Moving boundary problems on Earth’s surface

In a moving boundary problem one or more of the domain boundaries is an unknown function of time. The classic moving boundary problem, the Stefan problem, is related to the tracking of a sharp liquid/solid front during the melting of ice. A major societal problem of our time is the current rapid rate of degradation of Earth environments and resources through climate change and other anthropogenic forcing. Finding mitigation strategies and solutions to these problems requires large scale trans-disciplinary research efforts. A central plank in this research is the need to develop an understanding of the transport processes that govern how materials and resources are moved through and over the Earth’s surface. Two relevant examples, which can be seen as generalizations of the classic Stefan problem, are the sub-surface movement of resources such as water and the surface transport of sediment which builds and maintains landscapes and ocean shorelines. A feature in these problems is that the transport domain, i.e., the Earth’s surface, is highly heterogeneous exhibiting a wide ranges of length scales that are often power-law distributed. The consequence of this is that the transport processes are non-local in space and time, i.e., fluxes of conserved quantities cannot be determined from instantaneous and local conditions alone. Recently there has been much interest in the geomorphodynamics research community of exploring how fractional calculus representations of elements in the transport equations can be used to model non-locality, [3], [8], [7], [4]. Phenomenological fractional calculus models of geomorphic and geology transport processes that provide sound qualitative comparisons with filed and experimental observations have been constructed. These models have suggested many interesting hypothesis on how the controlling mechanisms of transport processes shape our environment. In this work, however, there is an almost complete absence of rigorous mathematical analysis. Let us recall the statement of the so-called one-phase Stefan, where u is the temperature of the melting solid and s is position of the interface, (x = 0 is the fixed boundary of a container). For the sake of simplicity we consider the one-dimensional case,