On a Cahn-Hilliard system with source term and thermal memory

Abstract

A nonisothermal phase field system of Cahn-Hilliard type is introduced and analyzed mathematically. The system constitutes an extension of the classical Caginalp model for nonisothermal phase transitions with a conserved order parameter. It couples a Cahn-Hilliard type equation with source term for the order parameter with the universal balance law of internal energy. In place of the standard Fourier form, the constitutive law of the heat flux is assumed in the form given by the theory developed by Green and Naghdi, which accounts for a possible thermal memory of the evolution. This has the consequence that the balance law of internal energy becomes a second-order in time equation for the thermal displacement or freezing index, that is, a primitive with respect to time of the temperature. Another particular feature of our system is the presence of the source term in the equation for the order parameter, which entails additional mathematical difficulties because the mass conservation of the order parameter is lost. We provide several mathematical Colli -Gilardi -Signori -Sprekels results under general assumptions on the source term and the double-well nonlinearity governing the evolution: existence and continuous dependence results are shown for weak and strong solutions to the corresponding initial-boundary value problem.

Optimal Temperature Distribution for a Nonisothermal Cahn-Hilliard System in Two Dimensions With Source Term and Double Obstacle Potential

“…Moreover, since the solution components ϕ and w are uniquely determined, the cost functional ( 6) is well defined on U ad . Besides, let us remark that the exponent σ appearing above is more general than σ = 6 which was stated in [9,Thm. 2.2].…”

“…The state system (1)-( 5) was recently analyzed in [9] concerning wellposedness and regularity (see the results cited below in Section 2); in [10] the corresponding optimal control problem (P) has been analyzed for the simpler differentiable case when the indicator function 2) is replaced by either a regular function or by a logarithmic expression of the form…”

“…The analysis of the systems (1)-( 5) and ( 1), ( 17), ( 3)-( 5) has been the subject of investigation in [9]. As a special case of [9, Thm.…”

Optimal temperature distribution for a nonisothermal Cahn-Hilliard system in two dimensions with source term and double obstacle potential

“…We pick an arbitrary sequence {α n } ⊂ (0, 1] such that α n ց 0 as n → ∞. By virtue of [9,Thm. 4.1], the problem (P αn ) has a solution u αn ∈ U ad with associated state (ϕ αn , µ αn , w αn ) and ξ αn := h ′ αn (ϕ αn ) for n ∈ N. Since U ad is bounded in L ∞ (Q), we may without loss of generality assume that u αn → u weakly star in L ∞ (Q) for some u ∈ U ad .…”

“…The state system (1)-( 5) was recently analyzed in [9] concerning well-posedness and regularity (see the results cited below in Section 2); in [10] the corresponding optimal control problem (P) has been analyzed for the simpler differentiable case when the indicator function 2) is replaced by either a regular function or by a logarithmic expression of the form…”