On Solvability of the First Boundary Value Problem for One-Dimensional Internal Erosion
This paper deals with a mathematical model of isothermal internal erosion without deformation of a porous medium. Underground water filtration occurs in the aquifer being in contact with frozen sandy soil. During soil thawing and at a certain magnitude of the filtration velocity, soil particles are removed from the flow, and underground cavities are created. These cavities increase in sizes and reach their critical sizes that result in a permafrost arch collapse. A mathematical model is based on mass conservation equations for water, moving solids particles and stationary porous skeleton along with Darcy''s law for water and moving solid particles (similar to a classical Muskat-Leverett model), and the equation for the intensity of suffusion flow. The problem statement and supporting information are provided in Paragraph 1 along with the statement of a theorem of unique classical solvability. Seven lemmas and physical principles for maxima of water saturation and porosity are presented in Paragraph 2. A key moment is to prove Holder''s continuity of saturation. Then, the conditions of Schauder''s theorem of a fixed point are verified.