1,720,974 research outputs found
Two-phase entropy solutions of forward-backward parabolic problems with unstable phase
Full text linkIn this paper we study a two-phase problem for a forward-backward parabolic equation with diffusion function of cubic type. Existence and uniqueness for these kind of problems were obtained in literature in the case in which the phases are both stable. Here we consider the situation in which the unstable phase is taken in account, obtaining not trivial solutions of the problem. It is interesting to note that such solutions are given by solving generalized Abel's equations
Entropy formulation for forward-backward parabolic equations
No full textWe give a brief overview of the results obtained for forward--backward parabolic equations of cubic type. In particular we summarize an approach that has some analogies with hyperbolic conservation laws. More precisely we give a concept of entropy solution and analyze in more details admissibility conditions satisfied by an interface that separates different phases. Finally we state uniqueness and existence results that appear in literature
Qualitative behavior of the two-phase entropy solution of a forward-backward parabolic problem
Full text linkWe discuss some qualitative aspects of a forward-backward parabolic problem that has been introduced in [L. C. Evans and M. Portilheiro, Math. Models Methods Appl. Sci., 14 (2004), pp. 1599-1620], [C. Mascia, A. Terracina, and A. Tesei, Evolution of stable phases in forwardbackward parabolic equations, in Asymptotic Analysis and Singularities, Mathematical Society of Japan, Tokyo, 2007, pp. 451-478] and further analyzed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887-925]. This problem arises in models of phase transition in which two stable phases are separated by an interface. In particular, we consider here the problem of the extension in time of the solution constructed in [C. Mascia, A. Terracina, and A. Tesei, Arch. Ration. Mech. Anal., 194 (2009), pp. 887-925]. We analyze the regularity of the solution u defined in a domain R × (0, T) and give an estimate, depending on the initial datum, of the number of convex regions of the function u(·, t) for every t ε (0, T). Copyright © 2011 by SIAM
Convergence of a Relaxation Approximation to a Boundary Value Problem for Conservation Laws
No full textSobolev approximation for two-phase solutions of forward-backward parabolic problems
No full textWe 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
Non-uniqueness results for entropy two-phase solutions of forward-backward parabolic problems with unstable phase
No full textThis paper studies the well-posedness of the entropy formulation given by Plotnikov (1994) in [23] for forward-backward parabolic problem obtained as singular limit of a proper pseudoparabolic approximation. It was proved in Mascia et al. (2009) [20] that such a formulation gives uniqueness when the solution takes values in the stable phases. Here we consider the situation in which unstable phase is taken in account, proving that, in general, uniqueness does not hold. (C) 2014 Elsevier Inc. All rights reserved
Evolution of stable phases for forward-backward parabolic equations
No full textWe review some recent work concerning an ill–posed forward–backward parabolic equation, which arises e.g. in the theory of phase transitions. Some new results are also presented, concerning local existence and uniqueness of solutions within a certain class of physical interest, and a hint of their proofs is given
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