The Bondi-van der Burg-Metzner-Sachs (BMS) group is the asymptotic symmetry group of asymptotically flat spacetime. It is infinite dimensional and entails an infinite number of conservation laws. According to the black hole membrane paradigm, null infinity (in asymptotically flat spacetime) and black hole event horizons behave like fluid membranes. The fluid dynamics of the membrane is governed...

In this paper we discuss and discover several conservation and associated decay laws in distributed coordination control systems, in particular in formation shape control systems. Specifically, we reveal conservations of linear momentum and angular momentum for gradient-based multi-agent formation systems modelled by single integrators, and show several corresponding conservation/decay laws for...

Symmetry considerations provide an extremely powerful and useful tool in our efforts to discover and understand the order in nature. Gradually they have become the backbone of our theoretical formulation of the physical laws. One of the most important consequences of the existence of symmetry (or invariance) is that a corresponding conservation law, with a conserved charge, follows. We shall el...

Conservation of mass, momentum, energy, helicity, and enstrophy in fluid flow are important because these quantities organize a flow, and characterize change in the flow’s structure over time. Thus, if a simulation of a turbulent flow is to be qualitatively correct, these quantities should be conserved in the simulation. However, such simulations are typically based on turbulence models whose c...

Newtonian physics is characterized by the existence of various conservation laws, that is, principles that tell us that however complicated the interactions among the components making up the system, certain quantities never change. Perhaps the most familiar such law is the conservation of the total mass of the system, or “amount of matter” in it: as described in Feynman, §3, we believe that in...

We consider an hyperbolic conservation law with discontinuous flux. Such partial differential equation arises in different applicative problems, 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...

Irreversible aggregation processes underly many natural phenomena including, e. g., polymerization [1], gelation [2], island growth [3-4] and aerosols [5]. The classical rate theory of Smoluchowski describes the kinetics of such processes [6-9]. Recently, scaling [10-16] and exact [17-23] theoretical studies showed that spatial correlations play a crucial role in low dimensions. While the above...

Its purpose is to provide an account of some recent advances in the mathematical theory of hyperbolic systems of conservation laws in one space dimension. After a brief review of basic concepts, we describe in detail the method of wave-front tracking approximation and present some of the latest results on uniqueness and stability of entropy weak solutions. 1-Review of basic theory. This chapter...

We discuss some recent developments and trends of applying measure-theoretic analysis to the study of nonlinear conservation laws. We focus particularly on entropy solutions without bounded variation and Cauchy fluxes on oriented surfaces which are used to formulate the balance law. Our analysis employs the Gauss-Green formula and normal traces for divergence-measure fields, Young measures and ...

We present a streamline diffusion shock capturing spacetime discontinuous Galerkin (DG) method to approximate nonlinear systems of conservation laws in several space dimensions. The degrees of freedom are in terms of the entropy variables and the numerical flux functions are the entropy stable finite volume fluxes. We show entropy stability of the (formally) arbitrarily high order accurate meth...