Well–posedness Of The Second Order Quasilinear Anisotropic Degenerate Parabolic-hyperbolic Equations

The present Ph.D. dissertation is mainly concerned with the well-posedness of theCauchy problem to the second order quasilinear anisotropic degenerate parabolic-hyperbolicequations. This kind of equations models many natural phenomenons, such as diffusion-convection process in porous medium, sedimentation-consolution process, convection of thebiome in nature, financial decision etc. Since this kind of equations has important and wideapplications in practice, it has received many attentions for many years. These works aremostly focused on isotropic case, and the solution function spaces are BV spaces. Foranisotropic case whose coefficients don't explicitly depend on time and spatial variables, thewell-posedness was first given in [21] by Chen-Perthame for kinetic solutions in L1 space.Based on these results, this dissertation discusses the existence and uniqueness of solutionsto the Cauchy problem of the anisotropic degenerate parabolic-hyperbolic equation when thecoefficients explicitly depend on time variable. The whole contents are organized as follows.In the first part, we will introduce several mathematical models which can be describedby degenerate parabolic-hyperbolic equations. These models include convection-diffusionequation of the pollutant transportation in porous medium, nonlinear heat transport equa-tion, financial decision equation etc. Meanwhile, we can also observe that this kind ofequations has different forms in practical applications. The coefficients can depend on theunknown function or the time and space variables as well. The equations with explicitly-time-dependent coefficients are the major concerns of this dissertation.In the second part, we will prove the uniqueness of entropy solutions to the Cauchyproblem of a special simple degenerate parabolic-hyperbolic equation by kuznetsov's device.From the process of the proof, we can see that the key point making the Kuznetsov's methodfeasible for degenerate parabolic-hyperbolic equation is to introduce the term of parabolic dissipation measure, which is the main difference from hyperbolic equations. This methodcan also be extended to more general degenerate parabolic-hyperbolic equations.In the third part, we study the uniqueness and existence of solutions to the Cauchy prob-lem with explicitly-time-dependent coefficients. Uniqueness is proved by kinetic method.For this end, we first deduce the kinetic formulation for general anisotropic degenerateparabolic-hyperbolic equations and give the definition of kinetic solutions. Meanwhile, weprove the equivalence between entropy inequality and kinetic formulation for entropy so-lutions. Thus, we know that entropy solutions and kinetic solutions are equivalent in L∞spaces. We use vanishing viscosity method to prove the existence. We calculate the a prioriestimates of smooth solutions to the viscous equation, from which we deduce that smooth so-lutions converge strongly to a function in C([0, +∞); L1(Rn)) space. By this convergence,we prove that the limit function is the kinetic solution or the entropy solution.In the fourth part, we study the error estimates between entropy solutions and thesmooth solutions to the corresponding viscous equations. We adopt kinetic technique, butwe modify the kinetic function and the kinetic formulation. By this formulation, We achievethat the error has the order of square root of viscosity coefficient.In the fifth part, we will brie?y introduce other research results obtained in my Ph.D.study– the global limits of entropy solutions to relativistic Euler equations in the ?uid dy-namics and the pointwise estimates to solutions for 1-Dimensional linear Thermoviscoelasticsystem .