arXiv subjects

Olga S. Rozanova

Publications and source records attributed to Olga S. Rozanova.

At least 19 recordsLinked to original sources

arXiv2026

A model of a multiphase medium based on the closure of moment chains for the Vlasov-Poisson equations

We propose a method for closing moment chains for the Vlasov-Poisson kinetic system (the Landau fluid model), which yields a hyperbolic system of equations at each subsequent step. This system is interpreted as a multicomponent medium, where each component has its own density, velocity, and pressure. The coupling between the components is mediated by an electric field. The principle of closing moment chains is that the final component of the fluid is assumed to be pressureless.

math.AP↗

arXiv2026

Euler-Poisson equations with velocity-dependent damping

We consider repulsive Euler-Poisson equations in both one-dimensional space and in the multidimensional case with radial symmetry, assuming a power-law velocity-dependent damping coefficient. We show that under this assumption, in the one-dimensional case the set of initial data corresponding to a globally time-smooth solution expands, whereas in other dimensions (except for dimension 4) the damping does not influence on improving of smooth properties of the solution. Namely, any arbitrary small perturbation of the steady state blow up, despite of its amplitude of oscillations decays.

math.AP↗

arXiv2026

The Kolmogorov forward equation for a distributed model of regime-switching diffusions

For the regime-switching diffusion process with and without advection term we propose an integro-differential equation describing the densities of states continuously distributed over a segment. We demonstrate that there exists a constructive algorithm for solving the Cauchy problem. We then show that for some initial distributions of states, the solution can be found explicitly. We also discuss how a model with a discrete number of hidden states can be approximated by a model with continuously distributed states.

math.AP↗

arXiv2026

Numerical study of loss of hyperbolicity using a cold plasma model

We study a one-dimensional system of cold plasma equations taking into account electron-ion collisions in both relativistic and nonrelativistic cases. It is known that for a constant collision coefficient $ν$, the solution to the Cauchy problem for such a system can lose smoothness. However, if the dependence of $ν$ on the electron density $N$ is more than linear, then the solution remains globally smooth for any initial data. However, the appearance of the dependence $ν(N)$ leads to a change in the type of the system, it loses hyperbolicity, which leads to computational problems. In this paper, we propose a new implicit solution method in Euler variables that overcomes these difficulties. It can be used in both nonrelativistic and relativistic cases and is tested for the threshold case of a linear dependence $ν(N)=ν_1+ν_0 N$, when smoothness can still be lost. The computational experiments carried out are in full agreement with the available theoretical results.

physics.comp-ph↗

arXiv2025

A sufficient condition for the existence of smooth solutions of the relativistic cold plasma equations on any given time interval

In terms of initial data, a sufficient condition for the smoothness of the solution to the Cauchy problem for one-dimensional relativistic cold plasma equations over any given time interval is found. Unlike the non-relativistic case, such sufficient conditions take into account the smallness properties of not only the derivatives of the initial data but also the initial data themselves. The accuracy of the obtained initial condition is investigated using a numerical experiment. The structure of the emerging singularities is also studied.

math-ph↗

arXiv2025

On non-uniqueness in the option valuation problem

It is known that the value of a call option in the case of constant elasticity processes (CEV) with the indicator $α$ exceeding the critical $α=1$ is determined in a non-unique way. We show how, based on an already existing mathematical theory concerning the correctness of boundary conditions for degenerate parabolic equations on the semi-axis $[0,\infty)$, this phenomenon can be explained. Namely, for $1<α\le \frac32$ the non-uniqueness is due to the fact that the initial data of the call option are outside the Täcklind class, and for $α> \frac32$ it is due to the absence boundary condition for $x=\infty$.

math.AP↗

arXiv2025

The simplest solutions of cold plasma equations: change in properties from a hydrodynamic to a kinetic model

We consider the transition from the kinetic model of Landau cold plasma to the hydrodynamic one by constructing a "multi-speed" moment chain in the case of one spatial variable. Closing this chain at the first step leads to the standard hydrodynamic system of cold plasma. The change in the properties of the solution when closing the chain at the second step is discussed using the example of two classes of solutions - affine in space and traveling waves, and it is shown that their properties change significantly compared to the hydrodynamic model.

math.AP↗

arXiv2025

Linearization method and sharp thresholds for spherically symmetric multidimensional pressureless Euler-Poisson equations

We show that the question about the criterion of a singularity formation for radially symmetric solutions to the Cauchy problem for a fairly wide class of equations related to the pressureless Euler-Poisson equations can be reduced to the study of solutions to a linear homogeneous ordinary differential equation. In some cases, such a criterion can be obtained in terms of the initial data. In the remaining cases, it is possible to construct a simple numerical procedure, on the basis of which the question about preserving smoothness for any set of initial data can be solved.

math.AP↗

arXiv2024

Criterion of singularity formation for radial solutions of the pressureless Euler-Poisson equations in exceptional dimension

The spatial dimensions 1 and 4 play an exceptional role for radial solutions of the pressureless repulsive Euler-Poisson equations. Namely, for any spatial dimension except 1 and 4, any nontrivial solution of the Cauchy problem blows up in a finite time (except in special cases), whereas for dimensions 1 and 4 there exists a neighborhood of trivial initial data in the $C^1$ - norm such that the respective solution preserves the initial smoothness globally. For dimension 1, the criterion of the singularity formation in terms of initial data was known, i.e. this neighborhood can be found exactly. For the case of dimension 4, there was no similar result. In this paper, we close this gap and obtain such a criterion for the case of a more technically complicated case of dimension 4.

math.AP↗

arXiv2024

Regularizing factors for the Euler-Poisson equations

he Cauchy problem for the Euler-Poisson equations without pressure is considered and the question of what additional terms added to the system can delay or completely prevent the loss of smoothness of the solution in a finite time is studied. We review already published and recent results in this field.

math.AP↗

arXiv2024

The repulsive Euler-Poisson equations with variable doping profile

We prove that arbitrary smooth perturbations of the zero equilibrium state of the repulsive pressureless Euler-Poisson equations, which describe the behavior of cold plasma, blow up for any non-constant doping profile already in one-dimensional space. Further, we study small perturbations of the equilibrium to determine which properties of the doping profile contribute to the blow-up. We also propose a numerical procedure that allows one to find the blow-up time for any initial data and present examples of such calculations for various doping profiles for standard initial data, corresponding to the laser pulse.

math.AP↗

arXiv2024

An explicit form of the fundamental solution of the master equation for a jump-diffusion Ornstein-Uhlenbeck process

An integro-differential equation for the probability density of the generalized stochastic Ornstein-Uhlenbeck process with jump diffusion is considered. It is shown that for a certain ratio between the intensity of jumps and the speed of reversion, the fundamental solution can be found explicitly. The properties of this solution are investigated. The fundamental solution allows one to obtain explicit formulas for the densities at each moment of time.

math-ph↗

arXiv2024

On Multidimensional Axisymmetric Oscillations of a Collisional Cold Plasma

We study the influence of the friction term on the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations in many spatial dimensions. It is shown that for any arbitrarily small non-negative constant friction coefficient, there exists a neighborhood of the zero equilibrium in the $C^1$ norm such that the solution of the Cauchy problem with initial data belonging to this neighborhood remains globally smooth in time. Moreover, this solution stabilizes to zero as $t\to\infty$. This result contrasts with the situation of zero friction, where any small deviation from the zero equilibrium generally leads to a blow-up. Our method allows us to estimate the lifetime of smooth solutions. Further, we prove that for any initial data, one can find such coefficient of friction that the respective solution to the Cauchy problem keeps smoothness for all $t>0$ and stabilizes to zero. We also present the results of numerical experiments for physically reasonable situations, which allows us to estimate the value of the friction coefficient, which makes it possible to suppress the formation of singularities of solutions.

math.AP↗

arXiv2023

On plane oscillations of the cold plasma in a constant magnetic field

We consider a class of two-dimensional solutions of the cold plasma equations compatible with a constant magnetic field and a constant electric field. For this class, under various assumptions about the electric field, we study the conditions on the initial data that guarantee the global existence of the classical solution of the Cauchy problem for a given period of time or a finite blowup. Particular attention is paid to the class of solutions with axial symmetry.

math-ph↗

arXiv2023

On the solution of the Kolmogorov-Feller equation arising in the model of biological evolution

The Kolmogorov-Feller equation for the probability density of a Markov process on a half-axis, which arises in important problems of biology, is considered. This process consists of random jumps distributed according to Laplace's law and a deterministic return to zero. It is shown that Green's function for such an equation can be found both in the form of a series and in explicit form for some ratios of the parameters. This allows one to explicitly find solutions to the Kolmogorov-Feller equation for many initial data.

math-ph↗

arXiv2023

The Riemann problem for equations of a cold plasma

A solution of the Riemann problem is constructed for a nonstrictly hyperbolic inhomogeneous system of equations describing one-dimensional cold plasma oscillations. Each oscillation period includes one rarefaction wave and one shock wave containing a delta singularity. The rarefaction wave can be constructed in a non-unique way, the admissibility principle is proposed.

math.AP↗

arXiv2023

Properties of moments of density for nonlocal mean field game equations with a quadratic cost function

We consider mean field game equations with an underlying jump-diffusion process $X_t$ for the case of a quadratic cost function and show that the expectation and variance of $X_t$ obey second-order ordinary differential equations with coefficients depending on the parameters of the cost function. Moreover, for the case of pure diffusion, the characteristic function and the fundamental solution of the equation for the probability density can only be expressed in terms of the expectation ${\mathbb E}$ and the variance ${\mathbb V}$ of the process $X_t$, so that the moments of any order depend only on ${\mathbb E}$ and ${\mathbb V}$.

math.AP↗

arXiv2022

On the behavior of multidimensional axisymmetric solutions of the repulsive Euler-Poisson equations

It is proved that the radially symmetric solutions of the repulsive Euler-Poisson equations with a non-zero background, corresponding to cold plasma oscillations blow up in many spatial dimensions except for $\bd=4$ for almost all initial data. The initial data, for which the solution may not blow up, correspond to simple waves. Moreover, if a solution is globally smooth in time, then it is either affine or tends to affine as $t\to\infty$.

math-ph↗