Continuity Equation

Markov Processes from K. Ito's Perspective by Daniel W. Stroock,

Kiyosi Ito's greatest contribution to probability theory may be his introduction of stochastic differential equations to explain the Kolmogorov-Feller theory of Markov processes. Starting with the geometric ideas that guided him, this book gives an account of Ito's program. The modern theory of Markov processes was initiated by A. N. Kolmogorov. However, Kolmogorov's approach was too analytic to reveal the probabilistic foundations on which it rests. In particular, it hides the central role played by the simplest Markov processes: those with independent, identically distributed increments. To remedy this defect, Ito interpreted Kolmogorov's famous forward equation as an equation that describes the integral curve of a vector field on the space of probability measures. Thus, in order to show how Ito's thinking leads to his theory of stochastic integral equations, Stroock begins with an account of integral curves on the space of probability measures continuity equation and then arrives at stochastic integral equations when he moves to a pathspace setting. In the first half of the book, everything is done in the context of general independent increment processes continuity equation and without explicit use of Ito's stochastic integral calculus. In the second half, the author provides a systematic development of Ito's theory of stochastic integration: first for Brownian motion continuity equation and then for continuous martingales. The final chapter presents Stratonovich's variation on Ito's theme continuity equation and ends with an application to the characterization of the paths on which a diffusion is supported. The book should be accessible to readers who have mastered the essentials of modern probability theory continuity equation and should provide such readers with areasonably thorough introduction to continuous-time, stochastic processes.
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Incompressible Flow & the Finite Element Method: Isothermal Laminar Flow by P. M. Gresho,

This comprehensive two volume reference work is devoted to the important details regarding the application of the finite element method to incompressible flows, addressing the theoretical background continuity equation and the detailed development of appropriate numerical methods applied to their solution. Volume One provides extensive coverage of the prototypical fluid mechanics equation: the advection-diffusion equation. In addition, for both this equation continuity equation and the equations of principal interest the Navier-Stokes equations (covered in detail in Volume Two), a discussion of both the continuous continuity equation and discrete equations is presented. Also addressed are explanations of how to properly march the time-dependent equations using smart implicit methods. Boundary continuity equation and initial conditions, so important in applications, are thoroughly described continuity equation and discussed, including well-posedness. The important role played by the pressure, so confusing in the past, is carefully explained. Together, this two volume work explains continuity equation and emphasizes consistency in six areas: consistent mass matrix consistent pressure Poisson equation consistent penalty methods consistent normal direction consistent heat flux consistent forces Fully indexed continuity equation and referenced, these two volumes form an essential reference tool for all research students continuity equation and applied scientists in incompressible fluid mechanics.
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Continuity equation - All the examples of continuity equations below express the same idea; they are all really examples of the same concept. Continuity equations are the (stronger) local form of conservation laws.

Infraparticle - ... in an arbitrarily small neighborhood of the point in question, there corresponds a conserved charge called the Noether charge which is the space integral of a Nother density (assuming that the integral converges) and there is a Noether current satisfying the continuity equation.

Hilleā€“Yosida theorem - In functional analysis, the Hilleā€“Yosida theorem characterizes one-parameter semigroups of linear operators on Banach spaces satisfying certain continuity restrictions. The theorem is useful for solving certain differential equations such as the heat equation.

Hill equation - The Hill equation is an equation used in enzyme characterization, which should not be confused with the Hill differential equation that is also sometimes refered to as simply the Hill equation. It describes the fraction of the enzyme saturated by ligand as a function of the ligand concentration; it is used in determining the degree of cooperativity of the enzyme.

continuityequation

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