Well-posedness Of Entropy Solutions To Mixed Type Parabolic-hyperbolic Equations

The well-posedness of degenerate, even mixed partial differential equations has be-come one of the main concern in the field of nonlinear partial differential equations.The present Ph. D. dissertation is denoted to some analysis on three kinds of mixedtype parabolic-hyperbolic equations. The author uses Kruzkov device of doubling vari-ables and vanishing viscosity method to obtain the uniqueness and existence of entropysolutions.Chapter1, preface. This chapter is denoted to describing the physical backgroundand the mathematical models. The author also recalls the known results. The maindifficulties, results and innovations of this Ph. D. dissertation are briefly introduced inthe second part of this chapter. The contents of the last four chapters are organized asfollows.In chapter2, the author considers the nonhomogeneous Dirichlet problem of cou-pling parabolic-hyperbolic equations in two adjacent domains. The fluid satisfies thetransmission condition along the interface. The author focus on the hyperbolic equation,since the theory for parabolic equation is classical. The author puts forward a suitableboundary entropy condition to prove the uniqueness of entropy solutions.In chapter3, the author compares with and unifies the two different methods for ho-mogeneous and nonhomogeneous Dirichlet problems for isotropic degenerate parabolic-hyperbolic equations. For the general case that the coefficients depend on (t, x), the au-thor arrives at the well-posedness of L entropy solutions and entropy process solutions.For anisotropic equations, it is a big challenge since the equation may be neitherparabolic nor hyperbolic type. The main difficulty is how to handle the boundary con-dition. The author sets up the “weak trace” condition for the weak solution, and firstintroduces the boundary entropy-entropy flux triple to define an appropriate boundaryentropy inequality and then obtain the well-posedness of L entropy solutions in chapter 4.In chapter5, the Cauchy problem for multi-dimensional stochastic degenerate parabolic-hyperbolic equation is considered. With the help of spatial BV-estimate and tempo-ral L1continuity, the author proves the existence of BV\Lpstrong entropy solution.Moreover, the author gets the continuous dependence on initial data, nonlinearity andstochastic term, as well as the error estimate of approximate solutions.In Appendix One, we will introduce the regularity result of the sonic line for twodimensional Riemann problem of pressure gradient system.