Well-posedness Of Unbounded Entropy Solution Of Anistropic Degenerate Parabolic-hyperbolic Equations

Degenerate parabolic-hyperbolic equations model many natural phenomena,such as pollutant transportation in porous medium, convection difusion process inmulti-phase, heat conduction process, sedimentation-consolidation process, convec-tion of the biome in nature, financial decision etc. Since this kind of equations arevery important, it has received much attention for many years. The present Ph. D.dissertation is mainly concerned with the well-posedness of unbound entropy solu-tions for anisotropic degenerate parabolic-hyperbolic equations. The main resultsare as follows.1. Renormalized entropy solutions of Cauchy problems. GuiqiangChen and K. H. Karlsen [20] proved the well-posedness of L∞entropy solutions forCauchy problems of anisotropic degenerate parabolic-hyperbolic equations withtime-space dependent coefcients. If the initial data u0∈L1, entropy solutionsof the Cauchy problems may be unbounded. Since the convection flux functionand difusion function are Lipschitz continuous, it is possible that the convectionflux function and difusion function are not locally integrable. Thus, we considerrenormalized entropy solutions, and use Kruzkov device of doubling variables andvanishing viscosity method to prove the well-posedness of renormalized entropysolutions for the Cauchy problems.2. Renormalized entropy solutions of homogenous Dirichlet prob-lems. Yachun Li and Qi Wang [57] obtained the well-posedness of L∞entropysolutions for homogenous Dirichlet problems of anisotropic degenerate parabolic-hyperbolic equations. When the initial data u0∈L1, entropy solutions of thehomogenous Dirichlet problems may be unbounded. Since the convection fluxfunction and difusion function are Lipschitz continuous, it is possible that the convection flux function and difusion function are not locally integrable. Thus, wealso introduce renormalized entropy solutions, and use Kruzkov device of doublingvariables and vanishing viscosity method to prove the well-posedness of renormal-ized entropy solutions for the homogenous Dirichlet problems.3. Lpentropy solutions of homogenous Dirichlet problems. We con-tinue to consider homogenous Dirichlet boundary problems of anisotropic degen-erate parabolic-hyperbolic equations. When the initial data u0∈Lp(p>1) andthe convection flux function and difusion function satisfies some growth condi-tions, the convection flux function and difusion function are locally integrable.Therefore, Similarly as definition of L∞entropy solutions, we introduce Lpentropysolutions, and use modified Kruzkov device of doubling variables and vanishingviscosity method to prove the well-posedness of Lpentropy solutions for the ho-mogenous Dirichlet problems.