Nonlocal to local convergence of phase field systems with inertial term. (English) Zbl 07898810
Summary: This paper deals with a nonlocal model for a hyperbolic phase field system coupling the standard energy balance equation for temperature with a dynamic for the phase variable: the latter includes an inertial term and a nonlocal convolution-type operator where the family of kernels depends on a small parameter. We rigorously study the asymptotic convergence of the system as the approximating parameter tends to zero and we obtain at the limit the local system with the elliptic laplacian operator acting on the phase variable. Our analysis is based on some asymptotic properties on nonlocal-to-local convergence that have been recently and successfully applied to families of Cahn-Hilliard models.
MSC:
| 35B40 | Asymptotic behavior of solutions to PDEs |
| 35D30 | Weak solutions to PDEs |
| 35G61 | Initial-boundary value problems for systems of nonlinear higher-order PDEs |
| 35R09 | Integro-partial differential equations |
| 80A22 | Stefan problems, phase changes, etc. |
Keywords:
nonlocal-to-local convergence; hyperbolic phase-field systems; parabolic-hyperbolic system; asymptotic analysis; existence of solutionsReferences:
| [1] | Abels, H.; Bosia, S.; Grasselli, M., Cahn-Hilliard equation with nonlocal singular free energies, Ann. Mat. Pura Appl., 194, 4, 1071-1106, 2015 · Zbl 1320.35083 · doi:10.1007/s10231-014-0411-9 |
| [2] | Abels, H.; Hurm, C., Strong nonlocal-to-local convergence of the Cahn-Hilliard equation and its operator, J. Differ. Equ., 402, 593-624, 2024 · Zbl 07894002 · doi:10.1016/j.jde.2024.05.022 |
| [3] | Abels, H.; Terasawa, Y., Convergence of a nonlocal to a local diffuse interface model for two-phase flow with unmatched densities, Discrete Contin. Dyn. Syst. Ser. S, 15, 8, 1871-1881, 2022 · Zbl 1513.76149 · doi:10.3934/dcdss.2022117 |
| [4] | Bourgain, J., Brezis, H., Mironescu, P.: Another look at Sobolev spaces. In: Optimal control and partial differential equations, pp. 439-455. IOS, Amsterdam (2001) · Zbl 1103.46310 |
| [5] | Carrillo, J.A., Elbar, C., Skrzeczkowski, J.: Degenerate Cahn-Hilliard systems: From nonlocal to local. [math.AP], pp. 1-30 (2023). Preprint arXiv:2303.11929 |
| [6] | Colli, P., Grasselli, M., Ito, A.: On a parabolic-hyperbolic Penrose-Fife phase-field system. Electron. J. Differ. Equ. 100, 30 (2002) (Erratum: Electron. J. Differ. Equ. 100, 32 (2002) · Zbl 1011.35038 |
| [7] | Davoli, E.; Ranetbauer, H.; Scarpa, L.; Trussardi, L., Degenerate nonlocal Cahn-Hilliard equations: Well-posedness, regularity and local asymptotics, Ann. Inst. H. Poincaré C Anal. Non Linéaire, 37, 3, 627-651, 2020 · Zbl 1467.45017 · doi:10.1016/j.anihpc.2019.10.002 |
| [8] | Davoli, E., Rocca, E., Scarpa, L., Trussardi, L.: Local asymptotics and optimal control for a viscous Cahn-Hilliard-reaction-diffusion model for tumor growth. [math.AP], pp. 1-30 (2023). Preprint arXiv:2311.10457 |
| [9] | Davoli, E.; Scarpa, L.; Trussardi, L., Nonlocal-to-local convergence of Cahn-Hilliard equations: Neumann boundary conditions and viscosity terms, Arch. Ration. Mech. Anal., 239, 1, 117-149, 2021 · Zbl 1456.76139 · doi:10.1007/s00205-020-01573-9 |
| [10] | Davoli, E.; Scarpa, L.; Trussardi, L., Local asymptotics for nonlocal convective Cahn-Hilliard equations with \(W^{1, 1}\) kernel and singular potential, J. Differ. Equ., 289, 35-58, 2021 · Zbl 1465.45014 · doi:10.1016/j.jde.2021.04.016 |
| [11] | Elbar, C.; Skrzeczkowski, J., Degenerate Cahn-Hilliard equation: From nonlocal to local, J. Differ. Equ., 364, 576-611, 2023 · Zbl 1514.35045 · doi:10.1016/j.jde.2023.03.057 |
| [12] | Frigeri, S.; Gal, CG; Grasselli, M., Regularity results for the nonlocal Cahn-Hilliard equation with singular potential and degenerate mobility, J. Differ. Equ., 287, 295-328, 2021 · Zbl 1462.35077 · doi:10.1016/j.jde.2021.03.052 |
| [13] | Gal, CG; Giorgini, A.; Grasselli, M., The nonlocal Cahn-Hilliard equation with singular potential: Well-posedness, regularity and strict separation property, J. Differ. Equ., 263, 9, 5253-5297, 2017 · Zbl 1400.35178 · doi:10.1016/j.jde.2017.06.015 |
| [14] | Gal, CG; Giorgini, A.; Grasselli, M., The separation property for 2D Cahn-Hilliard equations: Local, nonlocal and fractional energy cases, Discrete Contin. Dyn. Syst., 43, 6, 2270-2304, 2023 · Zbl 1514.35039 · doi:10.3934/dcds.2023010 |
| [15] | Gal, CG; Grasselli, M., Longtime behavior of nonlocal Cahn-Hilliard equations, Discrete Contin. Dyn. Syst., 34, 1, 145-179, 2014 · Zbl 1274.35399 · doi:10.3934/dcds.2014.34.145 |
| [16] | Grasselli, M.; Miranville, A.; Pata, V.; Zelik, S., Well-posedness and long time behavior of a parabolic-hyperbolic phase-field system with singular potentials, Math. Nachr., 280, 13-14, 1475-1509, 2007 · Zbl 1133.35017 · doi:10.1002/mana.200510560 |
| [17] | Grasselli, M.; Petzeltová, H.; Schimperna, G., A nonlocal phase-field system with inertial term, Q. Appl. Math., 65, 3, 451-469, 2007 · Zbl 1140.35352 · doi:10.1090/S0033-569X-07-01070-9 |
| [18] | Jiang, J., Convergence to equilibrium for a fully hyperbolic phase-field model with Cattaneo heat flux law, Math. Methods Appl. Sci., 32, 9, 1156-1182, 2009 · Zbl 1180.35107 · doi:10.1002/mma.1092 |
| [19] | Kurima, S., Time discretization of an initial value problem for a simultaneous abstract evolution equation applying to parabolic-hyperbolic phase-field systems, ESAIM Math. Model. Numer. Anal., 54, 3, 977-1002, 2020 · Zbl 1437.65120 · doi:10.1051/m2an/2019086 |
| [20] | Kurima, S., Time discretization of a nonlocal phase-field system with inertial term, Matematiche (Catania), 77, 1, 47-66, 2022 · Zbl 1496.35025 |
| [21] | Kurima, S., Existence for a singular nonlocal phase field system with inertial term, Acta Appl. Math., 178, 10, 2022 · Zbl 1486.35143 · doi:10.1007/s10440-022-00482-1 |
| [22] | Kurima, S., Nonlocal to local convergence of singular phase field systems of conserved type, Adv. Math. Sci. Appl., 31, 2, 481-500, 2022 · Zbl 1507.35083 |
| [23] | Lions, JL, Quelques méthodes de résolution des problèmes aux limites non linéaires, 1968, Paris: Dunod Gauthier-Villas, Paris · Zbl 0179.41801 |
| [24] | Melchionna, S.; Ranetbauer, H.; Scarpa, L.; Trussardi, L., From nonlocal to local Cahn-Hilliard equation, Adv. Math. Sci. Appl., 28, 2, 197-211, 2019 · Zbl 1471.45009 |
| [25] | Miranville, A.; Quintanilla, R., A generalization of the Caginalp phase-field system based on the Cattaneo law, Nonlinear Anal., 71, 5-6, 2278-2290, 2009 · Zbl 1167.35304 · doi:10.1016/j.na.2009.01.061 |
| [26] | Poiatti, A.: The 3d strict separation property for the nonlocal Cahn-Hilliard equation with singular potential. [math.AP], pp. 1-27 (2023). Preprint arXiv:2303.07745 |
| [27] | Ponce, AC, A new approach to Sobolev spaces and connections to \(\Gamma \)-convergence, Calc. Var. Partial Differ. Equ., 19, 3, 229-255, 2004 · Zbl 1352.46037 · doi:10.1007/s00526-003-0195-z |
| [28] | Rocca, E.; Schimperna, G., Global attractor for a parabolic-hyperbolic Penrose-Fife phase field system, Discrete Contin. Dyn. Syst., 15, 4, 1193-1214, 2006 · Zbl 1116.35028 · doi:10.3934/dcds.2006.15.1193 |
| [29] | Sandier, E.; Serfaty, S., Gamma-convergence of gradient flows with applications to Ginzburg-Landau, Commun. Pure Appl. Math., 57, 12, 1627-1672, 2004 · Zbl 1065.49011 · doi:10.1002/cpa.20046 |
| [30] | Simon, J., Compact sets in the space \(L^p(0, T;B)\), Ann. Mat. Pura Appl., 4, 146, 65-96, 1987 · Zbl 0629.46031 |
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.
