Geometric inequalities from phase space translations
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The dynamics of transients of fluid-rock temperature, pore pressure, pollutants in porous rocks are of vivid interest for fundamental problems in hydrological, volcanic, hydrocarbon systems, deep oil drilling. This can concern rapid landslides or the fault weakening during coseismic slips and also a new field of research about stability of classical buildings. Here we analyze the transient evolution of temperature and pressure in a thin boundary layer between two adjacent homogeneous media for various types of rocks. In previous models, this boundary was often assumed to be a sharp mathematical plane. Here we consider a non-sharp, physical boundary between two adjacent rocks, where also local steady pore pressure and/or temperature fields are present. To obtain a more reliable model we also investigate the role of nonlinear effects as convection and fluid-rock “frictions”, often disregarded in early models: these nonlinear effects in some cases can give remarkable quick and sharp transients. All of this implies a novel model, whose solutions describe large, sharp and quick fronts. We also rapidly describe transients moving through a particularly irregular boundary layer.