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Well-posedness for non-isotropic degenerate parabolic-hyperbolic equations. (English) Zbl 1031.35077
Summary: We develop a well-posedness theory for solutions in \(L^1\) to the Cauchy problem of general degenerate parabolic-hyperbolic equations with non-isotropic nonlinearity. A new notion of entropy and kinetic solutions and a corresponding kinetic formulation are developed which extends the hyperbolic case. The notion of kinetic solutions applies to more general situations than that of entropy solutions; and its advantage is that the kinetic equations in the kinetic formulation are well defined even when the macroscopic fluxes are not locally integrable, so that \(L^1\) is a natural space on which the kinetic solutions are posed. Based on this notion, we develop a new, simpler, more effective approach to prove the contraction property of kinetic solutions in \(L^1\), especially including entropy solutions. It includes a new ingredient, a chain rule type condition, which makes it different from the isotropic case.

MSC:
35K65 Degenerate parabolic equations
35B30 Dependence of solutions to PDEs on initial and/or boundary data and/or on parameters of PDEs
35D05 Existence of generalized solutions of PDE (MSC2000)
35K15 Initial value problems for second-order parabolic equations
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