Narrow Results

We consider a class of doubly nonlinear degenerate hyperbolic-parabolic equations with homogeneous Dirichlet boundary conditions, for which we first establish the existence and uniqueness of entropy solutions. We then turn to the construction and analysis of discrete duality finite volume schemes (in the spirit of Domelevo and Omnès [43]) for these problems in two and three spatial dimensions. We derive a series of discrete duality formulas and entropy dissipation inequalities for the schemes. We establish the existence of solutions to the discrete problems, and prove that sequences of approximate solutions generated by the discrete duality finite volume schemes converge strongly to the entropy solution of the continuous problem. The proof revolves around basic a priori estimates, the discrete duality features, Minty–Browder type arguments, and "hyperbolic" L^∞ weak-⋆ compactness arguments (i.e. propagation of compactness along the lines of Tartar, DiPerna, …). Our results cover the case of non-Lipschitz nonlinearities.

In 2011, Düntsch and Orłowska obtained a discrete duality for regular double Stone algebras. On the other hand, it is well known that regular double Stone algebras are polinominally equivalent to $3$-valued Łukasiewicz–Moisil algebras (or LM₃-algebras). In [R. Cignoli, Injective De Morgan and Kleene algebra, Proc. Amer. Math. Soc.47 (1975) 269–278], LM₃-algebras are considered as a Kleene algebras $〈L,\vee ,\wedge ,\sim ,0,1〉$ endowed with a unary operation $\square :L\to L$, satisfying the properties: $a\vee \sim \square a=1,$$\sim a\wedge a=a\wedge \sim \square a$ and $\square a\vee \square b\le \square (a\vee b).$ Motivated by this result, in this paper, we determine another discrete duality for LM₃-algebras, extending the discrete duality to De Morgan algebras described in [W. Dzik, E. Orłowska and C. van Alten, Relational representation theorems for general lattices with negations, in Relations and Kleene Algebra in Computer Science, Lecture Notes in Computer Science, Vol. 4136 (Springer, Berlin, 2006), pp. 162–176].

A discrete duality is a relationship between classes of algebras and classes of relational systems (frames). In this paper, discrete dualities are presented for De Morgan algebras with various kind of unary operators. To do this, we will extend the discrete duality given in [W. Dzik, E. Orłowska and C. van Alten, Relational representation theorems for general lattices with negations, in Relations and Kleene Algebra in Computer Science, Lecture Notes in Computer Science, Vol. 4136 (Springer, Berlin, 2006), pp. 162–176], for De Morgan algebras.

In [Tense operators on De Morgan algebras, Log. J. IGPL22(2) (2014) 255–267], Figallo and Pelaitay introduced the notion of tense operators on De Morgan algebras. Also, other notions of tense operators on De Morgan algebras were given by Chajda and Paseka in [De Morgan algebras with tense operators, J. Mult.-Valued Logic Soft Comput.1 (2017) 29–45; The Poset-based logics for the De Morgan negation and set representation of partial dynamic De Morgan algebras, J. Mult.-Valued Logic Soft Comput.31(3) (2018) 213–237; Set representation of partial dynamic De Morgan algebras, in 2016 IEEE 46th Int. Symp. Multiple-Valued Logic (IEEE Computer Society, 2016), pp. 119–124]. In this paper, we introduce a new notion of tense operators on De Morgan algebras and define the class of tense De Morgan $S4$-algebras. The main purpose of this paper is to give a discrete duality for tense De Morgan $S4$-algebras. To do this, we will extend the discrete duality given in [W. Dzik, E. Orłowska and C. van Alten, Relational Representation Theorems for Lattices with Negations: A Survey, Lecture Notes in Computer Science (2006), pp. 245–266], for De Morgan algebras.