January  2021, 14(1): 41-55. doi: 10.3934/dcdss.2020303

Cahn-Hilliard equation with capillarity in actual deforming configurations

1. 

Mathematical Institute, Math.-Phys. Faculty, Charles University, Sokolovská 83, CZ-186 75 Praha 8, Czech Republic

2. 

Institute of Thermomechanics,Czech Academy of Sciences, Dolejškova 5, CZ-182 00 Praha 8, Czech Republic

Dedicated to Alexander Mielke on the occasion of his sixtieth birthday.

Received  March 2019 Revised  August 2019 Published  March 2020

Fund Project: * This research has been supported from the grants 17-04301S (regarding the focus on the dissipative evolution of internal variables), 19-04956S (regarding the focus on the dynamic and nonlinear behaviour), and 19-29646L (especially regarding the focus on the large strains in materials science) of Czech Science Foundation, and from the FWF grant I 4052 N3 with BMBWF through the OeAD-WTZ project CZ04/2019, and also from the institutional support RVO: 61388998 (ČR).

The diffusion driven by the gradient of the chemical potential (by the Fick/Darcy law) in deforming continua at large strains is formulated in the reference configuration with both the Fick/Darcy law and the capillarity (i.e. concentration gradient) term considered at the actual configurations deforming in time. Static situations are analysed by the direct method. Evolution (dynamical) problems are treated by the Faedo-Galerkin method, the actual capillarity giving rise to various new terms as e.g. the Korteweg-like stress and analytical difficulties related to them. Some other models (namely plasticity at small elastic strains or damage) with gradients at an actual configuration allow for similar models and analysis.

Citation: Tomáš Roubíček. Cahn-Hilliard equation with capillarity in actual deforming configurations. Discrete & Continuous Dynamical Systems - S, 2021, 14 (1) : 41-55. doi: 10.3934/dcdss.2020303
References:
[1]

E. K. Agiasofitou and M. Lazar, Conservation and balance laws in linear elasticity of grade three, J. Elasticity, 94 (2009), 69-85.  doi: 10.1007/s10659-008-9185-x.  Google Scholar

[2]

S. M. Allen and J. W. Cahn, Ground state structures in ordered binary alloys with second neighbor interactions, Acta Metall., 20 (1972), 423-433.  doi: 10.1016/0001-6160(72)90037-5.  Google Scholar

[3]

L. Anand, A Cahn-Hilliard-type theory for species diffusion coupled with large elastic-plastic deformations, J. Mech. Phys. Solids, 60 (2012), 1983-2002.  doi: 10.1016/j.jmps.2012.08.001.  Google Scholar

[4]

P. AreiasE. Samaniego and T. Rabczuk, A staggered approach for the coupling of Cahn-Hilliard type diffusion and finite strain elasticity, Comput. Mech., 57 (2016), 339-351.  doi: 10.1007/s00466-015-1235-1.  Google Scholar

[5]

A. Bedford, Hamilton's Principle in Continuum Mechanics, Pitman, Boston, 1985. doi: 10.13140/2.1.1603.4887.  Google Scholar

[6]

E. BonettiP. ColliW. DreyerG. GilardiG. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65.  doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[8]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807.  Google Scholar

[9]

H. Dal and C. Miehe, Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains, Comput. Mech., 55 (2015), 303-325.  doi: 10.1007/s00466-014-1102-5.  Google Scholar

[10]

C. Di LeoE. Rejovitzky and L. Anand, A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials, J. Mech. Phys. Solids, 70 (2014), 1-29.  doi: 10.1016/j.jmps.2014.05.001.  Google Scholar

[11]

F. P. DudaA. C. Souza and E. Fried, A theory for species migration in a finitely strained solid with application to polymer network swelling, J. Mech. Phys. Solids, 58 (2010), 515-529.  doi: 10.1016/j.jmps.2010.01.009.  Google Scholar

[12]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Arch. Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.  Google Scholar

[13]

H. Garcke, On Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.  doi: 10.1017/S0308210500002419.  Google Scholar

[14]

S. Govindjee and J. C. Simo, Coupled stress-diffusion: Case II, J. Mech. Phys. Solids, 41 (1993), 863-887.  doi: 10.1016/0022-5096(93)90003-X.  Google Scholar

[15]

T. J. Healey and S. Krömer, Injective weak solutions in second-gradient nonlinear elasticity, ESAIM: Control Optim. Calc. Var., 15 (2009), 863-871.  doi: 10.1051/cocv:2008050.  Google Scholar

[16]

C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes, Springer Spektrum, Wiesbaden, 2014. doi: 10.1007/978-3-658-05252-2.  Google Scholar

[17]

C. HeschA. J. GilR. OrtigosaM. DittmannC. Bilgen and et al., A framework for polyconvex large strain phase-field methods to fracture, Comput. Methods Appl. Mech. Engrg., 317 (2017), 649-683.  doi: 10.1016/j.cma.2016.12.035.  Google Scholar

[18]

W. Hong and X. Wang, A phase-field model for systems with coupled large deformation and mass transport, J. Mech. Phys. Solids, 61 (2013), 1281-1294.  doi: 10.1016/j.jmps.2013.03.001.  Google Scholar

[19]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fuides si lón tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. Sci. Exactes Nat., 6 (1901), 1-24.   Google Scholar

[20]

S. Krömer and T. Roubíček, Quasistatic viscoelasticity with self-contact at large strains, preprint, arXiv: 1904.02423, 2019. Google Scholar

[21]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. doi: 10.1007/978-3-030-02065-1.  Google Scholar

[22]

F. C. Larché and J. W. Cahn, The effect of self–stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.  doi: 10.1016/0001-6160(82)90023-2.  Google Scholar

[23]

V. I. Levitas, Phase field approach to martensitic phase transformations with large strains and interface stresses, J. Mech. Phys. Solids, 70 (2014), 154-189.  doi: 10.1016/j.jmps.2014.05.013.  Google Scholar

[24]

C. MieheS. Mauthe and H. Ulmer, Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn-Hilliard-type and standard diffusion in elastic solids, Internat. J. Numer. Meth. Engrg., 99 (2014), 737-762.  doi: 10.1002/nme.4700.  Google Scholar

[25]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[26]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.  Google Scholar

[27]

A. Mielke and T. Roubíček, Rate-Independent Systems – Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[28]

A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, preprint, arXiv: 1903.11094, 2019. To appear: Arch. Ration. Mech. Anal.. Google Scholar

[29]

R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Internat. J. Solids Structures, 1 (1965), 417-438.  doi: 10.1016/0020-7683(65)90006-5.  Google Scholar

[30]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), 11pp. doi: 10.1007/s00526-017-1212-y.  Google Scholar

[31]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191.  doi: 10.3934/dcds.2006.15.1169.  Google Scholar

[32]

I. Pawłow and W. M. Zajaczkowski, Weak solutions to 3-D Cahn-Hilliard system in elastic solids, Topol. Methods Nonlinear Anal., 32 (2008), 347-377.   Google Scholar

[33]

T. Roubíček, Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids, ZAMM Z. Angew. Math. Mech., 97 (2017), 990-1002.  doi: 10.1002/zamm.201600269.  Google Scholar

[34]

T. Roubíček and U. Stefanelli, Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modeling, SIAM J. Appl. Math., 78 (2018), 2597-2625.  doi: 10.1137/17M1137656.  Google Scholar

[35]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Z. Angew. Math. Phys., 69 (2018), Art. no. 55, 34pp. doi: 10.1007/s00033-018-0932-y.  Google Scholar

[36]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.  Google Scholar

[37]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.  Google Scholar

[38]

T. WaffenschmidtC. PolindaraA. Menzel and S. Blanco, A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials, Comput. Methods Appl. Mech. Engrg., 268 (2014), 801-842.  doi: 10.1016/j.cma.2013.10.013.  Google Scholar

[39]

V. V. Yashin and A. C. Balazs, Theoretical and computational modeling of self-oscillating polymer gels, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2672951.  Google Scholar

[40]

V. V. Yashin, S. Suzuki, R. Yoshida and A. C. Balazs, Controlling the dynamic behavior of heterogeneous self-oscillating gels, J. Mater. Chem., 22 (2012). doi: 10.1039/c2jm32065g.  Google Scholar

show all references

References:
[1]

E. K. Agiasofitou and M. Lazar, Conservation and balance laws in linear elasticity of grade three, J. Elasticity, 94 (2009), 69-85.  doi: 10.1007/s10659-008-9185-x.  Google Scholar

[2]

S. M. Allen and J. W. Cahn, Ground state structures in ordered binary alloys with second neighbor interactions, Acta Metall., 20 (1972), 423-433.  doi: 10.1016/0001-6160(72)90037-5.  Google Scholar

[3]

L. Anand, A Cahn-Hilliard-type theory for species diffusion coupled with large elastic-plastic deformations, J. Mech. Phys. Solids, 60 (2012), 1983-2002.  doi: 10.1016/j.jmps.2012.08.001.  Google Scholar

[4]

P. AreiasE. Samaniego and T. Rabczuk, A staggered approach for the coupling of Cahn-Hilliard type diffusion and finite strain elasticity, Comput. Mech., 57 (2016), 339-351.  doi: 10.1007/s00466-015-1235-1.  Google Scholar

[5]

A. Bedford, Hamilton's Principle in Continuum Mechanics, Pitman, Boston, 1985. doi: 10.13140/2.1.1603.4887.  Google Scholar

[6]

E. BonettiP. ColliW. DreyerG. GilardiG. Schimperna and J. Sprekels, On a model for phase separation in binary alloys driven by mechanical effects, Phys. D, 165 (2002), 48-65.  doi: 10.1016/S0167-2789(02)00373-1.  Google Scholar

[7]

J. W. Cahn and J. E. Hilliard, Free energy of a uniform system. I. Interfacial free energy, J. Chem. Phys., 28 (1958), 258-267.   Google Scholar

[8]

P. G. Ciarlet and J. Nečas, Injectivity and self-contact in nonlinear elasticity, Arch. Rational Mech. Anal., 97 (1987), 171-188.  doi: 10.1007/BF00250807.  Google Scholar

[9]

H. Dal and C. Miehe, Computational electro-chemo-mechanics of lithium-ion battery electrodes at finite strains, Comput. Mech., 55 (2015), 303-325.  doi: 10.1007/s00466-014-1102-5.  Google Scholar

[10]

C. Di LeoE. Rejovitzky and L. Anand, A Cahn-Hilliard-type phase-field theory for species diffusion coupled with large elastic deformations: Application to phase-separating Li-ion electrode materials, J. Mech. Phys. Solids, 70 (2014), 1-29.  doi: 10.1016/j.jmps.2014.05.001.  Google Scholar

[11]

F. P. DudaA. C. Souza and E. Fried, A theory for species migration in a finitely strained solid with application to polymer network swelling, J. Mech. Phys. Solids, 58 (2010), 515-529.  doi: 10.1016/j.jmps.2010.01.009.  Google Scholar

[12]

E. Fried and M. E. Gurtin, Tractions, balances, and boundary conditions for nonsimple materials with application to liquid flow at small-length scales, Arch. Ration. Mech. Anal., 182 (2006), 513-554.  doi: 10.1007/s00205-006-0015-7.  Google Scholar

[13]

H. Garcke, On Cahn-Hilliard system with elasticity, Proc. Roy. Soc. Edinburgh Sect. A, 133 (2003), 307-331.  doi: 10.1017/S0308210500002419.  Google Scholar

[14]

S. Govindjee and J. C. Simo, Coupled stress-diffusion: Case II, J. Mech. Phys. Solids, 41 (1993), 863-887.  doi: 10.1016/0022-5096(93)90003-X.  Google Scholar

[15]

T. J. Healey and S. Krömer, Injective weak solutions in second-gradient nonlinear elasticity, ESAIM: Control Optim. Calc. Var., 15 (2009), 863-871.  doi: 10.1051/cocv:2008050.  Google Scholar

[16]

C. Heinemann and C. Kraus, Phase Separation Coupled with Damage Processes, Springer Spektrum, Wiesbaden, 2014. doi: 10.1007/978-3-658-05252-2.  Google Scholar

[17]

C. HeschA. J. GilR. OrtigosaM. DittmannC. Bilgen and et al., A framework for polyconvex large strain phase-field methods to fracture, Comput. Methods Appl. Mech. Engrg., 317 (2017), 649-683.  doi: 10.1016/j.cma.2016.12.035.  Google Scholar

[18]

W. Hong and X. Wang, A phase-field model for systems with coupled large deformation and mass transport, J. Mech. Phys. Solids, 61 (2013), 1281-1294.  doi: 10.1016/j.jmps.2013.03.001.  Google Scholar

[19]

D. J. Korteweg, Sur la forme que prennent les équations du mouvement des fuides si lón tient compte des forces capillaires causées par des variations de densité considérables mais continues et sur la théorie de la capillarité dans l'hypothèse d'une variation continue de la densité, Arch. Néerl. Sci. Exactes Nat., 6 (1901), 1-24.   Google Scholar

[20]

S. Krömer and T. Roubíček, Quasistatic viscoelasticity with self-contact at large strains, preprint, arXiv: 1904.02423, 2019. Google Scholar

[21]

M. Kružík and T. Roubíček, Mathematical Methods in Continuum Mechanics of Solids, Springer, Cham/Switzerland, 2019. doi: 10.1007/978-3-030-02065-1.  Google Scholar

[22]

F. C. Larché and J. W. Cahn, The effect of self–stress on diffusion in solids, Acta Metall., 30 (1982), 1835-1845.  doi: 10.1016/0001-6160(82)90023-2.  Google Scholar

[23]

V. I. Levitas, Phase field approach to martensitic phase transformations with large strains and interface stresses, J. Mech. Phys. Solids, 70 (2014), 154-189.  doi: 10.1016/j.jmps.2014.05.013.  Google Scholar

[24]

C. MieheS. Mauthe and H. Ulmer, Formulation and numerical exploitation of mixed variational principles for coupled problems of Cahn-Hilliard-type and standard diffusion in elastic solids, Internat. J. Numer. Meth. Engrg., 99 (2014), 737-762.  doi: 10.1002/nme.4700.  Google Scholar

[25]

A. Mielke, A gradient structure for reaction-diffusion systems and for energy-drift-diffusion systems, Nonlinearity, 24 (2011), 1329-1346.  doi: 10.1088/0951-7715/24/4/016.  Google Scholar

[26]

A. Mielke, Thermomechanical modeling of energy-reaction-diffusion systems, including bulk-interface interactions, Discrete Contin. Dyn. Syst. Ser. S, 6 (2013), 479-499.  doi: 10.3934/dcdss.2013.6.479.  Google Scholar

[27]

A. Mielke and T. Roubíček, Rate-Independent Systems – Theory and Application, Springer, New York, 2015. doi: 10.1007/978-1-4939-2706-7.  Google Scholar

[28]

A. Mielke and T. Roubíček, Thermoviscoelasticity in Kelvin-Voigt rheology at large strains, preprint, arXiv: 1903.11094, 2019. To appear: Arch. Ration. Mech. Anal.. Google Scholar

[29]

R. D. Mindlin, Second gradient of strain and surface-tension in linear elasticity, Internat. J. Solids Structures, 1 (1965), 417-438.  doi: 10.1016/0020-7683(65)90006-5.  Google Scholar

[30]

A. Z. Palmer and T. J. Healey, Injectivity and self-contact in second-gradient nonlinear elasticity, Calc. Var. Partial Differential Equations, 56 (2017), 11pp. doi: 10.1007/s00526-017-1212-y.  Google Scholar

[31]

I. Pawłow, Thermodynamically consistent Cahn-Hilliard and Allen-Cahn models in elastic solids, Discrete Contin. Dyn. Syst., 15 (2006), 1169-1191.  doi: 10.3934/dcds.2006.15.1169.  Google Scholar

[32]

I. Pawłow and W. M. Zajaczkowski, Weak solutions to 3-D Cahn-Hilliard system in elastic solids, Topol. Methods Nonlinear Anal., 32 (2008), 347-377.   Google Scholar

[33]

T. Roubíček, Variational methods for steady-state Darcy/Fick flow in swollen and poroelastic solids, ZAMM Z. Angew. Math. Mech., 97 (2017), 990-1002.  doi: 10.1002/zamm.201600269.  Google Scholar

[34]

T. Roubíček and U. Stefanelli, Thermodynamics of elastoplastic porous rocks at large strains towards earthquake modeling, SIAM J. Appl. Math., 78 (2018), 2597-2625.  doi: 10.1137/17M1137656.  Google Scholar

[35]

T. Roubíček and G. Tomassetti, A thermodynamically consistent model of magneto-elastic materials under diffusion at large strains and its analysis, Z. Angew. Math. Phys., 69 (2018), Art. no. 55, 34pp. doi: 10.1007/s00033-018-0932-y.  Google Scholar

[36]

T. Roubíček and G. Tomassetti, Thermomechanics of hydrogen storage in metallic hydrides: Modeling and analysis, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 2313-2333.  doi: 10.3934/dcdsb.2014.19.2313.  Google Scholar

[37]

R. A. Toupin, Elastic materials with couple stresses, Arch. Ration. Mech. Anal., 11 (1962), 385-414.  doi: 10.1007/BF00253945.  Google Scholar

[38]

T. WaffenschmidtC. PolindaraA. Menzel and S. Blanco, A gradient-enhanced large-deformation continuum damage model for fibre-reinforced materials, Comput. Methods Appl. Mech. Engrg., 268 (2014), 801-842.  doi: 10.1016/j.cma.2013.10.013.  Google Scholar

[39]

V. V. Yashin and A. C. Balazs, Theoretical and computational modeling of self-oscillating polymer gels, J. Chem. Phys., 126 (2007). doi: 10.1063/1.2672951.  Google Scholar

[40]

V. V. Yashin, S. Suzuki, R. Yoshida and A. C. Balazs, Controlling the dynamic behavior of heterogeneous self-oscillating gels, J. Mater. Chem., 22 (2012). doi: 10.1039/c2jm32065g.  Google Scholar

[1]

Shu-Yu Hsu. Existence and properties of ancient solutions of the Yamabe flow. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 91-129. doi: 10.3934/dcds.2018005

[2]

Feng Luo. A combinatorial curvature flow for compact 3-manifolds with boundary. Electronic Research Announcements, 2005, 11: 12-20.

[3]

Hong Seng Sim, Wah June Leong, Chuei Yee Chen, Siti Nur Iqmal Ibrahim. Multi-step spectral gradient methods with modified weak secant relation for large scale unconstrained optimization. Numerical Algebra, Control & Optimization, 2018, 8 (3) : 377-387. doi: 10.3934/naco.2018024

[4]

Luke Finlay, Vladimir Gaitsgory, Ivan Lebedev. Linear programming solutions of periodic optimization problems: approximation of the optimal control. Journal of Industrial & Management Optimization, 2007, 3 (2) : 399-413. doi: 10.3934/jimo.2007.3.399

[5]

Shihu Li, Wei Liu, Yingchao Xie. Large deviations for stochastic 3D Leray-$ \alpha $ model with fractional dissipation. Communications on Pure & Applied Analysis, 2019, 18 (5) : 2491-2509. doi: 10.3934/cpaa.2019113

[6]

Haiyan Wang. Existence and nonexistence of positive radial solutions for quasilinear systems. Conference Publications, 2009, 2009 (Special) : 810-817. doi: 10.3934/proc.2009.2009.810

[7]

Kin Ming Hui, Soojung Kim. Asymptotic large time behavior of singular solutions of the fast diffusion equation. Discrete & Continuous Dynamical Systems - A, 2017, 37 (11) : 5943-5977. doi: 10.3934/dcds.2017258

[8]

Luigi C. Berselli, Jishan Fan. Logarithmic and improved regularity criteria for the 3D nematic liquid crystals models, Boussinesq system, and MHD equations in a bounded domain. Communications on Pure & Applied Analysis, 2015, 14 (2) : 637-655. doi: 10.3934/cpaa.2015.14.637

[9]

Yanqin Fang, Jihui Zhang. Multiplicity of solutions for the nonlinear Schrödinger-Maxwell system. Communications on Pure & Applied Analysis, 2011, 10 (4) : 1267-1279. doi: 10.3934/cpaa.2011.10.1267

[10]

Lucas C. F. Ferreira, Jhean E. Pérez-López, Élder J. Villamizar-Roa. On the product in Besov-Lorentz-Morrey spaces and existence of solutions for the stationary Boussinesq equations. Communications on Pure & Applied Analysis, 2018, 17 (6) : 2423-2439. doi: 10.3934/cpaa.2018115

[11]

Zhihua Zhang, Naoki Saito. PHLST with adaptive tiling and its application to antarctic remote sensing image approximation. Inverse Problems & Imaging, 2014, 8 (1) : 321-337. doi: 10.3934/ipi.2014.8.321

[12]

Matthias Erbar, Jan Maas. Gradient flow structures for discrete porous medium equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (4) : 1355-1374. doi: 10.3934/dcds.2014.34.1355

[13]

Boris Kramer, John R. Singler. A POD projection method for large-scale algebraic Riccati equations. Numerical Algebra, Control & Optimization, 2016, 6 (4) : 413-435. doi: 10.3934/naco.2016018

[14]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[15]

Graziano Crasta, Philippe G. LeFloch. Existence result for a class of nonconservative and nonstrictly hyperbolic systems. Communications on Pure & Applied Analysis, 2002, 1 (4) : 513-530. doi: 10.3934/cpaa.2002.1.513

[16]

Thomas Y. Hou, Ruo Li. Nonexistence of locally self-similar blow-up for the 3D incompressible Navier-Stokes equations. Discrete & Continuous Dynamical Systems - A, 2007, 18 (4) : 637-642. doi: 10.3934/dcds.2007.18.637

[17]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[18]

M. Grasselli, V. Pata. Asymptotic behavior of a parabolic-hyperbolic system. Communications on Pure & Applied Analysis, 2004, 3 (4) : 849-881. doi: 10.3934/cpaa.2004.3.849

[19]

Elena Bonetti, Pierluigi Colli, Gianni Gilardi. Singular limit of an integrodifferential system related to the entropy balance. Discrete & Continuous Dynamical Systems - B, 2014, 19 (7) : 1935-1953. doi: 10.3934/dcdsb.2014.19.1935

[20]

Dmitry Treschev. A locally integrable multi-dimensional billiard system. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5271-5284. doi: 10.3934/dcds.2017228

2019 Impact Factor: 1.233

Metrics

  • PDF downloads (84)
  • HTML views (335)
  • Cited by (0)

Other articles
by authors

[Back to Top]