The Paint Layer’s One Phase Stefan Problem In The Solvent Evaporation Process

This article mainly studies the one-dimensional and two-dimensional free boundary problem of the the paint layer during the drying process and proves this kind of free boundary problems possesses a unique classical solution. Solutions constituted by a pair of functions:the volume fraction of solventu(x, t)、u(x,y,t) and boundary function t(t)、 h(y,t).Proving the solution existence has two fundamental technologies:First, by changing of variable, straighten the boundary, make the free boundary into a fixed boundary. The second is constructing a mapping, using the Schuader fixed point theorem to prove the existence of the mapping’s fixed point, so the equation classical solution exists, and then prove the classical solution is unique. This article selects the one-dimensional and two-dimensional paint model as an example researches their classical solutions existence and uniqueness, finally derived the paint model equations and studied the asymptotic behavior of a one-dimensional classical solution, the two-dimensional paint equation classical solution can not determine whether there is a whole solution, it does not exist the asymptotic behavior, this article is divided into the following chapters to illustrate:Chapter Ⅰ briefly introduces the free boundary and cites one classic model in two categories free boundary problem, and prepares the way for the study of the following sections, finally, introduces the main work of this article and the problem to be solved as well as domestic and foreign research situation.Chapter Ⅱ proves Stefen problem of one-dimensional paint system, the model equations are as follows:This equation describes the free boundary problem of one-dimensional paint layer in the solvent evaporation process, this equation shows the variation of the volume fraction of solvent u(x,t) and paint thickness h(t) as well as the relationship between the two, it is compared with the previous studiesthe paint model equations biggest difference is the diffusion coefficient D is no longer a constant but a function about u, which greatly increased the difficulty of the problem, in order to prove existence and uniqueness of the Classical solution, this chapter used variable substitution and Schauder fixed point theorem and maximum Norm Estimates and other skills proved:if one-dimensional paint system equations meet the conditions,then in the corresponding area exist a unique classical solution u∈C2-α,1-2/α, the thickness of the paint h(t) with u the uniquely determined,Chapter III, on the basis of the one-dimensional model’s point in the second chapter extended to two-dimensional model’s line, its boundary is a curve,2D paint system model equation:Proving the existence and uniqueness methods of the classical solution of the equation is similar to proving the classical solution of one-dimensional paint system in the chapter one, the method is also variable substitution and Schauder fixed point theorem.Chapter Ⅳderives the one-dimensional and two-dimensional paint model equations, and then study the asymptotic behavior of solutions of the one-dimensional paint system, that is, in one-dimensional paint system: corresponding to this conclusion and tangible:in the process of the solvent evaporation when the time enough long,all the paint layer to dryness, after all the solvent was evaporated to dryness the thickness of the paint is also determined to be a fixed constant, This result in the theoretical and practical organic unity and fully demonstrate the strong value of applied mathematics.