Convective Cahn-Hilliard Equation With Dynamic Boundary Conditions

This paper considers the convective Cahn-Hilliard equation with dynamic boundaryThe convective Cahn-Hilliard equation arises naturally as a continuousmodel for the formation of facets and corners in crystal growth. Here u(x, t) denotes the. slope of the interface. The convective term stem from the effect of kinetics (the finite rate of atoms or molecules attachment to the crystal surface) that provides an independent flux of the order parameter, similar to the effect of an external field in spinodal decomposition of a driven system.During the past years, many authors have paid much attention to the convective Cahn-Hilliard equationdue. Due to the existence of the convection, the dynamic boundary condition and tangential LaplacianΔ_‖in the bound, which make it difficult to proof the global existence and uniqueness of solutions, such that, we introduce the approximate problem (P_ε),andWe use contraction mapping theorem and the prior estimates to prove the local existence and uniqueness of the problem (P_ε),Theorem 1 Suppose that the initial dataφ₀, u₀ satisfy the conditions above. Then there is a positive constantδ, which may depend on the initial data and onεsuch that the problem (P_ε) admits a unique local solutionφ, u such thatφ∈C⁰([0,δ]; H³),φ_t∈C⁰([0,δ];V),φ_t∈L²([0,δ];H²),φ_tt∈L²([0,δ];H), u∈C⁰([0,δ];H¹)∩L²([0,δ];H²),u∈L²([0,δ];L²).We use energy estimate and Gaghiardo-Nirenberg interpolationinequality to get the uniform prior estimate ||φ(t)||_H³, ||u(t)|_H¹, ||φ_t(t)||_V. Furthermore, we get the existence and uniqueness of solutionsfor the approximate problem problem (P_ε), that isTheorem 2 Suppose that the initial dataφ₀, u₀ satisfy the conditions (3.1)-(3.3) in the Section 3. Then the problem admits a unique global solution such that for any T > 0,φ∈C⁰([0,T]; H³),φ_t∈C⁰([0,T];V),φ_t∈L²([0,T];H²),φ_tt∈L²([0,T];H), u∈C⁰([0,T];H¹)∩L²([0,T];H²),u∈L²([0,T];L²).Letε→0, the existence and uniqueness of the solution for the initial problem is proved, that isTheorem 3 Suppose that n = 2 or n = 3 and that the initial dataψ₀ satisfiesψ₀∈H³, Bφ₀∈H¹(Γ). Then the initial boundary value problem (5.1)-(5.5) admits a unique global solutionψsuch that for any T > 0,ψ∈C⁰([0, T];H¹)∩L²([0, T];H³),ψ_t∈L²([0,T];V),Δ_ψ∈L²([0,T];H³),μ= -Δ_ψ-ψ+ψ³âˆˆL²([0, T]; H³).