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Many science phenomena are described as interacting particle systems (IPS). The mean field limit (MFL) of large all-to-all coupled deterministic IPS is given by the solution of a PDE, the Vlasov Equation (VE). Yet, many applications demand IPS coupled on networks/graphs. In this paper, we are interested in IPS on directed graphs, or digraphs for short. It is interesting to know, how the limit of a sequence of digraphs associated with the IPS influences the macroscopic MFL of the IPS. This paper studies VEs on a generalized digraph, regarded as limit of a sequence of digraphs, which we refer to as a digraph measure (DGM) to emphasize that we work with its limit via measures. We provide (i) unique existence of solutions of the VE on continuous DGMs, and (ii) discretization of the solution of the VE by empirical distributions supported on solutions of an IPS via ODEs coupled on a sequence of digraphs converging to the given DGM. The result substantially extends results on one-dimensional Kuramoto-type models and we allow the underlying digraphs to be not necessarily dense. The technical contribution of this paper is a generalization of Neunzerts in-cell-particle approach from a measure-theoretic viewpoint, which is different from the known techniques in $L^p$-functions using graphons and their generalization via harmonic analysis of locally compact Abelian groups. Finally, we apply our results to various models in higher-dimensional Euclidean spaces in epidemiology, ecology, and social sciences.
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In this paper we introduce and study semigroups of operators on spaces of fuzzy-number-valued functions, and various applications to fuzzy differential equations are presented. Starting from the space of fuzzy numbers, many new spaces sharing the same properties are introduced. We derive basic operator theory results on these spaces and new results in the theory of semigroups of linear operators on fuzzy-number kind spaces. The theory we develop is used to solve classical fuzzy systems of differential equations, including, for example, the fuzzy Cauchy problem and the fuzzy wave equation. These tools allow us to obtain explicit solutions to fuzzy initial value problems which bear explicit formulas similar to the crisp case, with some additional fuzzy terms which in the crisp case disappear. The semigroup method displays a clear advantage over other methods available in the literature (i.e., the level set method, the differential inclusions method and other fuzzification methods of the real-valued solution) in the sense that the solutions can be easily constructed, and that the method can be applied to a larger class of fuzzy differential equations that can be transformed into an abstract Cauchy problem.
A characterization of a semilinear elliptic partial differential equation (PDE) on a bounded domain in $mathbb{R}^n$ is given in terms of an infinite-dimensional dynamical system. The dynamical system is on the space of boundary data for the PDE. This is a novel approach to elliptic problems that enables the use of dynamical systems tools in studying the corresponding PDE. The dynamical system is ill-posed, meaning solutions do not exist forwards or backwards in time for generic initial data. We offer a framework in which this ill-posed system can be analyzed. This can be viewed as generalizing the theory of spatial dynamics, which applies to the case of an infinite cylindrical domain.
In this work there are considered model problems for two nonlinear equations, which type depends on the solution. One of the equations may be called a nonlinear analog of the Lavrentev-Bitsadze equation.
We consider the $L_t^2L_x^r$ estimates for the solutions to the wave and Schrodinger equations in high dimensions. For the homogeneous estimates, we show $L_t^2L_x^infty$ estimates fail at the critical regularity in high dimensions by using stable Levy process in $R^d$. Moreover, we show that some spherically averaged $L_t^2L_x^infty$ estimate holds at the critical regularity. As a by-product we obtain Strichartz estimates with angular smoothing effect. For the inhomogeneous estimates, we prove double $L_t^2$-type estimates.
In this survey, our aim is to emphasize the main known limitations to the use of Wigner measures for Schrodinger equations. After a short review of successful applications of Wigner measures to study the semi-classical limit of solutions to Schrodinger equations, we list some examples where Wigner measures cannot be a good tool to describe high frequency limits. Typically, the Wigner measures may not capture effects which are not negligible at the pointwise level, or the propagation of Wigner measures may be an ill-posed problem. In the latter situation, two families of functions may have the same Wigner measures at some initial time, but different Wigner measures for a larger time. In the case of systems, this difficulty can partially be avoided by considering more refined Wigner measures such as two-scale Wigner measures; however, we give examples of situations where this quadratic approach fails.
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