Abstract
In this paper, we study the pointwise estimates of solutions to the viscous Cahn-Hilliard equation with the inertial term in multidimensions. We use Green’s function method. Our approach is based on a detailed analysis on the Green’s function of the linear system. And we get the solution’s Lp convergence rate.
Highlights
In this paper, we study the pointwise estimates of the solution ρðx, tÞ to the Cauchy problem:( ηρtt + ρt + Δ2ρ − kΔρt − Δf ðρÞ = 0, ðx, tÞ ∈ Rn × ð0,∞Þ, ρðx, tÞjt=0 = ρ0ðxÞ, ρtðx, tÞjt=0 = ρ1ðxÞ, ð1Þ where n ≥ 4. f ðρÞ is the intrinsic chemical potential which is smooth in the small neighborhood of the origin, and f ðρÞ = Oðρ1+θÞ when jρj ≤ 1 and θ is a positive integer
The Cahn-Hilliard equations with inertial term model nonequilibrium decompositions caused by deep supercooling in certain glasses
The wellknown Cahn-Hilliard equation is a parabolic equation, but the Cahn-Hilliard equation with the inertial term is a hyperbolic equation with relaxation which brings many mathematical difficulties to study
Summary
We study the pointwise estimates of the solution ρðx, tÞ to the Cauchy problem:. Without smallness assumption on initial data, [7] got the global existence of the classical solution. Wang and Wu [9] obtained the global existence and optimal decay rate of the classical solution by long wave-short wave method. For the viscous Cahn-Hilliard equation with the inertial term, it describes the early stages of spinodal decomposition in certain glasses (see [15,16]). We are interested in the viscous Cahn-Hilliard equation with the inertial term. Under the smallness assumption on initial data, based on the detailed analysis of the Green’s function, we get the pointwise estimates of solutions. In this paper, we have the same assumptions for the spacial dimension
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