
Previous Article
Sample average approximation method for stochastic complementarity problems with applications to supply chain supernetworks
 JIMO Home
 This Issue

Next Article
New passivity analysis of continuoustime recurrent neural networks with multiple discrete delays
Optimal control of electrorheological clutch described by nonlinear parabolic equation with nonlocal boundary conditions
1.  Institute of Mathematics, University of Augsburg, Universitätsstr. 14, D86159 Augsburg, Germany 
References:
[1] 
R. A. Adams, "Sobolev Spaces,", Academic Press, (1978). Google Scholar 
[2] 
J.P. Aubin, "Approximation of Elliptic BoundaryValue Problems,", Pure and Applied Mathematics, XXVI (1972). Google Scholar 
[3] 
Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566,", Bayer AG, (1260). Google Scholar 
[4] 
O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems,", Nauka, (1975). Google Scholar 
[5] 
G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions,, in, (2002), 9. Google Scholar 
[6] 
P. Dreyfuss and N. Hungerbühler, Results on a NavierStokes system with applications to electrorheological fluid flow,, Int. J. Pure Appl. Math., 14 (2004), 241. Google Scholar 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor,, Int. J. Differ. Egu., 9 (2004), 59. Google Scholar 
[8] 
D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using twodimensional Bingham plastic analysis and computational fluid dynamics,, J. Power and Energy, 219 (2005), 639. Google Scholar 
[9] 
A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications,", Translated from the 1999 Russian original by Tamara Rozhkovskaya, (1999). Google Scholar 
[10] 
R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows,, Communication on Pure and Applied Analysis, 3 (2004), 809. doi: 10.3934/cpaa.2004.3.809. Google Scholar 
[11] 
R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system,, SIAM J., 65 (2005), 1633. doi: 10.1137/S0036139903432883. Google Scholar 
[12] 
V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory,", Mathematics in Science and Engineering, (1980). Google Scholar 
[13] 
K. Josida, "Functional Analysis,", Springer, (1965). Google Scholar 
[14] 
L. V. Kantorovich and G. P.Akilov, "Functional Analysis,", Nauka, (1977). Google Scholar 
[15] 
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968). Google Scholar 
[16] 
L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media,", Pergamon, (1984). Google Scholar 
[17] 
J.L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", (French) Dunod; GauthierVillars, (1969). Google Scholar 
[18] 
W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid,", Nauka, (1982). Google Scholar 
[19] 
W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics,", Birkh\, (2000). Google Scholar 
[20] 
W. G. Litvinov, A problems on non steady flow of a nonlinear viscous fluid in a deformable pipe,, Methods of Functional Analysis and Topology, 2 (1996), 85. Google Scholar 
[21] 
W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary nonisothermal flow of electrorheological fluids,, Com. Pure Appl. Anal., 4 (2005), 779. doi: 10.3934/cpaa.2005.4.779. Google Scholar 
[22] 
W. G. Litvinov, Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary,, Com. Pure Appl. Anal., 6 (2007), 247. doi: 10.3934/cpaa.2007.6.247. Google Scholar 
[23] 
W. G. Litvinov, Dynamics of electrorheological clutch and a problem for nonlinear parabolic equation with nonlocal boundary conditions,, IMA Journal of Applied Mathematics, 73 (2008), 619. doi: 10.1093/imamat/hxn008. Google Scholar 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods,", Nauka, (1981). Google Scholar 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models,, Material Science and Engineering, R17 (1996), 57. Google Scholar 
[26] 
B. N. Pshenichnyi, "Necessary Optimality Conditions,", Nauka, (1969). Google Scholar 
[27] 
Y. A. Roitberg, "Elliptic Boundary Value Problems in the Spaces of Distributions,", Translated from the Russian by Peter Malyshev and Dmitry Malyshev. Mathematics and its Applications, (1996). Google Scholar 
[28] 
Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect,", Nauka i Technika, (1982). Google Scholar 
[29] 
L. Schwartz, "Analyse Mathématique 1,", Hermann, (1967). Google Scholar 
[30] 
V. A. Solonnikov, A priori estimates for parabolic equations of the second order,, Trudy Mat. Inst. Steklov, 70 (1964), 133. Google Scholar 
[31] 
M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch,, Int. J. Mod. Phys., 13 (1999), 2119. doi: 10.1142/S0217979299002216. Google Scholar 
[32] 
M. Whittle, R. J. Atkin and W. A. Bullough, Fluid dynamic limitations on the performance of an electrorheological clutch,, J. NonNewtonian Fluid Mech., 57 (1995), 61. doi: 10.1016/03770257(94)01296T. Google Scholar 
show all references
References:
[1] 
R. A. Adams, "Sobolev Spaces,", Academic Press, (1978). Google Scholar 
[2] 
J.P. Aubin, "Approximation of Elliptic BoundaryValue Problems,", Pure and Applied Mathematics, XXVI (1972). Google Scholar 
[3] 
Bayer AG, "Provisional Product Information, Rheobay TP AI 3565 and Rheobay TP AI 3566,", Bayer AG, (1260). Google Scholar 
[4] 
O. V. Besov, V. P. Il'in and S. M. Nikolsky, "Integral Representation of Functions and Embedding Theorems,", Nauka, (1975). Google Scholar 
[5] 
G. Bossis (Ed), Electrorheological Fluids and Magnetorheological suspensions,, in, (2002), 9. Google Scholar 
[6] 
P. Dreyfuss and N. Hungerbühler, Results on a NavierStokes system with applications to electrorheological fluid flow,, Int. J. Pure Appl. Math., 14 (2004), 241. Google Scholar 
[7] 
P. Dreyfuss and N. Hungerbühler, NavierStokes systems with quasimonotone viscosity tensor,, Int. J. Differ. Egu., 9 (2004), 59. Google Scholar 
[8] 
D. J. Ellam, R. J. Atkin and W. A. Bullough, Analysis of a smart clutch with cooling flow using twodimensional Bingham plastic analysis and computational fluid dynamics,, J. Power and Energy, 219 (2005), 639. Google Scholar 
[9] 
A. V. Fursikov, "Optimal Control of Distributed System. Theory and Applications,", Translated from the 1999 Russian original by Tamara Rozhkovskaya, (1999). Google Scholar 
[10] 
R. H. W. Hoppe and W. G. Litvinov, Problems on electrorheological fluid flows,, Communication on Pure and Applied Analysis, 3 (2004), 809. doi: 10.3934/cpaa.2004.3.809. Google Scholar 
[11] 
R. H. W. Hoppe, W. G. Litvinov and T. Rahman, Problems on stationary flow of electrorheological fluids in cylindrical coordinate system,, SIAM J., 65 (2005), 1633. doi: 10.1137/S0036139903432883. Google Scholar 
[12] 
V. C. L. Hutson and J. S. Pym, "Applications of Functional Analysis and Operator Theory,", Mathematics in Science and Engineering, (1980). Google Scholar 
[13] 
K. Josida, "Functional Analysis,", Springer, (1965). Google Scholar 
[14] 
L. V. Kantorovich and G. P.Akilov, "Functional Analysis,", Nauka, (1977). Google Scholar 
[15] 
O. A. Ladyzhenskaya, V. A. Solonnikov and N. N. Uraltseva, "Linear and Quasilinear Equations of Parabolic Type,", Amer. Math. Soc., (1968). Google Scholar 
[16] 
L. D. Landau and E. M. Lifschitz, "Electrodynamics of Continuous Media,", Pergamon, (1984). Google Scholar 
[17] 
J.L. Lions, "Quelques Méthodes de Résolution des Problèmes aux Limites non Linéaires,", (French) Dunod; GauthierVillars, (1969). Google Scholar 
[18] 
W. G. Litvinov, "Motion of Nonlinearly Viscous Fluid,", Nauka, (1982). Google Scholar 
[19] 
W. G. Litvinov, "Optimization in Elliptic Problems with Applications to Mechanics of Deformable Bodies and Fluid Mechanics,", Birkh\, (2000). Google Scholar 
[20] 
W. G. Litvinov, A problems on non steady flow of a nonlinear viscous fluid in a deformable pipe,, Methods of Functional Analysis and Topology, 2 (1996), 85. Google Scholar 
[21] 
W. G. Litvinov and R. H. W. Hoppe, Coupled problems on stationary nonisothermal flow of electrorheological fluids,, Com. Pure Appl. Anal., 4 (2005), 779. doi: 10.3934/cpaa.2005.4.779. Google Scholar 
[22] 
W. G. Litvinov, Problem on stationary flow of electrorheological fluids at the generalized conditions of slip on the boundary,, Com. Pure Appl. Anal., 6 (2007), 247. doi: 10.3934/cpaa.2007.6.247. Google Scholar 
[23] 
W. G. Litvinov, Dynamics of electrorheological clutch and a problem for nonlinear parabolic equation with nonlocal boundary conditions,, IMA Journal of Applied Mathematics, 73 (2008), 619. doi: 10.1093/imamat/hxn008. Google Scholar 
[24] 
G. I. Marchuk, V. I. Agoshkov, "Introduction in ProjectiveNet Methods,", Nauka, (1981). Google Scholar 
[25] 
M. Parthasarathy and D. J. Kleingenberg, Electrorheology: mechanisms and models,, Material Science and Engineering, R17 (1996), 57. Google Scholar 
[26] 
B. N. Pshenichnyi, "Necessary Optimality Conditions,", Nauka, (1969). Google Scholar 
[27] 
Y. A. Roitberg, "Elliptic Boundary Value Problems in the Spaces of Distributions,", Translated from the Russian by Peter Malyshev and Dmitry Malyshev. Mathematics and its Applications, (1996). Google Scholar 
[28] 
Z. P. Shulman and V. I. Kordonskii, "Magnetorheological Effect,", Nauka i Technika, (1982). Google Scholar 
[29] 
L. Schwartz, "Analyse Mathématique 1,", Hermann, (1967). Google Scholar 
[30] 
V. A. Solonnikov, A priori estimates for parabolic equations of the second order,, Trudy Mat. Inst. Steklov, 70 (1964), 133. Google Scholar 
[31] 
M. Whittle, R. J. Atkin and W. A. Bullough, Dynamic of a radial electrorheological clutch,, Int. J. Mod. Phys., 13 (1999), 2119. doi: 10.1142/S0217979299002216. Google Scholar 
[32] 
M. Whittle, R. J. Atkin and W. A. Bullough, Fluid dynamic limitations on the performance of an electrorheological clutch,, J. NonNewtonian Fluid Mech., 57 (1995), 61. doi: 10.1016/03770257(94)01296T. Google Scholar 
[1] 
JongShenq Guo. Blowup behavior for a quasilinear parabolic equation with nonlinear boundary condition. Discrete & Continuous Dynamical Systems  A, 2007, 18 (1) : 7184. doi: 10.3934/dcds.2007.18.71 
[2] 
Alain Hertzog, Antoine Mondoloni. Existence of a weak solution for a quasilinear wave equation with boundary condition. Communications on Pure & Applied Analysis, 2002, 1 (2) : 191219. doi: 10.3934/cpaa.2002.1.191 
[3] 
R.G. Duran, J.I. Etcheverry, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition. Discrete & Continuous Dynamical Systems  A, 1998, 4 (3) : 497506. doi: 10.3934/dcds.1998.4.497 
[4] 
Alexander Gladkov. Blowup problem for semilinear heat equation with nonlinear nonlocal Neumann boundary condition. Communications on Pure & Applied Analysis, 2017, 16 (6) : 20532068. doi: 10.3934/cpaa.2017101 
[5] 
Hiroko Morimoto. Survey on time periodic problem for fluid flow under inhomogeneous boundary condition. Discrete & Continuous Dynamical Systems  S, 2012, 5 (3) : 631639. doi: 10.3934/dcdss.2012.5.631 
[6] 
Yueding Yuan, Zhiming Guo, Moxun Tang. A nonlocal diffusion population model with age structure and Dirichlet boundary condition. Communications on Pure & Applied Analysis, 2015, 14 (5) : 20952115. doi: 10.3934/cpaa.2015.14.2095 
[7] 
G. Acosta, Julián Fernández Bonder, P. Groisman, J.D. Rossi. Numerical approximation of a parabolic problem with a nonlinear boundary condition in several space dimensions. Discrete & Continuous Dynamical Systems  B, 2002, 2 (2) : 279294. doi: 10.3934/dcdsb.2002.2.279 
[8] 
Mahamadi Warma. Parabolic and elliptic problems with general Wentzell boundary condition on Lipschitz domains. Communications on Pure & Applied Analysis, 2013, 12 (5) : 18811905. doi: 10.3934/cpaa.2013.12.1881 
[9] 
Gaoxi Li, Zhongping Wan, Jiawei Chen, Xiaoke Zhao. Necessary optimality condition for trilevel optimization problem. Journal of Industrial & Management Optimization, 2017, 13 (5) : 116. doi: 10.3934/jimo.2018140 
[10] 
Zhiming Guo, ZhiChun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a nonlocal differential equation with homogeneous Dirichlet boundary conditionA nonmonotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 18251838. doi: 10.3934/cpaa.2012.11.1825 
[11] 
Hakima Bessaih, Yalchin Efendiev, Florin Maris. Homogenization of the evolution Stokes equation in a perforated domain with a stochastic Fourier boundary condition. Networks & Heterogeneous Media, 2015, 10 (2) : 343367. doi: 10.3934/nhm.2015.10.343 
[12] 
Hongwei Zhang, Qingying Hu. Asymptotic behavior and nonexistence of wave equation with nonlinear boundary condition. Communications on Pure & Applied Analysis, 2005, 4 (4) : 861869. doi: 10.3934/cpaa.2005.4.861 
[13] 
Cong He, Hongjun Yu. Large time behavior of the solution to the Landau Equation with specular reflective boundary condition. Kinetic & Related Models, 2013, 6 (3) : 601623. doi: 10.3934/krm.2013.6.601 
[14] 
Marek Fila, Kazuhiro Ishige, Tatsuki Kawakami. Convergence to the Poisson kernel for the Laplace equation with a nonlinear dynamical boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (3) : 12851301. doi: 10.3934/cpaa.2012.11.1285 
[15] 
Keng Deng, Zhihua Dong. Blowup for the heat equation with a general memory boundary condition. Communications on Pure & Applied Analysis, 2012, 11 (5) : 21472156. doi: 10.3934/cpaa.2012.11.2147 
[16] 
Jaeyoung Byeon, Sangdon Jin. The Hénon equation with a critical exponent under the Neumann boundary condition. Discrete & Continuous Dynamical Systems  A, 2018, 38 (9) : 43534390. doi: 10.3934/dcds.2018190 
[17] 
Muhammad I. Mustafa. On the control of the wave equation by memorytype boundary condition. Discrete & Continuous Dynamical Systems  A, 2015, 35 (3) : 11791192. doi: 10.3934/dcds.2015.35.1179 
[18] 
Khadijah Sharaf. A perturbation result for a critical elliptic equation with zero Dirichlet boundary condition. Discrete & Continuous Dynamical Systems  A, 2017, 37 (3) : 16911706. doi: 10.3934/dcds.2017070 
[19] 
Kazuhiro Ishige, Ryuichi Sato. Heat equation with a nonlinear boundary condition and uniformly local $L^r$ spaces. Discrete & Continuous Dynamical Systems  A, 2016, 36 (5) : 26272652. doi: 10.3934/dcds.2016.36.2627 
[20] 
R. H.W. Hoppe, William G. Litvinov. Problems on electrorheological fluid flows. Communications on Pure & Applied Analysis, 2004, 3 (4) : 809848. doi: 10.3934/cpaa.2004.3.809 
2018 Impact Factor: 1.025
Tools
Metrics
Other articles
by authors
[Back to Top]