- Previous Article
- DCDS-S Home
- This Issue
-
Next Article
Effective diffusion in thin structures via generalized gradient systems and EDP-convergence
Stochastic homogenization of $ \Lambda $-convex gradient flows
1. | Fakultät für Mathematik, Technische Universität München, Garching, Germany |
2. | Weierstrass Institute for Applied Analysis and Stochastics, Berlin, Germany |
3. | Faculty of Mathematics, Technische Universität Dresden, Dresden, Germany |
In this paper we present a stochastic homogenization result for a class of Hilbert space evolutionary gradient systems driven by a quadratic dissipation potential and a $ \Lambda $-convex energy functional featuring random and rapidly oscillating coefficients. Specific examples included in the result are Allen-Cahn type equations and evolutionary equations driven by the $ p $-Laplace operator with $ p\in (1, \infty) $. The homogenization procedure we apply is based on a stochastic two-scale convergence approach. In particular, we define a stochastic unfolding operator which can be considered as a random counterpart of the well-established notion of periodic unfolding. The stochastic unfolding procedure grants a very convenient method for homogenization problems defined in terms of ($ \Lambda $-)convex functionals.
References:
[1] |
C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Studies in Economic Theory, 4. Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-662-03004-2. |
[2] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[3] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. |
[4] |
K. T. Andrews and S. Wright,
Stochastic homogenization of elliptic boundary-value problems with $L^{p}$-data, Asymptot. Anal., 17 (1998), 165-184.
|
[5] |
H. Attouch, Convergence de fonctionnelles convexes, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 1–40. |
[6] |
H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
[7] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[8] |
V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[9] |
A. Bourgeat, A. Mikelić and S. Wright,
Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.
|
[10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. |
[11] |
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[12] |
D. Cioranescu, A. Damlamian and R. De Arcangelis,
Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math. Acad. Sci. Paris, 339 (2004), 77-82.
doi: 10.1016/j.crma.2004.03.028. |
[13] |
D. Cioranescu, A. Damlamian and G. Griso,
Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104.
doi: 10.1016/S1631-073X(02)02429-9. |
[14] |
D. Cioranescu, A. Damlamian and G. Griso,
The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.
doi: 10.1137/080713148. |
[15] |
S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal transportation, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 413 (2014), 100–144. arXiv: 1009.3737. |
[16] |
F. Delarue and R. Rhodes,
Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.
doi: 10.3233/ASY-2008-0925. |
[17] |
Y. Efendiev and A. Pankov,
Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260.
|
[18] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010.
doi: 10.1090/gsm/019. |
[19] |
A. Faggionato,
Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.
doi: 10.1214/EJP.v13-591. |
[20] |
T. Fatima, A. Muntean and M. Ptashnyk,
Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.
doi: 10.1080/00036811.2011.625016. |
[21] |
G. Griso,
Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286.
|
[22] |
H. Hanke and D. Knees,
A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.
doi: 10.3233/ASY-161396. |
[23] |
M. Heida,
An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.
doi: 10.3233/ASY-2010-1022. |
[24] |
M. Heida,
Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, 91 (2012), 1243-1264.
doi: 10.1080/00036811.2011.567191. |
[25] |
V. V. Jikov, S. M. Kozlov and O. A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-642-84659-5. |
[26] |
A. Y. Kruger,
On Fréchet subdifferentials, J. Math. Sci. (N.Y.), 116 (2003), 3325-3358.
doi: 10.1023/A:1023673105317. |
[27] |
M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary $\Gamma$-convergence, Nonlinear Differ. Equat. Appl., 25 (2018), Art. 6, 31 pp.
doi: 10.1007/s00030-018-0495-9. |
[28] |
D. Lukkassen, G. Nguetseng and P. Wall,
Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.
|
[29] |
P. Mathieu and A. Piatnitski,
Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287-2307.
doi: 10.1098/rspa.2007.1876. |
[30] |
A. Mielke,
Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.
doi: 10.3934/dcds.2015.35.2679. |
[31] |
A. Mielke,
On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.
doi: 10.1007/978-3-319-26883-5_3. |
[32] |
A. Mielke, S. Reichelt and M. Thomas,
Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.
doi: 10.3934/nhm.2014.9.353. |
[33] |
A. Mielke, R. Rossi and G. Savaré,
Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), 253-310.
doi: 10.1007/s00526-011-0482-z. |
[34] |
A. Mielke and A. M. Timofte,
Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.
doi: 10.1137/060672790. |
[35] |
S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. |
[36] |
S. Neukamm and M. Varga,
Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.
doi: 10.1137/17M1141230. |
[37] |
S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. |
[38] |
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), 687-720.
doi: 10.1137/060665452. |
[39] |
G. Nguetseng,
A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
[40] |
G. C. Papanicolaou and S. R. S. Varadhan,
Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873.
|
[41] |
R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236. |
[42] |
R. Rossi and G. Savaré,
Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.
doi: 10.1051/cocv:2006013. |
[43] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. |
[44] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
[45] |
M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409.
doi: 10.4310/DPDE.2011.v8.n4.a1. |
[46] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
[47] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. |
[48] |
U. Stefanelli,
The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.
doi: 10.1137/070684574. |
[49] |
M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019. Available from: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342. |
[50] |
A. Visintin,
Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.
doi: 10.1051/cocv:2006012. |
[51] |
J. L. Woukeng,
Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.
doi: 10.1007/s10231-009-0112-y. |
[52] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4612-5020-3. |
[53] |
V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236. |
[54] |
V. V. Zhikov and A. L. Pyatnitskiǐ,
Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.
doi: 10.1070/IM2006v070n01ABEH002302. |
show all references
References:
[1] |
C. D. Aliprantis and K. C. Border, Infinite Dimensional Analysis: A Hitchhiker's Guide, Studies in Economic Theory, 4. Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-662-03004-2. |
[2] |
G. Allaire,
Homogenization and two-scale convergence, SIAM J. Math. Anal., 23 (1992), 1482-1518.
doi: 10.1137/0523084. |
[3] |
L. Ambrosio, N. Gigli and G. Savaré, Gradient Flows in Metric Spaces and in the Space of Probability Measures, Second edition, Lectures in Mathematics ETH Zürich. Birkhäuser Verlag, Basel, 2008. |
[4] |
K. T. Andrews and S. Wright,
Stochastic homogenization of elliptic boundary-value problems with $L^{p}$-data, Asymptot. Anal., 17 (1998), 165-184.
|
[5] |
H. Attouch, Convergence de fonctionnelles convexes, Journées d'Analyse Non Linéaire, Lecture Notes in Math., Springer, Berlin, 665 (1978), 1–40. |
[6] |
H. Attouch, Variational Convergence for Functions and Operators, Applicable Mathematics Series, Pitman (Advanced Publishing Program), Boston, MA, 1984. |
[7] |
V. Barbu, Nonlinear Differential Equations of Monotone Types in Banach Spaces, Springer Monographs in Mathematics, Springer, New York, 2010.
doi: 10.1007/978-1-4419-5542-5. |
[8] |
V. I. Bogachev, Measure Theory. Vol. I, Springer-Verlag, Berlin, 2007.
doi: 10.1007/978-3-540-34514-5. |
[9] |
A. Bourgeat, A. Mikelić and S. Wright,
Stochastic two-scale convergence in the mean and applications, J. Reine Angew. Math., 456 (1994), 19-51.
|
[10] |
H. Brézis, Operateurs Maximaux Monotones et Semi-groupes de Contractions dans les Espaces de Hilbert, Vol. 5, Elsevier, 1973. |
[11] |
H. Brézis, Functional Analysis, Sobolev Spaces and Partial Differential Equations, Universitext, Springer, New York, 2011. |
[12] |
D. Cioranescu, A. Damlamian and R. De Arcangelis,
Homogenization of nonlinear integrals via the periodic unfolding method, C. R. Math. Acad. Sci. Paris, 339 (2004), 77-82.
doi: 10.1016/j.crma.2004.03.028. |
[13] |
D. Cioranescu, A. Damlamian and G. Griso,
Periodic unfolding and homogenization, C. R. Math. Acad. Sci. Paris, 335 (2002), 99-104.
doi: 10.1016/S1631-073X(02)02429-9. |
[14] |
D. Cioranescu, A. Damlamian and G. Griso,
The periodic unfolding method in homogenization, SIAM J. Math. Anal., 40 (2008), 1585-1620.
doi: 10.1137/080713148. |
[15] |
S. Daneri and G. Savaré, Lecture notes on gradient flows and optimal transport, Optimal transportation, London Math. Soc. Lecture Note Ser., Cambridge Univ. Press, Cambridge, 413 (2014), 100–144. arXiv: 1009.3737. |
[16] |
F. Delarue and R. Rhodes,
Stochastic homogenization of quasilinear PDEs with a spatial degeneracy, Asymptotic Analysis, 61 (2009), 61-90.
doi: 10.3233/ASY-2008-0925. |
[17] |
Y. Efendiev and A. Pankov,
Homogenization of nonlinear random parabolic operators, Advances in Differential Equations, 10 (2005), 1235-1260.
|
[18] |
L. C. Evans, Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI., 2010.
doi: 10.1090/gsm/019. |
[19] |
A. Faggionato,
Random walks and exclusion processes among random conductances on random infinite clusters: Homogenization and hydrodynamic limit, Electron. J. Probab., 13 (2008), 2217-2247.
doi: 10.1214/EJP.v13-591. |
[20] |
T. Fatima, A. Muntean and M. Ptashnyk,
Unfolding-based corrector estimates for a reaction-diffusion system predicting concrete corrosion, Appl. Anal., 91 (2012), 1129-1154.
doi: 10.1080/00036811.2011.625016. |
[21] |
G. Griso,
Error estimate and unfolding for periodic homogenization, Asymptot. Anal., 40 (2004), 269-286.
|
[22] |
H. Hanke and D. Knees,
A phase-field damage model based on evolving microstructure, Asymptot. Anal., 101 (2017), 149-180.
doi: 10.3233/ASY-161396. |
[23] |
M. Heida,
An extension of the stochastic two-scale convergence method and application, Asymptot. Anal., 72 (2011), 1-30.
doi: 10.3233/ASY-2010-1022. |
[24] |
M. Heida,
Stochastic homogenization of heat transfer in polycrystals with nonlinear contact conductivities, Applicable Analysis, 91 (2012), 1243-1264.
doi: 10.1080/00036811.2011.567191. |
[25] |
V. V. Jikov, S. M. Kozlov and O. A. Oleǐnik, Homogenization of Differential Operators and Integral Functionals, Springer-Verlag, Berlin, 1994.
doi: 10.1007/978-3-642-84659-5. |
[26] |
A. Y. Kruger,
On Fréchet subdifferentials, J. Math. Sci. (N.Y.), 116 (2003), 3325-3358.
doi: 10.1023/A:1023673105317. |
[27] |
M. Liero and S. Reichelt, Homogenization of Cahn-Hilliard-type equations via evolutionary $\Gamma$-convergence, Nonlinear Differ. Equat. Appl., 25 (2018), Art. 6, 31 pp.
doi: 10.1007/s00030-018-0495-9. |
[28] |
D. Lukkassen, G. Nguetseng and P. Wall,
Two-scale convergence, Int. J. Pure Appl. Math., 2 (2002), 35-86.
|
[29] |
P. Mathieu and A. Piatnitski,
Quenched invariance principles for random walks on percolation clusters, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 463 (2007), 2287-2307.
doi: 10.1098/rspa.2007.1876. |
[30] |
A. Mielke,
Deriving amplitude equations via evolutionary $\Gamma$-convergence, Discrete Contin. Dyn. Syst., 35 (2015), 2679-2700.
doi: 10.3934/dcds.2015.35.2679. |
[31] |
A. Mielke,
On evolutionary $\Gamma$-convergence for gradient systems, Macroscopic and Large Scale Phenomena: Coarse Graining, Mean Field Limits and Ergodicity, Springer, 3 (2016), 187-249.
doi: 10.1007/978-3-319-26883-5_3. |
[32] |
A. Mielke, S. Reichelt and M. Thomas,
Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion, Netw. Heterog. Media, 9 (2014), 353-382.
doi: 10.3934/nhm.2014.9.353. |
[33] |
A. Mielke, R. Rossi and G. Savaré,
Nonsmooth analysis of doubly nonlinear evolution equations, Calc. Var. Partial Differential Equations, 46 (2013), 253-310.
doi: 10.1007/s00526-011-0482-z. |
[34] |
A. Mielke and A. M. Timofte,
Two-scale homogenization for evolutionary variational inequalities via the energetic formulation, SIAM J. Math. Anal., 39 (2007), 642-668.
doi: 10.1137/060672790. |
[35] |
S. Neukamm, Homogenization, Linearization and Dimension Reduction in Elasticity with Variational Methods, PhD thesis, Technische Universität München, 2010. |
[36] |
S. Neukamm and M. Varga,
Stochastic unfolding and homogenization of spring network models, Multiscale Model. Simul., 16 (2018), 857-899.
doi: 10.1137/17M1141230. |
[37] |
S. Neukamm, M. Varga and M. Waurick, Two-scale homogenization of abstract linear time-dependent PDEs, arXiv: 1905.02945. |
[38] |
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), 687-720.
doi: 10.1137/060665452. |
[39] |
G. Nguetseng,
A general convergence result for a functional related to the theory of homogenization, SIAM J. Math. Anal., 20 (1989), 608-623.
doi: 10.1137/0520043. |
[40] |
G. C. Papanicolaou and S. R. S. Varadhan,
Boundary value problems with rapidly oscillating random coefficients, Random Fields, Vol. Ⅰ, Ⅱ (Esztergom, 1979), Colloq. Math. Soc. János Bolyai, North-Holland, Amsterdam-New York, 27 (1981), 835-873.
|
[41] |
R. T. Rockafellar, Convex integral functionals and duality, Contributions to Nonlinear Functional Analysis, Academic Press, New York, (1971), 215–236. |
[42] |
R. Rossi and G. Savaré,
Gradient flows of non convex functionals in Hilbert spaces and applications, ESAIM Control Optim. Calc. Var., 12 (2006), 564-614.
doi: 10.1051/cocv:2006013. |
[43] |
T. Roubíček, Nonlinear Partial Differential Equations with Applications, Vol. 153, Springer Science & Business Media, 2013. |
[44] |
E. Sandier and S. Serfaty,
Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Comm. Pure Appl. Math., 57 (2004), 1627-1672.
doi: 10.1002/cpa.20046. |
[45] |
M. Sango and J. L. Woukeng, Stochastic $\Sigma$-convergence and applications, Dyn. Partial Differ. Equ., 8 (2011), 261–310. arXiv: 1106.0409.
doi: 10.4310/DPDE.2011.v8.n4.a1. |
[46] |
S. Serfaty,
Gamma-convergence of gradient flows on Hilbert and metric spaces and applications, Discrete Contin. Dyn. Syst., 31 (2011), 1427-1451.
doi: 10.3934/dcds.2011.31.1427. |
[47] |
R. E. Showalter, Monotone Operators in Banach Space and Nonlinear Partial Differential Equations, Mathematical Surveys and Monographs, 49. American Mathematical Society, Providence, RI, 1997. |
[48] |
U. Stefanelli,
The Brézis-Ekeland principle for doubly nonlinear equations, SIAM J. Control Optim., 47 (2008), 1615-1642.
doi: 10.1137/070684574. |
[49] |
M. Varga, Stochastic Unfolding and Homogenization of Evolutionary Gradient Systems, Dissertation, TU Dresden, 2019. Available from: https://nbn-resolving.org/urn:nbn:de:bsz:14-qucosa2-349342. |
[50] |
A. Visintin,
Towards a two-scale calculus, ESAIM Control Optim. Calc. Var., 12 (2006), 371-397.
doi: 10.1051/cocv:2006012. |
[51] |
J. L. Woukeng,
Periodic homogenization of nonlinear non-monotone parabolic operators with three time scales, Ann. Mat. Pura Appl., 189 (2010), 357-379.
doi: 10.1007/s10231-009-0112-y. |
[52] |
E. Zeidler, Nonlinear Functional Analysis and its Applications. III. Variational Methods and Optimization, Springer-Verlag, New York, 1985.
doi: 10.1007/978-1-4612-5020-3. |
[53] |
V. V. Zhikov, S. M. Kozlov and O. A. Oleǐnik, Averaging of parabolic operators, Trudy Moskovskogo Matematicheskogo Obshchestva, 45 (1982), 182–236. |
[54] |
V. V. Zhikov and A. L. Pyatnitskiǐ,
Homogenization of random singular structures and random measures, Izv. Math., 70 (2006), 19-67.
doi: 10.1070/IM2006v070n01ABEH002302. |
[1] |
Martin Heida, Stefan Neukamm, Mario Varga. Stochastic two-scale convergence and Young measures. Networks and Heterogeneous Media, 2022, 17 (2) : 227-254. doi: 10.3934/nhm.2022004 |
[2] |
Robert E. Miller. Homogenization of time-dependent systems with Kelvin-Voigt damping by two-scale convergence. Discrete and Continuous Dynamical Systems, 1995, 1 (4) : 485-502. doi: 10.3934/dcds.1995.1.485 |
[3] |
Aurore Back, Emmanuel Frénod. Geometric two-scale convergence on manifold and applications to the Vlasov equation. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 223-241. doi: 10.3934/dcdss.2015.8.223 |
[4] |
Alexander Mielke, Sina Reichelt, Marita Thomas. Two-scale homogenization of nonlinear reaction-diffusion systems with slow diffusion. Networks and Heterogeneous Media, 2014, 9 (2) : 353-382. doi: 10.3934/nhm.2014.9.353 |
[5] |
Erik Kropat, Silja Meyer-Nieberg, Gerhard-Wilhelm Weber. Bridging the gap between variational homogenization results and two-scale asymptotic averaging techniques on periodic network structures. Numerical Algebra, Control and Optimization, 2017, 7 (3) : 223-250. doi: 10.3934/naco.2017016 |
[6] |
Fang Liu, Aihui Zhou. Localizations and parallelizations for two-scale finite element discretizations. Communications on Pure and Applied Analysis, 2007, 6 (3) : 757-773. doi: 10.3934/cpaa.2007.6.757 |
[7] |
Alexandre Mouton. Expansion of a singularly perturbed equation with a two-scale converging convection term. Discrete and Continuous Dynamical Systems - S, 2016, 9 (5) : 1447-1473. doi: 10.3934/dcdss.2016058 |
[8] |
Ibrahima Faye, Emmanuel Frénod, Diaraf Seck. Two-Scale numerical simulation of sand transport problems. Discrete and Continuous Dynamical Systems - S, 2015, 8 (1) : 151-168. doi: 10.3934/dcdss.2015.8.151 |
[9] |
Jingwei Hu, Shi Jin, Li Wang. An asymptotic-preserving scheme for the semiconductor Boltzmann equation with two-scale collisions: A splitting approach. Kinetic and Related Models, 2015, 8 (4) : 707-723. doi: 10.3934/krm.2015.8.707 |
[10] |
Alexandre Mouton. Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. Kinetic and Related Models, 2009, 2 (2) : 251-274. doi: 10.3934/krm.2009.2.251 |
[11] |
Xu Yang, François Golse, Zhongyi Huang, Shi Jin. Numerical study of a domain decomposition method for a two-scale linear transport equation. Networks and Heterogeneous Media, 2006, 1 (1) : 143-166. doi: 10.3934/nhm.2006.1.143 |
[12] |
Shi Jin, Xu Yang, Guangwei Yuan. A domain decomposition method for a two-scale transport equation with energy flux conserved at the interface. Kinetic and Related Models, 2008, 1 (1) : 65-84. doi: 10.3934/krm.2008.1.65 |
[13] |
Nils Svanstedt. Multiscale stochastic homogenization of monotone operators. Networks and Heterogeneous Media, 2007, 2 (1) : 181-192. doi: 10.3934/nhm.2007.2.181 |
[14] |
Ioana Ciotir, Nicolas Forcadel, Wilfredo Salazar. Homogenization of a stochastic viscous transport equation. Evolution Equations and Control Theory, 2021, 10 (2) : 353-364. doi: 10.3934/eect.2020070 |
[15] |
Zhiqiang Yang, Junzhi Cui, Qiang Ma. The second-order two-scale computation for integrated heat transfer problem with conduction, convection and radiation in periodic porous materials. Discrete and Continuous Dynamical Systems - B, 2014, 19 (3) : 827-848. doi: 10.3934/dcdsb.2014.19.827 |
[16] |
Hengrong Du, Changyou Wang. Global weak solutions to the stochastic Ericksen–Leslie system in dimension two. Discrete and Continuous Dynamical Systems, 2022, 42 (5) : 2175-2197. doi: 10.3934/dcds.2021187 |
[17] |
Luca Lussardi, Stefano Marini, Marco Veneroni. Stochastic homogenization of maximal monotone relations and applications. Networks and Heterogeneous Media, 2018, 13 (1) : 27-45. doi: 10.3934/nhm.2018002 |
[18] |
Hakima Bessaih, Yalchin Efendiev, Razvan Florian Maris. Stochastic homogenization for a diffusion-reaction model. Discrete and Continuous Dynamical Systems, 2019, 39 (9) : 5403-5429. doi: 10.3934/dcds.2019221 |
[19] |
Junhao Hu, Chenggui Yuan. Strong convergence of neutral stochastic functional differential equations with two time-scales. Discrete and Continuous Dynamical Systems - B, 2019, 24 (11) : 5831-5848. doi: 10.3934/dcdsb.2019108 |
[20] |
Theodore Tachim Medjo. On the convergence of a stochastic 3D globally modified two-phase flow model. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 395-430. doi: 10.3934/dcds.2019016 |
2021 Impact Factor: 1.865
Tools
Metrics
Other articles
by authors
[Back to Top]