Existence results for diffuse interface models describing phase separation and damage

摘要: In this paper, we analytically investigate multi-component Cahn-Hilliard and Allen-Cahn systems which are coupled with elasticity uni-directional damage processes. The free energy of the system is form $\int_\Omega\frac{1}{2}\mathbf\Gamma\nabla c:\nabla c+\frac{1}{2}|\nabla z|^2+W^\mathrm{ch}(c)+W^\mathrm{el}(e,c,z)\,\mathrm dx$ a polynomial or logarithmic chemical density $W^\mathrm{ch}$, an inhomogeneous elastic $W^\mathrm{el}$ quadratic structure gradient variable $z$. For corresponding processes, present appropriate notion weak solutions prove existence results based on certain regularization methods higher integrability result for strain $e$.