Sobolev approximation for two-phase solutions of forward-backward parabolic problems
Flavia Smarrazzo Andrea Terracina
Discrete & Continuous Dynamical Systems - A 2013, 33(4): 1657-1697 doi: 10.3934/dcds.2013.33.1657
We discuss some properties of a forward-backward parabolic problem that arises in models of phase transition in which two stable phases are separated by an interface. Here we consider a formulation of the problem that comes from a Sobolev approximation of it. In particular we prove uniqueness of the previous problem extending to nonlinear diffusion function a result obtained in [21] in the piecewise linear case. Moreover, we analyze the third order partial differential problem that approximates the forward-backward parabolic one. In particular, for some classes of initial data, we obtain a priori estimates that generalize that proved in [22]. Using these results we study the singular limit of the Sobolev approximation, as a consequence we obtain existence of the forward-backward problem for a class of initial data.
keywords: Forward-backward parabolic equations a priori estimates uniqueness results. two-phase solutions pseudo parabolic regularization
Conservation laws with discontinuous flux
Mauro Garavello Roberto Natalini Benedetto Piccoli Andrea Terracina
Networks & Heterogeneous Media 2007, 2(1): 159-179 doi: 10.3934/nhm.2007.2.159
We consider a hyperbolic conservation law with discontinuous flux. Such a partial differential equation arises in different applications, in particular we are motivated by a model of traffic flow. We provide a new formulation in terms of Riemann Solvers. Moreover, we determine the class of Riemann Solvers which provide existence and uniqueness of the corresponding weak entropic solutions.
keywords: discontinuous flux traffic flow Conservation laws Riemann Solvers front tracking

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