Mathematical Models and Methods in Applied Sciences

We present adaptive procedures in space and time for the numerical study of positive solutions to the following problem:

In this paper we consider a system of integrodifferential equations describing, in a long wave approximation, plane-parallel rotational motions of an ideal incompressible liquid with a free surface. Using special identities, we calculate integral Riemann invariants which are conserved along the trajectories and characteristics. A new class of solutions for this system is found. The description of these special solutions simplifies essentially since in this case the integrodifferential system reduces to a hyperbolic system of two differential equations admitting formulation in the Riemann invariants. The solutions describing propagation of simple waves are obtained in an analytical form. Numerical simulations of rotational flows are performed using both the system describing the special class of the solutions and shallow water equations for rotational flows. In order to describe discontinuous rotational flows, the equations of motion are written in a special conservation form and jump conditions are derived. A non-oscillatory, second-order central scheme is used in the calculations. Some illustrative solutions are presented for smooth and discontinuous rotational flows.

In this paper, we study the asymptotic behavior in time and the existence of the modified scattering operator of the globally defined smooth solutions to the Cauchy problem for the bipolar nonlinear Schrödinger–Poisson equations with small data in the space ℝ³.

As it follows from the results of C. H. Waddigton, F. E. Yates, A. S. Iberall, and other well-known bio-physicists, living fluids cannot be modelled within the frames of the fundamental assumptions of the statistical-mechanics formalism. One has to go beyond them. The present work does it by means of the generalized kinetics (GK), the theory enabling one to allow for the complex stochasticity of internal properties and parameters of the fluid particles. This is one of the key features which distinguish living fluids from the nonliving ones. It creates the disparity of the particles and hence breaks the each-fluid-component-uniformity requirement underlying statistical mechanics.

The work deals with the corresponding modification of common kinetic equations which is in line with the GK theory and is the complement to the latter. This complement allows a subdivision of a fluid into the fluid components in terms of nondiscrete probability distributions. The treatment leads to one more equation that describes the above internal parameters. The resulting model is the system of these two equations. It appears to be always nonlinear in case of living fluids. In case of nonliving fluids, the model can be linear. Moreover, the living-fluid model, as a whole, cannot have the thermodynamic equilibrium, only partial equilibriums (such as the motional one) are possible. In contrast to this, in case of nonliving fluids, the thermodynamic equilibrium is, of course, possible. The number of the fluid components is treated as the number of the modes of the particle-characteristic probability density. In so doing, a fairly general extension of the notion of the mode from the one-dimensional case to the multidimensional case is proposed. The work also discusses the variety of the time-scales in a living fluid, the simplest quantum-mechanical equation relevant to living fluids, and the non-equilibrium nonlinear stochastic hydrodynamics option. The latter is simpler than, but conceptually comparable to, stochastic kinetic equations. A few directions for future research are suggested. The work notes a cohesion of mathematical physics and fluid mechanics with the living-fluid-related fields as a complex interdisciplinary problem.

In this paper we study the motion of a magnetic field H in a conductive medium Ω⊂R³ under the influence of a system generator. By neglecting displacement currents, the magnetic field satisfies a nonlinear Maxwell's system: H_t+∇×[ρ(x,t)∇×H]=f(|H|)H, where f(|H|)H represents the magnetic currents depending upon the strength of H. We prove that under appropriate initial and boundary conditions, the system has a global solution and the solution is also unique. Moreover, we show that the solution H will blow up in finite time if f(s) satisfies certain growth conditions. Finally, we generalize the results to the problem associated with a nonlinear boundary condition.

This paper aims to consider user equilibrium problems in transportation networks in the most complete and realistic situations. In fact, the presented model allows for the dependence of data on time, the presence of elastic travel demands, the capacity restrictions and delay effects. The equilibrium conditions for such a model are given and the equivalent formulation in terms of a quasi-variational inequality is discussed. Moreover, a theorem for the existence of solutions is shown and a numerical example is provided. Finally, some questions of stability are studied.

In the Langevin or Ornstein–Uhlenbeck approach to diffusion, stochastic increments are applied to the velocity rather than to the space variable. The density of this process satisfies a linear partial differential equation of the general form of a transport equation which is hyperbolic with respect to the space variable but parabolic with respect to the velocity variable, the Klein–Kramers or simply Kramers equation. This modeling approach allows for a more detailed description of individual movement and orientation dependent interaction than the frequently used reaction diffusion framework.

For the Kramers equation, moments are computed, the infinite system of moment equations is closed at several levels, and telegraph and diffusion equations are derived as approximations. Then nonlinearities are introduced such that the semi-linear reaction Kramers equation describes particles which move and interact on the same time-scale. Also for these nonlinear problems a moment approach is feasible and yields nonlinear damped wave equations as limiting cases.

We apply the moment method to the Kramers equation for chemotactic movement and obtain the classical Patlak–Keller–Segel model. We discuss similarities between chemotactic movement of bacteria and gravitational movement of physical particles.