  Google Scholar

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L. DieningM. Ruzicka and J. Wolf, Existence of weak solutions for unsteady motions of generalized newtonian fluids, Ann. Sc. Norm. Super. Pisa cl. Sci. (5), 9 (2010), 1-46.   Google Scholar

[13]

M. Fuchs and G. A. Seregin, A global nonlinear evolution problem for generalized Newtonian fluids: Local initial regularity of the strong solution, Comput. Math. Appl., 53 (2007), 509-520.  doi: 10.1016/j.camwa.2006.02.039.  Google Scholar

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M. Fuchs and G. A. Seregin, Existence of global solutions for a parabolic system related to the nonlinear Stokes problem, Zap. Nauchn. Sem. S. -Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 348 (2007), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 38,254–271,306; translation in J. Math. Sci. (N. Y. ) 152 (2008), 769–779. doi: 10.1007/s10958-008-9088-1.  Google Scholar

[15]

M. Fuchs and G. Zhang, Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model, calc. Var. Partial Differential Equations, 44 (2012), 271-295.  doi: 10.1007/s00526-011-0434-7.  Google Scholar

[16]

M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214.   Google Scholar

[17]

B. J. Jin, On the caccioppoli inequality of the unsteady Stokes system, Int. J. Numer. Anal. Model. Ser. B, 4 (2013), 215-223.   Google Scholar

[18]

O. A. Ladyženskaya, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Trudy Mat. Inst. Steklov., 102 (1967), 85-104.   Google Scholar

[19]

O. A. Ladyženskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zapiski Naukhnych Seminarov LOMI, 7 (1968), 126-154.   Google Scholar

[20]

O. A. Ladyženskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Beach, New York, 1969.  Google Scholar

[21]

J. L. Lions, Quelques Methodes de Résolution de Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar

[22]

J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical computation, 13. chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[23]

J. MálekJ. Nečas and M. Ružička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.  doi: 10.1142/S0218202593000047.  Google Scholar

[24]

J. MálekJ. Nečas and M. Ružička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2, Adv. Differential Equations, 6 (2001), 257-302.   Google Scholar

[25]

J. MálekD. Pražák and M. Steinhauer, On the Existence and Regularity of solutions for degenerate power-law fluids, Differential Integral Equations, 19 (2006), 449-462.   Google Scholar

[26]

J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138.  doi: 10.1007/s00021-006-0219-5.  Google Scholar

[27]

J. Wolf, A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations, in Advances in mathematical fluid mechanics, Springer, Berlin, (2010), 613–630.  Google Scholar

[28]

J. Wolf, On the local regularity of suitable weak solutions to the generalized Navier-Stokes equations, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 61 (2015), 149-171.  doi: 10.1007/s11565-014-0203-6.  Google Scholar

show all references

References:
[1]

H. Amman, Stability of the rest state of viscous incompressible fluid, Arch. Rat. Mech. Anal., 126 (1994), 231-242.  doi: 10.1007/BF00375643.  Google Scholar

[2]

H.-O. Bae and J.-B. Jin, Regularity of Non-Newtonian fluids, J. Math. Fluid Mech., 16 (2014), 225-241.  doi: 10.1007/s00021-013-0149-y.  Google Scholar

[3]

H. Beirão da Veiga, On the regularity of flows with Ladyzhenskaya Shear-dependent viscosity and slip or nonslip boundary conditions, Comm. Pure Appl. Math., 58 (2005), 552-577.  doi: 10.1002/cpa.20036.  Google Scholar

[4]

H. Beirão da Veiga, On some boundary value problems for incompressible viscous flows with Shear dependent viscosity, Progress in Nonlinear Differentail Equations, 63 (2005), 23-32.  doi: 10.1007/3-7643-7384-9_3.  Google Scholar

[5]

H. Beirão da VeigaP. Kaplický and M. Ružička, Boundary regularity of shear thickening flows, J. Math. Fluid Mech., 13 (2011), 387-404.  doi: 10.1007/s00021-010-0025-y.  Google Scholar

[6]

H. BelloutF. Bloom and J. Nečas, Young Measure-Valued Solutions for Non-Newtonian Incompressible Fluids, Comm. in PDE, 19 (1994), 1763-1803.  doi: 10.1080/03605309408821073.  Google Scholar

[7]

L. C. BerselliL. Diening and M. Ružička, Existence of strong solutions for incompressible fluids with shear dependent viscosities, J. Math. Fluid Mech., 12 (2010), 101-132.  doi: 10.1007/s00021-008-0277-y.  Google Scholar

[8]

D. Bothe and J. Prüss, Lp-theory for a class of non-Newtonian fluids, SIAM J. Math. Anal., 39 (2007), 379-421.  doi: 10.1137/060663635.  Google Scholar

[9]

M. BuličekF. EttweinP. Kaplický and D. Pražák, On uniqueness and time regularity of flows of power-law like non-Newtonian fluids, Math. Methods Appl. Sci., 33 (2010), 1995-2010.  doi: 10.1002/mma.1314.  Google Scholar

[10]

H. J. Choe and M. Yang, Hausdorff measure of the singular set in the incompressible magnetohydrodynamic equations, Comm. Math. phys., 336 (2015), 171-198.  doi: 10.1007/s00220-015-2307-y.  Google Scholar

[11]

L. Diening and M. Ružička, Strong solutions for generalized Newtonian fluids, J. Math. Fluid Mech., 7 (2005), 413-450.  doi: 10.1007/s00021-004-0124-8.  Google Scholar

[12]

L. DieningM. Ruzicka and J. Wolf, Existence of weak solutions for unsteady motions of generalized newtonian fluids, Ann. Sc. Norm. Super. Pisa cl. Sci. (5), 9 (2010), 1-46.   Google Scholar

[13]

M. Fuchs and G. A. Seregin, A global nonlinear evolution problem for generalized Newtonian fluids: Local initial regularity of the strong solution, Comput. Math. Appl., 53 (2007), 509-520.  doi: 10.1016/j.camwa.2006.02.039.  Google Scholar

[14]

M. Fuchs and G. A. Seregin, Existence of global solutions for a parabolic system related to the nonlinear Stokes problem, Zap. Nauchn. Sem. S. -Peterburg. Otdel. Mat. Inst. Steklov. (POMI), 348 (2007), Kraevye Zadachi Matematicheskoi Fiziki i Smezhnye Voprosy Teorii Funktsii. 38,254–271,306; translation in J. Math. Sci. (N. Y. ) 152 (2008), 769–779. doi: 10.1007/s10958-008-9088-1.  Google Scholar

[15]

M. Fuchs and G. Zhang, Liouville theorems for entire local minimizers of energies defined on the class L log L and for entire solutions of the stationary Prandtl-Eyring fluid model, calc. Var. Partial Differential Equations, 44 (2012), 271-295.  doi: 10.1007/s00526-011-0434-7.  Google Scholar

[16]

M. Giaquinta and G. Modica, Nonlinear systems of the type of the stationary Navier-Stokes system, J. Reine Angew. Math., 330 (1982), 173-214.   Google Scholar

[17]

B. J. Jin, On the caccioppoli inequality of the unsteady Stokes system, Int. J. Numer. Anal. Model. Ser. B, 4 (2013), 215-223.   Google Scholar

[18]

O. A. Ladyženskaya, New equations for the description of the motions of viscous incompressible fluids, and global solvability for their boundary value problems, Trudy Mat. Inst. Steklov., 102 (1967), 85-104.   Google Scholar

[19]

O. A. Ladyženskaya, Modifications of the Navier-Stokes equations for large gradients of the velocities, Zapiski Naukhnych Seminarov LOMI, 7 (1968), 126-154.   Google Scholar

[20]

O. A. Ladyženskaya, The Mathematical Theory of Viscous Incompressible Flow, Gordon and Beach, New York, 1969.  Google Scholar

[21]

J. L. Lions, Quelques Methodes de Résolution de Problèmes aux Limites Non Linéaires, Dunod, Paris, 1969.  Google Scholar

[22]

J. Málek, J. Nečas, M. Rokyta and M. Ružička, Weak and Measure-Valued Solutions to Evolutionary PDEs, Applied Mathematics and Mathematical computation, 13. chapman & Hall, London, 1996. doi: 10.1007/978-1-4899-6824-1.  Google Scholar

[23]

J. MálekJ. Nečas and M. Ružička, On the non-Newtonian incompressible fluids, Math. Models Methods Appl. Sci., 3 (1993), 35-63.  doi: 10.1142/S0218202593000047.  Google Scholar

[24]

J. MálekJ. Nečas and M. Ružička, On weak solutions to a class of non-Newtonian incompressible fluids in bounded three-dimensional domains: the case p ≥ 2, Adv. Differential Equations, 6 (2001), 257-302.   Google Scholar

[25]

J. MálekD. Pražák and M. Steinhauer, On the Existence and Regularity of solutions for degenerate power-law fluids, Differential Integral Equations, 19 (2006), 449-462.   Google Scholar

[26]

J. Wolf, Existence of weak solutions to the equations of non-stationary motion of non-Newtonian fluids with shear rate dependent viscosity, J. Math. Fluid Mech., 9 (2007), 104-138.  doi: 10.1007/s00021-006-0219-5.  Google Scholar

[27]

J. Wolf, A new criterion for partial regularity of suitable weak solutions to the Navier-Stokes equations, in Advances in mathematical fluid mechanics, Springer, Berlin, (2010), 613–630.  Google Scholar

[28]

J. Wolf, On the local regularity of suitable weak solutions to the generalized Navier-Stokes equations, Ann. Univ. Ferrara Sez. Ⅶ Sci. Mat., 61 (2015), 149-171.  doi: 10.1007/s11565-014-0203-6.  Google Scholar

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