Initial Value Problems For Two Types Of Evolution Equations

In this thesis, we study two types of nonlinear evolution equations: the fractal parabolicconservation laws and the pressure-gradient system. Firstly, we take the Cauchy problem ofthe fractal parabolic conservation laws as an example to consider how the solutions behave inthe L2space and the homogeneous Sobolev spaces. For the classical parabolic conservationlaws, the Green's function method is applied to the initial value problem. Secondly, weconsider the pressure-gradient system which is a simplified model of Euler equation. Westudy a special Riemann problem. The thesis is arranged as follows:In Chapter1, we review the physical background of the fractal conservation laws andthe pressure gradient system. We state the history of study for the systems. We also introducethe problems we will study and the main results.In Chapter2, we study the time-asymptotic behavior of solutions to the Cauchy problemfor multi-dimensional parabolic conservation laws with fractional dissipation. For arbitrar-ily large initial data, we obtain the optimal decay rates in L2and homogeneous Sobolevspaces for solutions to the equation with the power of Laplacian12<α≤1by using thetime-frequency decomposition method and the energy method. The argument is based on amaximum principle.In Chapter3, we consider the parabolic conservation laws which is the case α=1inthe last chapter. We consider the large initial data case which has a big perturbation. We usethe method of Green's function to obtain the pointwise estimate of the solution. Combingthe conclusion of the last Chapter, we finally obtain the pointwise estimate of the solution.In Chapter4, we consider two dimensional Riemann problems for the pressure-gradientsystem. The pressure-gradient system is interesting and important as models for similarproblems for the Euler systems. In this chapter, we construct a global continuous solutionfor the interaction of four rarefaction waves in the bi-symmetric class for the two dimensionalpressure-gradient system. These solutions are for initial planar waves that are larger than acritical strength which is more explicitly calculated here than the theoretical value for the Euler case. The solutions do not have sonic points while the vacuum is a single point locatedat the origin. We provide detailed descriptions of the simple wave patch (resulted from theplanar wave interaction), especially the behavior of the characteristic curves near the vacuumpoint. We also discuss the interaction of two simple waves using the method of characteristicdecomposition on the inclination angle of corresponding characteristic curves.