Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
-
Ning Duan
Abstract
The main purpose of this paper is to study the existence of local and global strong solutions for a diffuse-interface model that describes the dynamics of incompressible two-phase viscous flows with surfactant in 3D whole space. We first establish the local existence of strong solutions by using the mollifier technique. Then, applying pure energy method and the standard continuity argument, one proves the global existence of strong solutions provided that initial data is sufficiently small.
(Communicated by Giuseppe Di Fazio)
References
[1] Abels, H.—Depner, D.—Garcke, H.: On an incompressible Navier-Stokes/Cahn-Hilliard system with degenerate mobility, Ann. Inst. H. Poincaré Anal. Non Lineairé 30 (2013), 1175–1190.Search in Google Scholar
[2] Bae, H.—Lee, W.: Existence, Gevrey regularity, and decay properties of solutions to Active models in critial spaces, J. Math. Anal. Appl. 506 (2022), Art. 125700.Search in Google Scholar
[3] Cao, C.—Gal, C. G.: Global solutions for the 2D NS- CH model for a two-phsed flow of viscous, incmpressible fluids with mixed partial viscosity and mobility, Nonlinearity 25 (2012), 3211–3234.Search in Google Scholar
[4] Cherfil, L.—Feireisl, E.—Michálek, M.—Miranville, A.—Petcu, M.—Pražák, D.: The compressible Navier-Stokes-Cahn-Hilliard equations with dynamic boundary conditions, Math. Model. Methods Appl. Sci. 29 (2019), 2557–2584.Search in Google Scholar
[5] Frigeri, S.—Gal, C. G.—Grasselli, M.—Sprekels, J.: Two-dimensional nonlocal Cahn-Hilliard-Navier-Stokes systems with variable viscosity, degenerate mobility and singular potential, Nonlinearity 32 (2019), 678–727.Search in Google Scholar
[6] Frigeri, S.—Grasselli, M.—Rocca, E.: A diffuse interface model for two-phase incompressible flows with non-local interactions and non-constant mobility, Nonlinearity 28 (2015), 1257–1293.Search in Google Scholar
[7] Gal, C. G.—Grasselli, M.: Asymptotic behavior of a Cahn-Hilliard-Navier-Stokes system in 2D, Ann. Inst. H. Poincaré Anal. Non Lineairé 27 (2010), 401–436.Search in Google Scholar
[8] Giorgina, A.—Miranville, A.—Temam, R.: Uniqueness and regularity for the Navier-Stokes-Cahn-Hilliard system, SIAM J. Math. Anal. 51 (2019), 2535–2574.Search in Google Scholar
[9] Gompper, G.—Schick, M.: Self-assembling Amphiphilic Systems, In: Phase Trasitions and Critical Phenomena (C. Domb, J. Lebowitz, eds.), Vol. 16, Academic Press, London, 1994.Search in Google Scholar
[10] Guo, Y.—Wang, Y.: Decay of dissipative equations and negative Sobolev spaces, Commun. Partial. Diff. Equ. 37 (2012), 2165–2208.Search in Google Scholar
[11] Ju, N.: Existence and uniqueness of the solution to the dissipative 2D Quasi-Geostrophic equations in the Sobolev space, Comm. Math. Phys. 251 (2004), 365–376.Search in Google Scholar
[12] Kato, T.—Ponce, G.: Commutator estimates and the Euler and Navier-Stokes equations, Comm. Pure Appl. Math. 41 (1988), 891–907.Search in Google Scholar
[13] Komura, S.—Kodama, H.: Two-order-parameter model for an oil-water-surfactant system, Phys. Rev. E 55 (1997), 1722–1727.Search in Google Scholar
[14] Ladyzenskaja, O. A.: Solution “in the large” of the nonstationary boundary value problem for the Navier-Stokes system with two space variables, Comm. Pure Appl. Math. 12 (1959), 427–433.Search in Google Scholar
[15] Lam, K. F.—Wu, H: Thermodynamically consistent Navier-Stokes-Cahn-Hilliard models with mass transfer and chemotaxis, European J. Appl. Math. 29 (2018), 595–644.Search in Google Scholar
[16] Leray, J.: Study of various nonlinear integral equations and of some problems posed by hydrodynamics, J. Pure Appl. Math. 12 (1933), 1–82.Search in Google Scholar
[17] Majda, A. J.—Bertozzi, A. L.: Vorticity and Incompressible Flow, Cambridge University Press, 2001.Search in Google Scholar
[18] Di Primio, A.—Grasselli, M.—Wu, H: Well-posedness for a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant, Math. Models Methods Appl. Sci. 33 (2023), 755–828.Search in Google Scholar
[19] Simon, J.: Nonhomogeneous viscous incompressible fluids: Existence of velocity, density, and pressure, SIAM J. Math. Anal. 21 (1990), 1093–1117.Search in Google Scholar
[20] Stein, E. M.: Singular integrals and Differentiability Properties of Functions, Princeton Unversity Press: Princeton, NJ, 1970.Search in Google Scholar
[21] Wang, Y.: Decay of the Navier-Stokes-Poisson equations, J. Differential Equations 253 (2012), 273–297.Search in Google Scholar
[22] Xu, C.—Chen, C.—Yang, X.: Efficient, non-iterative, and decoupled numerical scheme for a new modified binary phase-field surfactant system, Numerical Algorithms 86 (2021), 863–885.Search in Google Scholar
[23] Yang, X.: Numerical approximations for the Cahn-Hilliard phase field model of the binary fluid-surfactant system, J. Sci. Comput. 74 (2018), 1533–1553.Search in Google Scholar
[24] Zhao, X.: Strong solutions to the density-dependent incompressible Cahn-Hilliard-Navier-Stokes system, J. Hyperbolic Differ. Equ. 16 (2019), 701–742.Search in Google Scholar
© 2025 Mathematical Institute Slovak Academy of Sciences
Articles in the same Issue
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
Articles in the same Issue
- Copies of monomorphic structures
- Endomorphism kernel property for extraspecial and special groups
- Sums of Tribonacci numbers close to powers of 2
- Multiplicative functions k-additive on hexagonal numbers
- Monogenic even cyclic sextic polynomials
- Elegant proofs for properties of normalized remainders of Maclaurin power series expansion of exponential function
- Degree of independence in non-archimedean fields
- Remarks on some one-ended groups
- Some new characterizations of weights for hardy-type inequalities with kernels on time scales
- Inequalities for Riemann–Liouville fractional integrals in co-ordinated convex functions: A Newton-type approach
- Radius estimates for functions in the class 𝒰r(λ)
- Sharp bounds on the logarithmic coefficients of inverse functions for certain classes of univalent functions
- More q-congruences from Singh’s quadratic transformation
- Stability and controllability of cycled dynamical systems
- Existence, uniqueness, and multiplicity of radially symmetric k-admissible solutions for k-hessian equations
- Strong solution of a Navier-Stokes-Cahn-Hilliard system for incompressible two-phase flows with surfactant
- Coincidence points via tri-simulation functions with an application in integral equations
- On Fong-Tsui conjecture and binormality of operators
- Riemannian maps of CR-submanifolds of Kaehler manifolds
- On structural numbers of topological spaces
- The κ-Fréchet-Urysohn property for Cp(X) is equivalent to baireness of B1(X)
- Weighted pseudo S-asymptotically (ω, c)-periodic solutions to fractional stochastic differential equations
