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A revisit to the consensus for linearized Vicsek model under joint rooted leadership via a special matrix
Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion
1. | Weierstrass Institute, Mohrenstraße 39, 10117 Berlin, Germany, Germany |
2. | Weierstrass Institute for Applied Analysis and Stochastics, Mohrenstraße 39, 10117 Berlin |
References:
[1] |
G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.
doi: 10.1137/0523084. |
[2] |
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978).
|
[3] |
D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125.
doi: 10.1016/S0022-247X(03)00457-8. |
[4] |
V. Chalupecký, T. Fatima and A. Muntean, Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit,, Journal of Math-for-Industry, 2B (2010), 171.
|
[5] |
V. Chalupecký and A. Muntean, Semi-discrete finite difference multiscale scheme for a concrete corrosion model: A priori estimates and convergence,, Jpn. J. Ind. Appl. Math., 29 (2012), 289.
doi: 10.1007/s13160-012-0060-6. |
[6] |
D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99.
doi: 10.1016/S1631-073X(02)02429-9. |
[7] |
D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM J. Math. Anal., 40 (2008), 1585.
doi: 10.1137/080713148. |
[8] |
D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999).
|
[9] |
D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method,, SIAM J. Math. Anal., 37 (2006), 1435.
doi: 10.1137/040620898. |
[10] |
A. Damlamian, An elementary introduction to periodic unfolding,, Math. Sci. Appl., 24 (2005), 119.
|
[11] |
C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1.
doi: 10.1137/S1540345903425177. |
[12] |
J. Elstrodt, Ma$\beta$- und Integrationstheorie,, 3rd edition, (2002). Google Scholar |
[13] |
E. K. Essel, K. Kuliev, G. Kulieva and L.-E. Persson, Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence,, Appl. Math., 55 (2010), 305.
doi: 10.1007/s10492-010-0023-7. |
[14] |
L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).
|
[15] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992).
|
[16] |
T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion,, Appl. Anal., 91 (2012), 1129.
doi: 10.1080/00036811.2011.625016. |
[17] |
B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies,, Adv. Differential Equations, 6 (2001), 1377.
|
[18] |
B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms,, Asymptot. Anal., 34 (2003), 159.
|
[19] |
L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators,, Can. Appl. Math. Q., 14 (2006), 149.
|
[20] |
A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity,, ESAIM Control Optim. Calc. Var., 17 (2011), 1035.
doi: 10.1051/cocv/2010036. |
[21] |
A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures,, Appl. Anal., 66 (1997), 205.
doi: 10.1080/00036819708840583. |
[22] |
H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651.
doi: 10.1142/S0218202511005520. |
[23] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[24] |
U. Hornung, W. Jäger and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 59.
|
[25] |
V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). Google Scholar |
[26] |
D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35.
|
[27] |
H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting,, Ph.D thesis, (2013). Google Scholar |
[28] |
V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations,, Birkhäuser Boston Inc., (2006).
|
[29] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).
doi: 10.1007/978-3-7091-6961-2. |
[30] |
A. Matache and C. Schwab, Two-scale FEM for homogenization problems,, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537.
doi: 10.1051/m2an:2002025. |
[31] |
S. A. Meier and A. Muntean, A two-scale reaction-diffusion system: Homogenization and fast-reaction limits,, in Current Advances in Nonlinear Analysis and Related Topics, (2010), 443.
|
[32] |
A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329.
doi: 10.1088/0951-7715/24/4/016. |
[33] |
A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions,, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479.
doi: 10.3934/dcdss.2013.6.479. |
[34] |
A. Mielke and E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model,, Math. Models Meth. Appl. Sci. (M$^3$AS), 23 (2013), 873.
doi: 10.1142/S0218202512500637. |
[35] |
A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation,, SIAM J. Math. Analysis, 39 (2007), 642.
doi: 10.1137/060672790. |
[36] |
A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media,, J. Math. Anal. Appl., 371 (2010), 705.
doi: 10.1016/j.jmaa.2010.05.056. |
[37] |
F. Murat and L. Tartar, $H$-convergence,, in Topics in the mathematical modelling of composite materials, (1997), 21.
|
[38] |
J. D. Murray, Mathematical Biology. I. An Introduction,, 3rd edition, (2002).
|
[39] |
S. Nesenenko, Homogenization in viscoplasticity,, SIAM J. Math. Anal., 39 (2007), 236.
doi: 10.1137/060655092. |
[40] |
M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface,, SIAM J. Math. Anal., 39 (2007), 687.
doi: 10.1137/060665452. |
[41] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.
doi: 10.1137/0520043. |
[42] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[43] |
J. Persson, Homogenization of monotone parabolic problems with several temporal scales,, Appl. Math., 57 (2012), 191.
doi: 10.1007/s10492-012-0013-z. |
[44] |
M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium,, Math. Meth. Appl. Sci., 31 (2008), 1257.
doi: 10.1002/mma.966. |
[45] |
M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey,, Milan J. Math., 78 (2010), 417.
doi: 10.1007/s00032-010-0133-4. |
[46] |
S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems,, in preparation, (2014). Google Scholar |
[47] |
B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740.
doi: 10.1137/060675472. |
[48] |
B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening,, J. Multiscale Model., 2 (2010), 69. Google Scholar |
[49] |
L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009).
doi: 10.1007/978-3-642-05195-1. |
[50] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[51] |
A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239.
doi: 10.1007/s00526-006-0068-3. |
[52] |
A. Visintin, Homogenization of a parabolic model of ferromagnetism,, J. Differential Equations, 250 (2011), 1521.
doi: 10.1016/j.jde.2010.09.016. |
[53] |
A. Visintin, Some properties of two-scale convergence,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 93.
|
[54] |
A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371.
doi: 10.1051/cocv:2006012. |
[55] |
A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity,, Roy. Soc. Edinb. Proc. A, 138 (2008), 1363.
doi: 10.1017/S0308210506000709. |
[56] |
J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales,, Ann. Mat. Pura Appl. (4), 189 (2010), 357.
doi: 10.1007/s10231-009-0112-y. |
show all references
References:
[1] |
G. Allaire, Homogenization and two-scale convergence,, SIAM J. Math. Anal., 23 (1992), 1482.
doi: 10.1137/0523084. |
[2] |
A. Bensoussan, J.-L. Lions and G. Papanicolaou, Asymptotic Analysis for Periodic Structures,, Studies in Mathematics and its Applications, (1978).
|
[3] |
D. Bothe and D. Hilhorst, A reaction-diffusion system with fast reversible reaction,, J. Math. Anal. Appl., 286 (2003), 125.
doi: 10.1016/S0022-247X(03)00457-8. |
[4] |
V. Chalupecký, T. Fatima and A. Muntean, Multiscale sulfate attack on sewer pipes: Numerical study of a fast micro-macro mass transfer limit,, Journal of Math-for-Industry, 2B (2010), 171.
|
[5] |
V. Chalupecký and A. Muntean, Semi-discrete finite difference multiscale scheme for a concrete corrosion model: A priori estimates and convergence,, Jpn. J. Ind. Appl. Math., 29 (2012), 289.
doi: 10.1007/s13160-012-0060-6. |
[6] |
D. Cioranescu, A. Damlamian and G. Griso, Periodic unfolding and homogenization,, C. R. Math. Acad. Sci. Paris, 335 (2002), 99.
doi: 10.1016/S1631-073X(02)02429-9. |
[7] |
D. Cioranescu, A. Damlamian and G. Griso, The periodic unfolding method in homogenization,, SIAM J. Math. Anal., 40 (2008), 1585.
doi: 10.1137/080713148. |
[8] |
D. Cioranescu and P. Donato, An Introduction to Homogenization,, Oxford Lecture Series in Mathematics and its Applications, (1999).
|
[9] |
D. Cioranescu, A. Damlamian and R. De Arcangelis, Homogenization of quasiconvex integrals via the periodic unfolding method,, SIAM J. Math. Anal., 37 (2006), 1435.
doi: 10.1137/040620898. |
[10] |
A. Damlamian, An elementary introduction to periodic unfolding,, Math. Sci. Appl., 24 (2005), 119.
|
[11] |
C. Eck, Homogenization of a phase field model for binary mixtures,, Multiscale Model. Simul., 3 (): 1.
doi: 10.1137/S1540345903425177. |
[12] |
J. Elstrodt, Ma$\beta$- und Integrationstheorie,, 3rd edition, (2002). Google Scholar |
[13] |
E. K. Essel, K. Kuliev, G. Kulieva and L.-E. Persson, Homogenization of quasilinear parabolic problems by the method of Rothe and two scale convergence,, Appl. Math., 55 (2010), 305.
doi: 10.1007/s10492-010-0023-7. |
[14] |
L. C. Evans, Partial Differential Equations,, Graduate Studies in Mathematics, (1998).
|
[15] |
L. C. Evans and R. F. Gariepy, Measure Theory and Fine Properties of Functions,, Studies in Advanced Mathematics, (1992).
|
[16] |
T. Fatima, A. Muntean and M. Ptashnyk, Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion,, Appl. Anal., 91 (2012), 1129.
doi: 10.1080/00036811.2011.625016. |
[17] |
B. Fiedler and M. Vishik, Quantitative homogenization of analytic semigroups and reaction-diffusion equations with Diophantine spatial frequencies,, Adv. Differential Equations, 6 (2001), 1377.
|
[18] |
B. Fiedler and M. Vishik, Quantitative homogenization of global attractors for reaction-diffusion systems with rapidly oscillating terms,, Asymptot. Anal., 34 (2003), 159.
|
[19] |
L. Flodén and M. Olsson, Reiterated homogenization of some linear and nonlinear monotone parabolic operators,, Can. Appl. Math. Q., 14 (2006), 149.
|
[20] |
A. Giacomini and A. Musesti, Two-scale homogenization for a model in strain gradient plasticity,, ESAIM Control Optim. Calc. Var., 17 (2011), 1035.
doi: 10.1051/cocv/2010036. |
[21] |
A. Glitzky and R. Hünlich, Global estimates and asymptotics for electro-reaction-diffusion systems in heterostructures,, Appl. Anal., 66 (1997), 205.
doi: 10.1080/00036819708840583. |
[22] |
H. Hanke, Homogenization in gradient plasticity,, Math. Models Meth. Appl. Sci. (M$^3$AS), 21 (2011), 1651.
doi: 10.1142/S0218202511005520. |
[23] |
D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Lecture Notes in Mathematics, (1981).
|
[24] |
U. Hornung, W. Jäger and A. Mikelić, Reactive transport through an array of cells with semi-permeable membranes,, RAIRO Modél. Math. Anal. Numér., 28 (1994), 59.
|
[25] |
V. V. Jikov, S. M. Kozlov and O. A. Oleĭnik, Homogenization of Differential Operators and Integral Functionals,, Springer-Verlag, (1994). Google Scholar |
[26] |
D. Lukkassen, G. Nguetseng and P. Wall, Two-scale convergence,, Int. J. Pure Appl. Math., 2 (2002), 35.
|
[27] |
H. S. Mahato, Homogenization of a System of Nonlinear Multi-Species Diffusion-Reaction Equations in an $H^{1,p}$ Setting,, Ph.D thesis, (2013). Google Scholar |
[28] |
V. A. Marchenko and E. Y. Khruslov, Homogenization of Partial Differential Equations,, Birkhäuser Boston Inc., (2006).
|
[29] |
P. A. Markowich, C. A. Ringhofer and C. Schmeiser, Semiconductor Equations,, Springer-Verlag, (1990).
doi: 10.1007/978-3-7091-6961-2. |
[30] |
A. Matache and C. Schwab, Two-scale FEM for homogenization problems,, Math. Model. Numer. Anal. (M2AN), 36 (2002), 537.
doi: 10.1051/m2an:2002025. |
[31] |
S. A. Meier and A. Muntean, A two-scale reaction-diffusion system: Homogenization and fast-reaction limits,, in Current Advances in Nonlinear Analysis and Related Topics, (2010), 443.
|
[32] |
A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems,, Nonlinearity, 24 (2011), 1329.
doi: 10.1088/0951-7715/24/4/016. |
[33] |
A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions,, Discr. Cont. Dynam. Systems Ser. S, 6 (2013), 479.
doi: 10.3934/dcdss.2013.6.479. |
[34] |
A. Mielke and E. Rohan, Homogenization of elastic waves in fluid-saturated porous media using the Biot model,, Math. Models Meth. Appl. Sci. (M$^3$AS), 23 (2013), 873.
doi: 10.1142/S0218202512500637. |
[35] |
A. Mielke and A. M. Timofte, Two-scale homogenization for evolutionary variational inequalities via the energetic formulation,, SIAM J. Math. Analysis, 39 (2007), 642.
doi: 10.1137/060672790. |
[36] |
A. Muntean and M. Neuss-Radu, A multiscale Galerkin approach for a class of nonlinear coupled reaction-diffusion systems in complex media,, J. Math. Anal. Appl., 371 (2010), 705.
doi: 10.1016/j.jmaa.2010.05.056. |
[37] |
F. Murat and L. Tartar, $H$-convergence,, in Topics in the mathematical modelling of composite materials, (1997), 21.
|
[38] |
J. D. Murray, Mathematical Biology. I. An Introduction,, 3rd edition, (2002).
|
[39] |
S. Nesenenko, Homogenization in viscoplasticity,, SIAM J. Math. Anal., 39 (2007), 236.
doi: 10.1137/060655092. |
[40] |
M. Neuss-Radu and W. Jäger, Effective transmission conditions for reaction-diffusion processes in domains separated by an interface,, SIAM J. Math. Anal., 39 (2007), 687.
doi: 10.1137/060665452. |
[41] |
G. Nguetseng, A general convergence result for a functional related to the theory of homogenization,, SIAM J. Math. Anal., 20 (1989), 608.
doi: 10.1137/0520043. |
[42] |
A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations,, Applied Mathematical Sciences, (1983).
doi: 10.1007/978-1-4612-5561-1. |
[43] |
J. Persson, Homogenization of monotone parabolic problems with several temporal scales,, Appl. Math., 57 (2012), 191.
doi: 10.1007/s10492-012-0013-z. |
[44] |
M. A. Peter and M. Böhm, Different choices of scaling in homogenization of diffusion and interfacial exchange in a porous medium,, Math. Meth. Appl. Sci., 31 (2008), 1257.
doi: 10.1002/mma.966. |
[45] |
M. Pierre, Global existence in reaction-diffusion systems with control of mass: A survey,, Milan J. Math., 78 (2010), 417.
doi: 10.1007/s00032-010-0133-4. |
[46] |
S. Reichelt, Multi-scale Analysis of Nonlinear Reaction-Diffusion Systems,, in preparation, (2014). Google Scholar |
[47] |
B. Schweizer, Homogenization of degenerate two-phase flow equations with oil trapping,, SIAM J. Math. Anal., 39 (2008), 1740.
doi: 10.1137/060675472. |
[48] |
B. Schweizer and M. Veneroni, Periodic homogenization of Prandtl-Reuss plasticity with hardening,, J. Multiscale Model., 2 (2010), 69. Google Scholar |
[49] |
L. Tartar, The General Theory of Homogenization. A Personalized Introduction,, Lecture Notes of the Unione Matematica Italiana, (2009).
doi: 10.1007/978-3-642-05195-1. |
[50] |
R. Temam, Infinite-dimensional Dynamical Systems in Mechanics and Physics,, Springer-Verlag, (1988).
doi: 10.1007/978-1-4684-0313-8. |
[51] |
A. Visintin, Two-scale convergence of some integral functionals,, Calc. Var. Partial Differential Equations, 29 (2007), 239.
doi: 10.1007/s00526-006-0068-3. |
[52] |
A. Visintin, Homogenization of a parabolic model of ferromagnetism,, J. Differential Equations, 250 (2011), 1521.
doi: 10.1016/j.jde.2010.09.016. |
[53] |
A. Visintin, Some properties of two-scale convergence,, Atti Accad. Naz. Lincei Cl. Sci. Fis. Mat. Natur. Rend. Lincei (9) Mat. Appl., 15 (2004), 93.
|
[54] |
A. Visintin, Towards a two-scale calculus,, ESAIM Control Optim. Calc. Var., 12 (2006), 371.
doi: 10.1051/cocv:2006012. |
[55] |
A. Visintin, Homogenization of the nonlinear Maxwell model of viscoelasticity and of the Prandtl-Reuss model of elastoplasticity,, Roy. Soc. Edinb. Proc. A, 138 (2008), 1363.
doi: 10.1017/S0308210506000709. |
[56] |
J. L. Woukeng, Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales,, Ann. Mat. Pura Appl. (4), 189 (2010), 357.
doi: 10.1007/s10231-009-0112-y. |
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