On the global solvability of the Cahn-Hilliard equation

Kh. S. Taramova

Čebyševskij sbornik, Tome 22 (2021) no. 3, pp. 467-473.

Résumé

The paper investigates the global in time solvability of the solution of the Cauchy problem for a nonlinear partial differential equation of Sobolev type that is not resolved with respect to the time derivative of the first order, the so-called Cahn-Hillard equation, in the Banach space of continuous bounded functions on the entire number axis, for which there are limits by minus and plus infinity. The existence of a classical solution is proved (by which we mean a sufficiently smooth function that has all continuous derivatives of the required order and satisfies the equation at each point of the domain of the considered Cauchy problem) on an arbitrary time interval. A priori estimates are obtained that ensure the existence of a global solution to the Cauchy problem for the pseudoparabolic Cahn-Hillard equation, since the classical solution $v\left(x,t\right)$ from the interval $\left[0,t_*\right]$, taking $v\left(x,t_*\right)$ as a new initial function, continues to the classical solution $v\left(x,t\right)$ on the interval $\left[0,t_*+\delta \right]$, where the value of $\delta$ depends only on the norm of the initial function and the parameters of the Cahn-Hillard equation. Repeating this process, a sufficiently large number of times, we obtain the classical solution of the considered Cauchy problem on an arbitrary time interval.

Détail

Keywords: Cahn-Hilliard equation, estimates for the solution of an equation, global solvability.

Bibliographie

[1] J.W. Cahn, J.E. Hilliard, “Free energy of a non-uniform system, Part I: Interfacial free energy”, J. Chemical Physics, 28:1 (1958), 258–267 | DOI | Zbl

[2] Ch. Elliot, S. Zheng, “On the Chan–Hilliard equation”, Arch. Rat. Mech. Anal., 96:4 (1986), 339–357 | DOI | MR | Zbl

[3] F. Bai, C.M. Elliott, A. Gardiner, A. Spence, A.M. Stuart, “The viscous Cahn-Hilliard equation I. Computations”, Nonlinearity, 1995, no. 8, 131–160 | DOI | MR | Zbl

[4] C.M. Elliott, A.M. Stuart, “Viscous Cahn-Hilliard Equation II. Analysis”, Journal of differential equations, 128 (1996), 387–414 | DOI | MR | Zbl

[5] Gal C., “Well-Posedness and Long Time Behavior of the Non-Isothermal Viscous Cahn-Hilliard Equation with Dynamic Boundary Conditions”, Dynamics of PDE, 5:1 (2008), 39–67 | MR | Zbl

[6] P.I. Plotnikov, “Predelnyi perekhod po malomu parametru v uravneniyakh Kana–Khillarda”, Sib. matem. zhurn., 38:3 (1997), 638–656 | MR | Zbl

[7] E.V. Radkevich, M.V. Zakharchenko, “Asimptoticheskoe reshenie rasshirennoi modeli Kana–Khillarda”, Sovremennaya matematika i ee prilozheniya, 2 (2003), 121–138

[8] E.V. Radkevich, “Korrektnost matematicheskikh modelei mekhaniki sploshnykh sred i termodinamika”, Sovremennaya matematika i ee prilozheniya, 3 (2003), 3–145

[9] I.S. Menshov, Ch.Chzhan, “Metod skvoznogo rascheta mezhfaznykh granits v dvukhfaznykh techeniyakh na osnove uravneniya Kana–Khillarda”, Zhurnal vychislitelnoi matematiki i matematicheskoi fiziki, 60:3 (2020), 476–488 | Zbl

[10] G.V. Demidenko, S.V. Uspenskii, Uravneniya i sistemy, ne razreshennye otnositelno starshei proizvodnoi, Nauch. kn., Novosibirsk, 1998, 436 pp. | MR

[11] A.G. Sveshnikov, A.B. Alshin, M.O. Korpusov, Yu.D. Pletner, Lineinye i nelineinye uravneniya sobolevskogo tipa, Fizmatlit, M., 2007, 736 pp.

[12] Kh.G. Umarov, “O razreshimosti odnomernogo uravneniya Kana-Khillarda s vyazkostyu v prostranstve nepreryvnykh ogranichennykh funktsii na vsei osi”, Nauchnye vedomosti BelGU. Seriya: Matematika. Fizika, 2012, no. 17(28), 91–101

[13] N. Danford, Dzh.T. Shvarts, Lineinye operatory. Obschaya teoriya, IL, M., 1962, 895 pp.

[14] T.B. Benjamin, J.L. Bona, J.J. Mahony, “Model equations for long waves in nonlinear dispersive systems”, Philos. Trans. R. Soc. London, 272 (1972), 47–78 | DOI | MR | Zbl

[15] B.P. Demidovich, Lektsii po matematicheskoi teorii ustoichivosti, MGU, M., 1998, 480 pp. | MR