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Shape optimization for non-Newtonian fluids in time-dependent domains
1. | Institut Élie Cartan Nancy, UMR 7502((Université Lorraine, CNRS, INRIA), Laboratoire de Mathématiques, Université de Lorraine, B.P.239, 54506 Vandoeuvre-lès-Nancy Cedex, France |
2. | Institute of Mathematics of the Academy of Sciences of the Czech Republic, Žitná 25, 110 00 Praha 1, Czech Republic |
References:
[1] |
N. Arada, Regularity of flows and optimal control of shear-thinning fluids,, Nonlinear Analysis: Theory, 89 (2013), 81.
doi: 10.1016/j.na.2013.04.015. |
[2] |
V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).
doi: 10.1090/memo/0852. |
[3] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011).
doi: 10.1137/1.9780898719826. |
[4] |
M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary,, SIAM Journal on Control and Optimization, 42 (2004), 2286.
doi: 10.1137/S0363012902411945. |
[5] |
L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids,, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 9 (2010), 1.
|
[6] |
R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations,, Journal of Convex Analysis, 6 (1999), 293.
|
[7] |
E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time-dependent domains,, Journal of Differential Equations, 254 (2013), 125.
doi: 10.1016/j.jde.2012.08.019. |
[8] |
E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows,, Journal of Differential Equations, 250 (2011), 596.
doi: 10.1016/j.jde.2010.09.031. |
[9] |
J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows,, Partial Differential Equations, 406 (2000), 121.
|
[10] |
J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the lipschitz truncation method,, SIAM Journal on Mathematical Analysis, 34 (2003), 1064.
doi: 10.1137/S0036141002410988. |
[11] |
O. A. Ladyzhenskaya, 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.
|
[12] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969).
|
[13] |
O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries,, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97.
|
[14] |
J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires,, (French) Dunod; Gauthier-Villars, (1969).
|
[15] |
J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,, in Evolutionary equations. Vol. II, (2005), 371. Google Scholar |
[16] |
M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277,, Chapman & Hall/CRC, (2006).
doi: 10.1201/9781420003246. |
[17] |
J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method,, Mathematical Methods in the Applied Sciences, 32 (2009), 653.
doi: 10.1002/mma.1059. |
[18] |
P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization,, Springer-Verlag, (2012).
doi: 10.1007/978-3-0348-0367-0. |
[19] |
K. Rajagopal, Mechanics of non-Newtonian fluids,, in Recent Developments in Theoretical Fluid Mechanics (Winter School, (1992), 129.
|
[20] |
W. Schowalter, Mechanics of Non-Newtonian Fluids,, Pergamon Press, (1978). Google Scholar |
[21] |
T. Slawig, Distributed control for a class of non-Newtonian fluids,, Journal of Differential Equations, 219 (2005), 116.
doi: 10.1016/j.jde.2005.03.009. |
[22] |
J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids,, Control and Cybernetics, 40 (2011), 1077.
|
[23] |
J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids,, in System Modeling and Optimization, (2013), 427. Google Scholar |
[24] |
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992). Google Scholar |
[25] |
C. Truesdell, W. Noll and S. Antman, The Non-linear Field Theories Of Mechanics,, Springer Verlag, (2004).
doi: 10.1007/978-3-662-10388-3. |
[26] |
D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids,, Z. Anal. Anwend., 29 (2010), 351.
doi: 10.4171/ZAA/1412. |
show all references
References:
[1] |
N. Arada, Regularity of flows and optimal control of shear-thinning fluids,, Nonlinear Analysis: Theory, 89 (2013), 81.
doi: 10.1016/j.na.2013.04.015. |
[2] |
V. Barbu, I. Lasiecka and R. Triggiani, Tangential boundary stabilization of Navier-Stokes equations,, Mem. Amer. Math. Soc., 181 (2006).
doi: 10.1090/memo/0852. |
[3] |
M. C. Delfour and J.-P. Zolésio, Shapes and Geometries: Metrics, Analysis, Differential Calculus, and Optimization,, Second edition, (2011).
doi: 10.1137/1.9780898719826. |
[4] |
M. C. Delfour and J. P. Zolésio, Oriented distance function and its evolution equation for initial sets with thin boundary,, SIAM Journal on Control and Optimization, 42 (2004), 2286.
doi: 10.1137/S0363012902411945. |
[5] |
L. Diening, M. Růžička and J. Wolf, Existence of weak solutions for unsteady motions of generalized Newtonian fluids,, Annali della Scuola Normale Superiore di Pisa. Classe di scienze, 9 (2010), 1.
|
[6] |
R. Dziri and J.-P. Zolésio, Dynamical shape control in non-cylindrical navier-stokes equations,, Journal of Convex Analysis, 6 (1999), 293.
|
[7] |
E. Feireisl, O. Kreml, Š. Nečasová, J. Neustupa and J. Stebel, Weak solutions to the barotropic Navier-Stokes system with slip boundary conditions in time-dependent domains,, Journal of Differential Equations, 254 (2013), 125.
doi: 10.1016/j.jde.2012.08.019. |
[8] |
E. Feireisl, J. Neustupa and J. Stebel, Convergence of a Brinkman-type penalization for compressible fluid flows,, Journal of Differential Equations, 250 (2011), 596.
doi: 10.1016/j.jde.2010.09.031. |
[9] |
J. Frehse, J. Málek and M. Steinhauer, On existence results for fluids with shear dependent viscosity-unsteady flows,, Partial Differential Equations, 406 (2000), 121.
|
[10] |
J. Frehse, J. Málek and M. Steinhauer, On analysis of steady flows of fluids with shear-dependent viscosity based on the lipschitz truncation method,, SIAM Journal on Mathematical Analysis, 34 (2003), 1064.
doi: 10.1137/S0036141002410988. |
[11] |
O. A. Ladyzhenskaya, 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.
|
[12] |
O. A. Ladyzhenskaya, The Mathematical Theory of Viscous Incompressible Flow,, Second English edition, (1969).
|
[13] |
O. A. Ladyzhenskaya, Initial-boundary problem for Navier-Stokes equations in domains with time-varying boundaries,, Zapiski Nauchnykh Seminarov LOMI, 11 (1968), 97.
|
[14] |
J.-L. Lions, Quelques Méthodes De Résolution Des Problèmes Aux Limites Non Linéaires,, (French) Dunod; Gauthier-Villars, (1969).
|
[15] |
J. Málek and K. R. Rajagopal, Mathematical issues concerning the Navier-Stokes equations and some of its generalizations,, in Evolutionary equations. Vol. II, (2005), 371. Google Scholar |
[16] |
M. Moubachir and J.-P. Zolésio, Moving Shape Analysis and Control, vol. 277,, Chapman & Hall/CRC, (2006).
doi: 10.1201/9781420003246. |
[17] |
J. Neustupa, Existence of a weak solution to the Navier-Stokes equation in a general time-varying domain by the Rothe method,, Mathematical Methods in the Applied Sciences, 32 (2009), 653.
doi: 10.1002/mma.1059. |
[18] |
P. Plotnikov and J. Sokolowski, Compressible Navier-Stokes Equations, Theory and Shape Optimization,, Springer-Verlag, (2012).
doi: 10.1007/978-3-0348-0367-0. |
[19] |
K. Rajagopal, Mechanics of non-Newtonian fluids,, in Recent Developments in Theoretical Fluid Mechanics (Winter School, (1992), 129.
|
[20] |
W. Schowalter, Mechanics of Non-Newtonian Fluids,, Pergamon Press, (1978). Google Scholar |
[21] |
T. Slawig, Distributed control for a class of non-Newtonian fluids,, Journal of Differential Equations, 219 (2005), 116.
doi: 10.1016/j.jde.2005.03.009. |
[22] |
J. Sokołowski and J. Stebel, Shape sensitivity analysis of time-dependent flows of incompressible non-Newtonian fluids,, Control and Cybernetics, 40 (2011), 1077.
|
[23] |
J. Sokołowski and J. Stebel, Shape sensitivity analysis of incompressible non-Newtonian fluids,, in System Modeling and Optimization, (2013), 427. Google Scholar |
[24] |
J. Sokołowski and J.-P. Zolésio, Introduction to Shape Optimization. Shape Sensitivity Analysis,, Springer Series in Computational Mathematics, (1992). Google Scholar |
[25] |
C. Truesdell, W. Noll and S. Antman, The Non-linear Field Theories Of Mechanics,, Springer Verlag, (2004).
doi: 10.1007/978-3-662-10388-3. |
[26] |
D. Wachsmuth and T. Roubíček, Optimal control of planar flow of incompressible non-Newtonian fluids,, Z. Anal. Anwend., 29 (2010), 351.
doi: 10.4171/ZAA/1412. |
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