A progress report on two unsolved problems that were presented at UD.

(a) Exact solution for unsaturated flow with Dirichlet boundary conditions.

From the mid 1980’s, the author and others adapted integrable nonlinear convection-diffusion equations to obtain realistic one-dimensional solutions for transient unsaturated flow in soil. The solution with constant-flux boundary conditions has been of great interest but the solution with Dirichlet boundary conditions has defied our best efforts. This problem can be transformed to a Stefan problem for solidification, with latent heat release, linear heat conduction and additional steady heat extraction occurring at the free boundary. If we choose independent coordinates to be canonical coordinates of the scaling symmetry, then separation of variables is admissible at all levels of correction for the non-invariant problem. The full solution is a power series in t^1/2 for which remarkably, each term satisfies the governing equation.

(b) PDEs for which entropy must increase.

A solution p(x,t) of an evolution PDE is stable if it minimises a suitable Liapunov functional such as energy, information or minus entropy. It is easily seen that the 2nd law of thermodynamics is equivalent to loss of Shannon information when p(x,t) satisfies a general nonlinear 2nd order diffusion equation. This is not so for 4^th order diffusion. We know from thin film flow and surface diffusion, that fourth order diffusion terms always generate ripples rather than a monotonic approach to equilibrium. Despite this, we can construct a non-trivial class of fourth order quasilinear diffusion equations that increase the Shannon entropy and maintain positivity. Work is in progress to find an entropy functional for physical 4^th order equations that is thermodynamically conjugate to the energy functional.