How to prove that weak solutions of an elliptic-parabolic system — linear elasticity coupled to a second-order parabolic equation whose leading coefficient is weighted by the gradient of the order parameter — gain regularity when the initial data is taken in H2 rather than H1. The central technique is a reciprocal-weight a priori estimate, obtained by multiplying the evolution equation by the inverse of the degenerate weight, which controls the time derivative even though the diffusion coefficient is neither bounded away from zero nor differentiable. Students also learn why writing the degenerate term in divergence form restricts the argument to one space dimension, and how pointwise convergence of the gradient can be recovered through a compact Sobolev embedding instead of an Aubin-Lions compactness lemma.
This arxiv paper by Peicheng Zhu investigates the regularity of weak solutions to a coupled system of PDEs consisting of linear elasticity equations and a nonlinear degenerate parabolic equation, wit…