Narrow Results

We consider a generalization of the classical results obtained by Freidlin and Wentzell to consider the case of time-dependent dissipative drifts. We show the convergence of diffusions with multiplicative noise in the zero limit of a diffusivity parameter to the related dynamical systems. We also derive the solution to the associated transport equation as an application.

We consider dynamical systems that are described by fractional power of coordinates and momenta. The fractional powers can be considered as a convenient way to describe systems in the fractional dimension space. For the usual space the fractional systems are non-Hamiltonian. Generalized transport equation is derived from Liouville and Bogoliubov equations for fractional systems. Fractional generalization of average values and reduced distribution functions are defined. Gasdynamic equations for fractional systems are derived from the generalized transport equation.

A new, wide class of relativistic stochastic processes is introduced. All relativistic processes considered so far in the literature (the Relativistic Ornstein–Uhlenbeck Process as well as the Franchi–Le Jan and the Dunkel–Hänggi processes) are members of this class. The stochastic equations of motion and the associated forward Kolmogorov equations are obtained for each process in the class. The corresponding manifestly covariant transport equation is also obtained. In particular, the manifestly covariant equations for the Franchi–Le Jan and the Dunkel–Hänggi processes are derived here for the first time. Finally, the manifestly covariant approach is used to prove a new H-theorem for all processes in the class.

The aim of this article is to prove some results regarding the existence of solutions on L₁ spaces to a nonlinear boundary value problem derived from a model introduced by Rotenberg (1983) describing the growth of a cell population. Each cell of this population is distinguished by two parameters: its degree of maturity μ and its maturation velocity v. The biological boundary of μ = 0 and μ = a (a > 0) are fixed and tightly coupled through the mitosis. At mitosis, daughter cells and mother cells are related by a general reproduction rule which covers all known biological ones. Our proofs, based on topological methods, use essentially the specific properties of weakly compact sets on L₁ spaces.

This paper deals with boundary value problems and spectral problems for neutron transport equations involving locally periodic (in space) collision frequencies and collision operators. We show strong convergence results to the solution of homogenized problems when the period goes to zero. The mathematical analysis relies mainly on smoothing effects of velocity averages.

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 study a new variant of mathematical prediction-correction model for crowd motion. The prediction phase is handled by a transport equation where the vector field is computed via an eikonal equation $\parallel \nabla \phi \parallel =f$, with a positive continuous function $f$ connected to the speed of the spontaneous travel. The correction phase is handled by a new version of the minimum flow problem. This model is flexible and can take into account different types of interactions between the agents, from gradient flow in Wassersetin space to granular type dynamics like in sandpile. Furthermore, different boundary conditions can be used, such as non-homogeneous Dirichlet (e.g. outings with different exit-cost penalty) and Neumann boundary conditions (e.g. entrances with different rates). Combining finite volume method for the transport equation and Chambolle–Pock’s primal dual algorithm for the eikonal equation and minimum flow problem, we present numerical simulations to demonstrate the behavior in different scenarios.

In this paper, we solve a Lévy driven linear stochastic first-order partial differential equation (transport equation) understood in the canonical (Marcus) form. The solution can be obtained with the help of the method of stochastic characteristics. It has the same form as a solution of a deterministic PDE or a solution of a stochastic PDE driven by a Brownian motion studied by Kunita [First order stochastic partial differential equations, in Stochastic Analysis. Proc. Taniguchi Int. Symp. Stochastic Analysis, Katata and Kyoto, 1982, North-Holland Mathematical Library, Vol. 32, ed. K. Itô (North-Holland Publishing Company, 1984), pp. 249–269; Stochastic Flows and Stochastic Differential Equations, Vol. 24 (Cambridge University Press, 1997)].

This paper presents an exact pointwise error estimator applied to the transport equation. The error estimate, which is represented by a combination of residual-free bubble and Green’s functions, is computed by postprocessing the finite element solution. The fine or unresolved scales are analyzed applying the variational multiscale method. The technology can be applied to linear and high-order elements. Numerical examples are presented in which the satisfactory results corroborate the theoretical foundations.

We consider coupled systems of a second order hyperbolic equation and a first order kinetic equation

For semi-linear dissipative wave equation □u + |u_t|^{p - 1}u_t = 0, we consider finite energy solutions with singularities propagating along a focusing light cone. At the tip of cone, the singularities are focused and partially smoothed out under strong nonlinear dissipation, i.e. the solution gets up to 1/2 more L² derivative after the focus. The smoothing phenomenon is in fact the result of simultaneous action of focusing and nonlinear dissipation.

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.

The object of this paper is to propose a model to simulate steady and unsteady cavitating flows. In the engineering practice, cavitation flow is often modeled as a single-phase flow (mixture), where the cavitation area is handled as an area with the pressure lower than the vapor pressure. This approach always leads to the result, and the requirement of computer time is many times lower in comparison with multiphase flow models. The Reynolds-averaged Navier–Stokes equations are solved for the mixture of liquid and vapor, which is considered as a single-phase with variable density. The vaporization and condensation processes are controlled by barocline low. A transport equation with source terms is implanted in the code Computational Fluid Dynamics (CFD) to compute the volume fraction of the vapor. The CFD code used is ANSYS CFX. The influence of numerical and the physical parameters are presented. The numerical results are compared to previous experimental measures. For steady flow, a SST turbulence model is adopted and LES for the unsteady flow.

This paper presents a technique for deriving least-squares-based polynomial bubble functions to enrich the standard linear finite elements, employed in the formulation of Galerkin weighted-residual statements. The element-level linear shape functions are enhanced using supplementary polynomial bubble functions with undetermined coefficients. The enhanced shape functions are inserted into the model equation and the residual functional is constructed and minimized by using the method of the least squares, resulting in an algebraic system of equations which can be solved to determine the unknown polynomial coefficients in terms of element-level nodal values. The stiffness matrices are subsequently formed with the standard finite elements assembly procedures followed by using these enriched elements which require no additional nodes to be introduced and no extra degree of freedom incurred. Furthermore, the proposed technique is tested on a number of benchmark linear transport equations where the quadratic and cubic bubble functions are derived and the numerical results are compared against the exact and standard linear element solutions. It is demonstrated that low order bubble enriched elements provide more accurate approximations for the exact analytical solutions than the standard linear elements at no extra computational cost in spite of using relatively crude meshes. On the other hand, it is observed that a satisfactory solution of the strongly convection-dominated transport problems may require element enrichment by using significantly higher order polynomial bubble functions in addition to the use of extremely fine computational meshes.

This paper is devoted to study spherically symmetric gravitational collapse with anisotropic fluid in f(R) theory which undergoes dissipation in the form of heat flux. We develop dynamical and transport equation and finally couple them. This yields different possibilities of collapse and explosions connected with supernovae events and immense of dark energy terms arising from modifying gravity.

In a recent work, we presented the first application of the Poisson-Wiseman-Anderson method of “matched expansions” to compute the self-force acting on a point particle moving in a curved spacetime. The method employs two expansions for the Green function which are respectively valid in the “quasilocal” and “distant past” regimes, and which may be matched together within the normal neighbourhood. In this article, we introduce the method of matched expansions and discuss transport equation methods for the calculation of the Green function in the quasilocal region. These methods allow the Green function to be evaluated throughout the normal neighborhood and are also relevant to a broad range of problems from radiation reaction to quantum field theory in curved spacetime and quantum gravity.