Math 2.0

Partial differential equations Definition

Cahn–Hilliard equation

Builds on Parabolic partial differential equationLaplacianAllen–Cahn equation

Definition

Let Ω⊆ℝn\Omega \subseteq \R^n be open and let γ>0\gamma > 0. The Cahn–Hilliard equation is the parabolic equation of fourth order for c ⁣:Ω×(0,T)→ℝc \colon \Omega \times (0, T) \to \R: ct=Δμ,μ=c3−c−γ Δc,c_t = \Delta \mu, \qquad \mu = c^3 - c - \gamma\, \Delta c, where Δ\Delta is the Laplacian in xx. The function μ\mu is the chemical potential. On a bounded Ω\Omega one usually imposes ∂νc=∂νμ=0\partial_\nu c = \partial_\nu \mu = 0 on ∂Ω\partial \Omega, where ν\nu is the outer normal.

Remarks

Example: every constant c≡c0c \equiv c_0 is a solution, since then μ\mu is constant and Δμ=0\Delta \mu = 0. Non-example: the Allen–Cahn equation ct=γΔc−(c3−c)c_t = \gamma \Delta c - (c^3 - c) is of second order and does not conserve mass, so it is not of this form. With the boundary conditions above, ddt∫Ωc dx=∫∂Ω∂νμ dS=0\frac{d}{dt}\int_\Omega c\, dx = \int_{\partial\Omega} \partial_\nu \mu \, dS = 0, so the mass is conserved. The energy E(c)=∫Ω(γ2∣∇c∣2+14(1−c2)2) dxE(c) = \int_\Omega \big(\frac{\gamma}{2}\abs{\nabla c}^2 + \frac{1}{4}(1 - c^2)^2\big)\, dx does not increase along smooth solutions.

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Allen–Cahn equation, concept, no evidence levelAllen–Cahn equationconceptEuclidean space, object, no evidence levelEuclidean spaceobjectFunction, concept, no evidence levelFunctionconceptParabolic partial differential equation, concept, no evidence levelParabolic partial dif…conceptTotal derivative, concept, no evidence levelTotal derivativeconceptFunction of several variables, concept, no evidence levelFunction of several v…conceptVector-valued function, concept, no evidence levelVector-valued functionconceptDifferentiable function of several variables, concept, no evidence levelDifferentiable functi…conceptPartial derivative, concept, no evidence levelPartial derivativeconceptVector field, concept, no evidence levelVector fieldconceptGradient, concept, no evidence levelGradientconceptDivergence, concept, no evidence levelDivergenceconceptLaplacian, concept, no evidence levelLaplacianconceptCahn–Hilliard equation, definition, no evidence levelCahn–Hilliard equationdefinition
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  1. Allen–Cahn equation concept
  2. Euclidean space object
  3. Function concept
  4. Parabolic partial differential equation concept
  5. Total derivative concept
  6. Function of several variables concept
  7. Vector-valued function concept
  8. Differentiable function of several variables concept
  9. Partial derivative concept
  10. Vector field concept
  11. Gradient concept
  12. Divergence concept
  13. Laplacian concept
  14. Cahn–Hilliard equation definition

Further reading

  1. Web https://en.wikipedia.org/wiki/Cahn%E2%80%93Hilliard_equation
About this page
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Kinddefinition
WritingW0 stub
MSC35K55, 35K25
DefinesCahn–Hilliard equation
UsesParabolic partial differential equation, Laplacian, Allen–Cahn equation
Changed2 November 2026

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