##
**On the viscous Cahn-Hilliard equation.**
*(English)*
Zbl 0632.76119

Material instabilities in continuum mechanics, Proc. Symp. Edinburgh/Scotl. 1985/86, 329-342 (1988).

[For the entire collection see Zbl 0627.00023.]

Two equations are derived and analyzed which model the dynamics of viscous first order phase transitions. When viscous and gradient energy terms are included, our derivation yields a viscous Cahn-Hilliard equation (1) \(c_ t=\Delta (f(c)+\nu c_ t-K\Delta c)\). Here c(x,t) is a concentration, f(c) is an intrinsic chemical potential, which typically is non-monotone, and \(\nu\) and K are, respectively, coefficients of viscosity and gradient energy. In the limit \(K\to 0\), equation (1) reduces to a viscous diffusion equation (2) \(c_ t=\Delta (f(c)+\nu c_ t)\) and in the limit \(\nu\to 0\), (1) reduces to the well-known Cahn- Hilliard equation (3) \(c_ t=\Delta (f(c)-K\Delta c).\)

Equations (1)-(3) can be viewed as viscous and/or gradient energy regularizations of the backwards-forwards heat equation. Early stages in the evolution of (1) follow closely the behaviour of equation (2) for small K, and in this manner phase separated transients probably evolve and decay in much the same way as is believed to occur in the Cahn- Hilliard equation, although such a similarity has yet to be proved. The newly derived equations are of lower order than the Cahn-Hilliard equation and to some extent lend themselves to simpler analysis. The derivation of equations (1) and (2) is outlined and several results are given. Full details are to appear.

Two equations are derived and analyzed which model the dynamics of viscous first order phase transitions. When viscous and gradient energy terms are included, our derivation yields a viscous Cahn-Hilliard equation (1) \(c_ t=\Delta (f(c)+\nu c_ t-K\Delta c)\). Here c(x,t) is a concentration, f(c) is an intrinsic chemical potential, which typically is non-monotone, and \(\nu\) and K are, respectively, coefficients of viscosity and gradient energy. In the limit \(K\to 0\), equation (1) reduces to a viscous diffusion equation (2) \(c_ t=\Delta (f(c)+\nu c_ t)\) and in the limit \(\nu\to 0\), (1) reduces to the well-known Cahn- Hilliard equation (3) \(c_ t=\Delta (f(c)-K\Delta c).\)

Equations (1)-(3) can be viewed as viscous and/or gradient energy regularizations of the backwards-forwards heat equation. Early stages in the evolution of (1) follow closely the behaviour of equation (2) for small K, and in this manner phase separated transients probably evolve and decay in much the same way as is believed to occur in the Cahn- Hilliard equation, although such a similarity has yet to be proved. The newly derived equations are of lower order than the Cahn-Hilliard equation and to some extent lend themselves to simpler analysis. The derivation of equations (1) and (2) is outlined and several results are given. Full details are to appear.

### MSC:

76T99 | Multiphase and multicomponent flows |

76M99 | Basic methods in fluid mechanics |

35Q99 | Partial differential equations of mathematical physics and other areas of application |