Free boundary problems arising in the freezing of soils in a bounded region. III: Hydrodynamics of the unfrozen phase. IV: Thermal stresses in the frozen phase
[For Part II see the author and \textit{R. B. Guenther}, ibid. 111, 475-512 (1985; Zbl 0592.35056).] The author utilizes the method of characteristics to construct unique solutions to problems II and III stated in Part I [ibid. 111, 1-13 (1985; Zbl 0592.35055)]. Part III deals with the hydrodynamics of the unfrozen phase. In Part III problem II is restated. Here the author finds a pair \(\{\) P(x,t),v(x,t)\(\}\) satisfying the conditions required in the problem in \(\Omega_ 2=\{(x,t):\) \(s(t)<x<b\), \(0<t<\tilde t\}\). In this problem s(t) is assumed to be known. The next section is devoted to proving the existence and uniqueness of solutions to Problem II. In section 3, the author derives an asymptotic estimate for the earliest time, \(t_ c\), of breakdown in the solution of Problem II and, in addition, the author establishes the continuous dependence of this solution on the initial and boundary data for all times up to \(t_ c.\) Part IV deals with thermal stresses in frozen phase. This part is devoted to the discussion of the displacement problem which was formulated in Part I and referred to as problem III. It is to find a function u(x,t) satisfying the stated conditions in \(\Omega_ 1=\{(x,t):\) \(0<x<s(t)\), \(0<t<\tilde t\}\). The author begins with reduction of governing differential equations to a system of first-order equations, and therefore introduces Problem V which is similar to problem II. Thus, the proof of existence, uniqueness, and continuous dependence on data of the solution of Problem V is carried out analogous to that of problem II.
scientific article; zbMATH DE number 3964581