Boundary Layer Problems For The One Dimensional Non-Newtonian Filtration Equation

The non-Newtonian filtration equation(i.e.,evolution -Laplace equation),as an important quasi-linear parabolic equation,arises from diffusion phenomenon widely existing in nature,filtration theory,phase transition theory,biochemistry,biological population dynamics and so on.In the past 60 years,the non-Newtonian filtration equation has been widely studied with many fruitful results.So far,however,the boundary layer theory of the equation has not been investigated.In fact,the theory of boundary layers has been one of the fundamental and important issues in fluid dynamics,and some problems such as the well posedness of the Prandtl equation are attracting attention of more and more scholars.In this thesis,we will study boundary layer behavior for the initial and boundary value problem of the 1-D non-Newtonian filtration equation with vanishing diffusion,and one of the aims is to extend Frid and sheulkhin's results published in Communications in mathematical physics(1999).The main work of this thesis is as follows.1.Existence of a boundary layer and estimation of boundary layer thickness for the problem with vanishing diffusion.Under the conditions that one of the boundary functions is not identically zero,we establish the existence of a boundary layer and give an almost optimal estimation of boundary layer thickness.The key point is to establish some uniform estimates independent of the diffusion coefficient of the solution.Because of its nonlinear structure and degeneracy,it is very difficult to study the non-Newtonian filtration equation.One of the main difficulties is that the required uniform estimates cannot be obtained directly.To overcome it,we first obtain a sequence of smooth solutions by using parabolic regularization method,and then establish some uniform estimates of smooth solutions by using energy estimation and other techniques.Finally,we obtain the existence of a continuous solution and the required uniform estimates by using weak convergence technique.It should be pointed out that Frid and Sheulkhin's techniques used to deal with Newtonian fluid could not be applied to this paper.2.Existence of an optimal thickness of boundary layer.For the corresponding problem with Newtonian fluid,although a tremendous amount of experimental data indicate that there should have been an optimal thickness,there is no a strict mathematical proof so far,and there is a few formal analysis to this conjecture at present.For the non-Newtonian filtration equation,in this thesis,we construct an optimal thickness of boundary layer by means of Barenblatt-type solutions.3.Asymptotic behavior of solutions as the diffusion vanishing,for instance,the optimal convergence rate and the optimal blow-up rate.Due to appearance of boundary layers,the structure of solutions near the boundary is extremely complex.Therefore,the study on asymptotic behavior of solutions will be helpful to understand the generation mechanism and internal state of boundary layers.In fact,one of our results shows that the solution must blow up in the boundary layer when a boundary layer occurs.In summary,we prove a new property of non-Newtonian filtration equation,namely,boundary layer phenomenon,and establish a relatively complete theory of boundary layer for the initial and boundary value problem of the equation in 1-D,which is is an important supplement to the mathematical theory of the non-Newtonian filtration equation.