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The continuity, equation relating the change in time of the position probability density to the gradient of the probability current density is generalized to PT-symmetric quantum mechanics. The normalization condition of eigenfunctions is modified in accordance with this new conservation law and illustrated with some detailed examples.

Starting with the usual definitions of octonions and split octonions in terms of Zorn vector matrix realization, we have made an attempt to write the continuity equation and other wave equations of dyons in split octonions. Accordingly, we have investigated the work energy theorem or "Poynting Theorem," Maxwell stress tensor and Lorentz invariant for generalized fields of dyons in split octonion electrodynamics. Our theory of dyons in split octonion formulations is discussed in term of simple and compact notations. This theory reproduces the dynamic of electric (magnetic) in the absence of magnetic (electric) charges.

This paper aims to apply the octonions to explore the torques and forces and so forth in the electromagnetic and gravitational fields, investigating the influences of material media on the equilibrium and continuity equations. The contemporary scholars utilize the quaternions and octonions to research the electromagnetic fields and gravitational fields and so forth. In this paper, the octonions are capable of surveying the electromagnetic and gravitational fields within material media, including the octonion field strength, field source, angular momentum, torque and force. Further, the octonion field strength and angular momentum can be combined together to become the octonion composite field strength, deducing the octonion composite torque and force and others. When the octonion composite force is equal to zero under certain circumstances, it is able to achieve eight equations independent of each other, including the force equilibrium equation, fluid continuity equation, current continuity equation and so on. The above reveals that the material media and octonion field strength can make a contribution to the eight equilibrium and continuity equations. And it is beneficial for deepening the understanding of equilibrium and continuity equations within material media.

A simple model to handle the flow of people in emergency evacuation situations is considered: at every point x, the velocity U(x) that individuals at x would like to realize is given. Yet, the incompressibility constraint prevents this velocity field to be realized and the actual velocity is the projection of the desired one onto the set of admissible velocities. Instead of looking at a microscopic setting (where individuals are represented by rigid discs), here the macroscopic approach is investigated, where the unknown is a density ρ(t,x). If a gradient structure is given, say U = -∇D where D is, for instance, the distance to the exit door, the problem is presented as a Gradient Flow in the Wasserstein space of probability measures. The functional which gives the Gradient Flow is neither finitely valued (since it takes into account the constraints on the density), nor geodesically convex, which requires for an ad hoc study of the convergence of a discrete scheme.

We study the problem of optimal mixing of a passive scalar $\rho$ advected by an incompressible flow on the two-dimensional unit square. The scalar $\rho$ solves the continuity equation with a divergence-free velocity field $u$ with uniform-in-time bounds on the homogeneous Sobolev semi-norm ${Ẇ}^{s,p}$, where $s>1$ and $1<p\le \infty$. We measure the degree of mixedness of the tracer $\rho$ via the two different notions of mixing scale commonly used in this setting, namely the functional and the geometric mixing scales. For velocity fields with the above constraint, it is known that the decay of both mixing scales cannot be faster than exponential. Numerical simulations suggest that this exponential lower bound is in fact sharp, but so far there is no explicit analytical example which matches this result. We analyze velocity fields of $cellulartype$, which is a special localized structure often used in constructions of explicit analytical examples of mixing flows and can be viewed as a generalization of the self-similar construction by Alberti, Crippa and Mazzucato [Exponential self-similar mixing by incompressible flows, arXiv:1605.02090]. We show that for any velocity field of cellular type both mixing scales cannot decay faster than polynomially.

In this paper, we consider a compressible dissipative Baer–Nunziato-type system for a mixture of two compressible heat conducting gases. We prove that the set of weak solutions is stable, meaning that any sequence of weak solutions contains a (weakly) convergent subsequence whose limit is again a weak solution to the original system. Such type of results is usually considered as the most essential step to the proof of the existence of weak solutions. This is the first result of this type in the mathematical literature. Nevertheless, the construction of weak solutions to this system however remains still an (difficult) open problem.

We prove quantitative estimates on flows of ordinary differential equations with vector field with gradient given by a singular integral of an L¹ function. Such estimates allow to prove existence, uniqueness, quantitative stability and compactness for the flow, going beyond the BV theory. We illustrate the related well-posedness theory of Lagrangian solutions to the continuity and transport equations.

When modelling a flow in the atmosphere and the processes strongly influenced by it (e.g., the dispersion of air pollution), it is important to appreciate that the properties of both the flow itself and the dispersion are affected by the flow regime; i.e., whether the flow is turbulent (as is almost always the case in the atmosphere) or laminar. A second factor that might complicate atmospheric flow is stability, which depends on the nature of vertical temperature stratification.

In the first part of this chapter, we demonstrate the impact of vertical temperature stratification on flow structure, modelled via the Boussinesq approximation and by varying the Froude number (Fr). The flow is assumed to be laminar and is modelled in 2D.

Next, we review several approaches to treating turbulence in modelling studies, with an emphasis on an implicit large-eddy simulation. The results of Taylor—Green vortex computations performed using this method are compared with the results of a direct numerical simulation at moderate Reynolds numbers. Several quantities are considered, including the kinetic energy dissipation rate, probability density functions of turbulent fluctuations, and 3D energy spectra.