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Qualitative analysis on positive steady-states for an autocatalytic reaction model in thermodynamics
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 |
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.
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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