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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

Bum Ja Jin 1, and  Kyungkeun Kang 2,,

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.

Keywords: Caccioppoli type inequality, non-Newtonian Stokes and Navier-Stokes equations, local energy inequality.
Mathematics Subject Classification: Primary: 76A05, 76D03; Secondary: 76D05, 76D07.
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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[5]

H. Beirão da Veiga, P. 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.

[6]

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

[7]

L. C. Berselli, L. 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.

[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.

[9]

M. Buliček, F. Ettwein, P. 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.

[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.

[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.

[12]

L. Diening, M. 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. 

[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.

[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.

[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.

[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. 

[17]

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

[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. 

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[24]

J. Málek, J. 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. 

[25]

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

[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.

[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.

[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.

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.

[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.

[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.

[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.

[5]

H. Beirão da Veiga, P. 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.

[6]

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

[7]

L. C. Berselli, L. 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.

[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.

[9]

M. Buliček, F. Ettwein, P. 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.

[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.

[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.

[12]

L. Diening, M. 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. 

[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.

[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.

[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.

[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. 

[17]

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

[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. 

[19]

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

[20]

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

[21]

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

[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.

[23]

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

[24]

J. Málek, J. 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. 

[25]

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

[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.

[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.

[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.

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