Research Of The Incompressible Porous Medium Equations

This paper studies the equation with a general fractional diffusion term of an incompressible fluid in a porous medium governed by Darcy's law. Under the assumption that the initial value T0 (x ) is small enough, it proves the global well-posedness of the small solution in pesudomeasure space through the properties of pesudomeasure space and the operator semigroup, and the contraction mapping principle. Moreover, the asymptotic behavior of the solutions is shown as t→∞. Finally, establish a new method to derive a decay estimate of the L2 -norm of the solution, which avoids using the Fourier splitting technique completely, and obtain the decay rate of the solutions to the incompressible porous medium equation in L2 (R 3).In Chapter 1, it introduces the backgrounds and developments of an incompressible fluid in a porous medium, presents the situation of study about the incompressible porous medium equation, and gives the study results of this paper.In Chapter 2, it introduces meanings of the incompressibility and well-posedness, and gives the definitions of the pesudomeasure space and proposition.In Chapter 3, give the integral form of the incompressible porous medium equation by applying Duhamel principle, and prove the global well-posedness of the solutions to the incompressible porous medium equation in pesudomeasure space by three lemma and one proposition. Moreover, the asymptotic behavior of the solutions is studied.In Chapter 4, the decay rate of the solutions to the incompressible porous medium equation in L2 (R 3) is established by using a new method .