Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials driven by multiplicative noise of jump type

References

[1] H. Abels, Longtime behavior of solutions of a Navier–Stokes/Cahn–Hilliard system, Nonlocal and Abstract Parabolic Equations and Their Applications, Banach Center Publ. 86, Polish Academy of Sciences, Warsaw (2009), 9–19. 10.4064/bc86-0-1Search in Google Scholar

[2] H. Abels, On a diffuse interface model for two-phase flows of viscous, incompressible fluids with matched densities, Arch. Ration. Mech. Anal. 194 (2009), no. 2, 463–506. 10.1007/s00205-008-0160-2Search in Google Scholar

[3] H. Abels and M. Röger, Existence of weak solutions for a non-classical sharp interface model for a two-phase flow of viscous, incompressible fluids, Ann. Inst. H. Poincaré C Anal. Non Linéaire 26 (2009), no. 6, 2403–2424. 10.1016/j.anihpc.2009.06.002Search in Google Scholar

[4] R. A. Adams, Sobolev Spaces, Pure Appl. Math. 65, Academic Press, New York, 1975. Search in Google Scholar

[5] Z. Brzeźniak and E. Hausenblas, Maximal regularity for stochastic convolutions driven by Lévy processes, Probab. Theory Related Fields 145 (2009), no. 3–4, 615–637. 10.1007/s00440-008-0181-7Search in Google Scholar

[6] Z. Brzeźniak and E. Hausenblas, Martingale solutions for stochastic equations of reaction diffusion type driven by Lévy noise or Poisson random measure, preprint (2010), https://arxiv.org/abs/1010.5933v1. Search in Google Scholar

[7] Z. Brzeźniak, U. Manna and A. A. Panda, Existence of weak martingale solution of nematic liquid crystals driven by pure jump noise, preprint (2017), https://arxiv.org/abs/1706.05056. Search in Google Scholar

[8] Z. Brzeźniak and E. Motyl, Existence of a martingale solution of the stochastic Navier–Stokes equations in unbounded 2D and 3D domains, J. Differential Equations 254 (2013), no. 4, 1627–1685. 10.1016/j.jde.2012.10.009Search in Google Scholar

[9] J. W. Cahn and J. E. Hilliard, Free energy of a nonuniform system. I. Interfacial free energy, J. Chem. Phys. 28 (1958), 258–267. 10.1063/1.1744102Search in Google Scholar

[10] P. Colli, S. Frigeri and M. Grasselli, Global existence of weak solutions to a nonlocal Cahn–Hilliard–Navier–Stokes system, J. Math. Anal. Appl. 386 (2012), no. 1, 428–444. 10.1016/j.jmaa.2011.08.008Search in Google Scholar

[11] A. Debussche and L. Dettori, On the Cahn–Hilliard equation with a logarithmic free energy, Nonlinear Anal. 24 (1995), no. 10, 1491–1514. 10.1016/0362-546X(94)00205-VSearch in Google Scholar

[12] G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Martingale solutions to stochastic nonlocal Cahn–Hilliard–Navier–Stokes equations with multiplicative noise of jump type, Phys. D 398 (2019), 23–68. 10.1016/j.physd.2019.05.012Search in Google Scholar

[13] G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, On the strong solutions for a stochastic 2D nonlocal Cahn–Hilliard–Navier–Stokes model, Dyn. Partial Differ. Equ. 17 (2020), no. 1, 19–60. 10.4310/DPDE.2020.v17.n1.a2Search in Google Scholar

[14] G. Deugoué, A. Ndongmo Ngana and T. Tachim Medjo, Strong solutions for the stochastic Cahn–Hilliard–Navier–Stokes system, J. Differential Equations 275 (2021), 27–76. 10.1016/j.jde.2020.12.002Search in Google Scholar

[15] C. M. Elliott and H. Garcke, On the Cahn–Hilliard equation with degenerate mobility, SIAM J. Math. Anal. 27 (1996), no. 2, 404–423. 10.1137/S0036141094267662Search in Google Scholar

[16] C. M. Elliot and S. Luckhaus, A generalised diffusion equation for phase separation of a multicomponent mixture with interfacial free energy, IMA Preprint Ser. 887 (1991), https://conservancy.umn.edu/handle/11299/1733. Search in Google Scholar

[17] S. Frigeri, C. G. Gal and M. Grasselli, On nonlocal Cahn–Hilliard–Navier–Stokes systems in two dimensions, J. Nonlinear Sci. 26 (2016), no. 4, 847–893. 10.1007/s00332-016-9292-ySearch in Google Scholar

[18] S. Frigeri and M. Grasselli, Global and trajectory attractors for a nonlocal Cahn–Hilliard–Navier–Stokes system, J. Dynam. Differential Equations 24 (2012), no. 4, 827–856. 10.1007/s10884-012-9272-3Search in Google Scholar

[19] S. Frigeri and M. Grasselli, Nonlocal Cahn–Hilliard–Navier–Stokes systems with singular potentials, Dyn. Partial Differ. Equ. 9 (2012), no. 4, 273–304. 10.4310/DPDE.2012.v9.n4.a1Search in Google Scholar

[20] S. Frigeri, M. Grasselli and P. Krejčí, Strong solutions for two-dimensional nonlocal Cahn–Hilliard–Navier–Stokes systems, J. Differential Equations 255 (2013), no. 9, 2587–2614. 10.1016/j.jde.2013.07.016Search in Google Scholar

[21] C. G. Gal and M. Grasselli, Asymptotic behavior of a Cahn–Hilliard–Navier–Stokes system in 2D, Ann. Inst. H. Poincaré C Anal. Non Linéaire 27 (2010), no. 1, 401–436. 10.1016/j.anihpc.2009.11.013Search in Google Scholar

[22] C. G. Gal and M. Grasselli, Trajectory attractors for binary fluid mixtures in 3D, Chinese Ann. Math. Ser. B 31 (2010), no. 5, 655–678. 10.1007/s11401-010-0603-6Search in Google Scholar

[23] A. Giorgini, A. Miranville and R. Temam, Uniqueness and regularity for the Navier–Stokes–Cahn–Hilliard system, SIAM J. Math. Anal. 51 (2019), no. 3, 2535–2574. 10.1137/18M1223459Search in Google Scholar

[24] E. Hausenblas, P. A. Razafimandimby and M. Sango, Martingale solution to equations for differential type fluids of grade two driven by random force of Lévy type, Potential Anal. 38 (2013), no. 4, 1291–1331. 10.1007/s11118-012-9316-7Search in Google Scholar

[25] M. Heida, J. Málek and K. R. Rajagopal, On the development and generalizations of Cahn–Hilliard equations within a thermodynamic framework, Z. Angew. Math. Phys. 63 (2012), no. 1, 145–169. 10.1007/s00033-011-0139-ySearch in Google Scholar

[26] P. C. Hohenberg and B. I. Halperin, Theory of dynamical critical phenomena, Rev. Modern Phys. 49 (1977), 435–479. 10.1103/RevModPhys.49.435Search in Google Scholar

[27] N. Ikeda and S. Watanabe, Stochastic Differential Equations and Diffusion Processes, 2nd ed., North-Holland Math. Libr. 24, North-Holland, Amsterdam, 1989. Search in Google Scholar

[28] A. Joffe and M. Métivier, Weak convergence of sequences of semimartingales with applications to multitype branching processes, Adv. in Appl. Probab. 18 (1986), no. 1, 20–65. 10.2307/1427238Search in Google Scholar

[29] J. Kim, Phase-field models for multi-component fluid flows, Commun. Comput. Phys. 12 (2012), no. 3, 613–661. 10.4208/cicp.301110.040811aSearch in Google Scholar

[30] H. J. Kushner, Numerical Methods for Controlled Stochastic Delay Systems, Systems Control Found. Appl., Birkhäuser, Boston, 2008. 10.1007/978-0-8176-4621-9Search in Google Scholar

[31] M. Métivier, Stochastic Partial Differential Equations in Infinite-Dimensional Spaces, Scuola Normale Superiore, Pisa, 1988. Search in Google Scholar

[32] A. Morro, Phase-field models of Cahn–Hilliard Fluids and extra fluxes, Adv. Theor. Appl. Mech. 3 (2010), 409–424. Search in Google Scholar

[33] E. Motyl, Martingale solutions to the 2D and 3D stochastic Navier–Stokes equations driven by the compensated poisson random measure, preprint 13, Department of Mathematics and Computer Sciences, Lodz University, 2011. Search in Google Scholar

[34] E. Motyl, Stochastic Navier–Stokes equations driven by Lévy noise in unbounded 3D domains, Potential Anal. 38 (2013), no. 3, 863–912. 10.1007/s11118-012-9300-2Search in Google Scholar

[35] M. Ondreját, Uniqueness for stochastic evolution equations in Banach spaces, Dissertationes Math. 426 (2004), 1–63. 10.4064/dm426-0-1Search in Google Scholar

[36] K. R. Parthasarathy, Probability Measures on Metric Spaces, Probab. Math. Statist. 3, Academic Press, New York, 1967. 10.1016/B978-1-4832-0022-4.50006-5Search in Google Scholar

[37] R. Situ, Theory of Stochastic Differential Equations with Jumps and Applications, Math. Anal. Tech. Appl. Eng., Springer, New York, 2005. Search in Google Scholar

[38] T. Tachim Medjo, Unique strong and 𝕍 -attractor of a three dimensional globally modified Cahn–Hilliard–Navier–Stokes model, Appl. Anal. 96 (2017), no. 16, 2695–2716. 10.1080/00036811.2016.1236924Search in Google Scholar

[39] R. Temam, Infinite-Dimensional Dynamical Systems in Mechanics and Physics, Appl. Math. Sci. 68, Springer, New York, 1997. 10.1007/978-1-4612-0645-3Search in Google Scholar