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Olga S. Rozanova

Publications and source records attributed to Olga S. Rozanova.

34 records · Page 2Linked to original sources
arXiv2021

Study of small perturbations of a stationary state in a model of upper hybrid plasma oscillations

It is shown that a constant external magnetic field, generally speaking, is not able to prevent breaking (loss of smoothness) of relativistic plasma oscillations, even if they are arbitrarily small perturbations of the zero steady-state. This result sharply differs from the non-relativistic case, for which it is possible to suppress the breaking of oscillations at any initial deviations by increasing the intensity of the magnetic field \cite {RCharx}. Nevertheless, even in the relativistic case, there are subclasses of solutions corresponding to solutions that are globally smooth in time.

math-ph↗
arXiv2021

The Interplay of Regularizing Factors in the Model of Upper Hybrid Oscillations of Cold Plasma

A one-dimensional nonlinear model of the so-called upper hybrid oscillations in a magnetoactive plasma is investigated taking into account electron-ion collisions. It is known that both the presence of an external magnetic field of strength $ B_0 $ and a sufficiently large collisional factor $ ν$ help suppress the formation of a finite-dimensional singularity in a solution (breaking of oscillations). Nevertheless, the suppression mechanism is different: an external magnetic field increases the oscillation frequency, and collisions tend to stabilize the medium and suppress oscillations. In terms of the initial data and the coefficients $ B_0 $ and $ ν$, we establish a criterion for maintaining the global smoothness of the solution. Namely, for fixed $ B_0 $ and $ ν\ge 0 $ one can precisely divide the initial data into two classes: one leads to stabilization to the equilibrium and the other leads to the destruction of the solution in a finite time. Next, we examine the nature of the stabilization. We show that for small $ B_0 $ an increase in the intensity factor first leads to a change in the oscillatory behavior of the solution to monotonic damping, which is then again replaced by oscillatory damping. At large values of $ B_0 $, the solution is characterized by oscillatory damping regardless of the value of the intensity factor $ ν$.

math-ph↗
arXiv2021

Suppression of singularities of solutions of the Euler-Poisson system with density-dependent damping

We find a sharp condition on the density-dependent coefficient of damping of a one-dimensional repulsive Euler-Poisson system, which makes it possible to suppress the formation of singularities in the solution of the Cauchy problem with arbitrary smooth data. In the context of plasma physics, this means the possibility of suppressing the breakdown of arbitrary oscillations of cold plasma.

math.AP↗
arXiv2021

Stabilization and blow-up in the relativistic model of cold collisional plasma

We study the influence of the factor of electron-ion collisions on the solution of the Cauchy problem in the one-dimensional relativistic model of cold plasma and show that, depending on their intensity and initial data, two scenarios are possible: either the solution remains smooth and stabilizes to a stationary state, or during a finite time the oscillations blowup. In contrast to the nonrelativistic model, when exact conditions can be obtained separating the two behaviors, in a much more complicated relativistic situation, it turns out to be possible to analytically estimate from below the time during which the existence of a smooth solution and the guaranteed number of oscillations during this time. In addition, we show that in contrast to the relativistic case without taking into account collisions, when oscillations corresponding to arbitrarily small deviations from the zero equilibrium position blow up, the presence of electron collisions can suppress the blow-up of sufficiently small oscillations. Further, based on the analysis of characteristics, a numerical algorithm is constructed, the order of accuracy of which is determined only by the smoothness of the initial data. Numerical experiments are presented to illustrate the theoretical results. The initial conditions are chosen as reasonably as possible from the point of view of full-scale physical experiments.

math-ph↗
arXiv2020

On the conditions for the breaking of oscillations in a cold plasma

The Cauchy problem for a quasilinear system of hyperbolic equations describing plane one-dimensional relativistic oscillations of electrons in a cold plasma is considered. For some simplified formulation of the problem, a criterion for the existence of a global in time solutions is obtained. For the original problem, a sufficient condition for the loss of smoothness is found, as well as a sufficient condition for the solution to remain smooth at least for time $ 2 π$. In addition, it is shown that in the general case, arbitrarily small perturbations of the trivial state lead to the formation of singularities in a finite time. It is further proved that there are special initial data such that the respective solution remains smooth for all time, even in the relativistic case. Periodic in space traveling wave gives an example of such a solution. In order for such a wave to be smooth, the velocity of the wave must be greater than a certain constant that depends on the initial data. Nevertheless, arbitrary small perturbation of general form destroys these global in time smooth solutions. The nature of the singularities of the solutions is illustrated by numerical examples.

math-ph↗
arXiv2020

Singularity formation for rotational gas dynamics

The Cauchy problem for the system of equations of two-dimensional rotational gas dynamics is considered. It is assumed that the Cauchy data are a smooth compact perturbation of a constant state. Integral conditions for the data sufficient for the loss of smoothness by a solution within a finite time are found. We analyze the possibility of fulfilling these conditions and compare them with the criterion of singularity formation, known for rotational gas dynamics without pressure.

math-ph↗
arXiv2019

Some analytically solvable problems of the mean-field games theory

We study the mean field games equations, consisting of the coupled Kolmogorov-Fokker-Planck and Hamilton-Jacobi-Bellman equations. The equations are complemented by initial and terminal conditions. It is shown that with some specific choice of data, this problem can be reduced to solving a quadratically nonlinear system of ODEs. This situation occurs naturally in economic applications. As an example, the problem of forming an investor's opinion on an asset is considered.

math.AP↗
arXiv2015

Nonlinear stability of two-dimensional axisymmetric vortices in compressible inviscid medium in a rotating reference frame

We study the stability of the vortex in a 2D model of continuous compressible media in a uniformly rotating reference frame. As it is known, the axisymmetric vortex in a fixed reference frame is stable with respect to asymmetric perturbations for the solution of the 2D incompressible Euler equations and basically instable for compressible Euler equations. We show that the situation is quite different for a compressible axisymmetric vortex in a rotating reference frame. First, we consider special solutions with linear profile of velocity (or with spatially-uniform velocity gradients), which are important because many real vortices have similar structure near their centers. We analyze both cyclonic and anticyclonic cases and show that the stability of the solution depends only on the ratio of the vorticity to the Coriolis parameter. Using a very delicate analysis along with computer aided proof, we show that the stability of solutions can take place only for a narrow range of this ratio. Our results imply that the rotation of the coordinate frame can stabilize the compressible vortex. Further, we perform both analytical and numerical analysis of stability for real-shaped vortices.

math-ph↗
arXiv2013

Stochastic perturbations method for a system of Riemann invariants

Basing on our results [1] on a representation of solutions to the Cauchy problem for multidimensional non-viscous Burgers equation obtained by a method of stochastic perturbation of the associated Langevin system, we deduce an explicit asymptotic formula for smooth solutions to the Cauchy problem for any genuinely nonlinear hyperbolic system of equations written in the Riemann invariants.

math.AP↗
arXiv2010

On trajectories of vortices in the compressible fluid on a two-dimensional manifold

For the model of a compressible barotropic fluid on a two dimensional rotating Riemmanian manifold we discuss a special class of smooth solutions having a form of a steady non-singular vortex moving with a bearing field. The model can be obtained from the system of primitive equations governing the motion of air over the Earth surface after averaging over the height and therefore the solution obtained can be interpreted as a tropical cyclone which is known as a long time existing stable vortex. We consider approximations of $l$- plane and $β$ - plane used in geophysics for modeling of middle scale processes and equations on the whole sphere as well. We show that the solutions of the mentioned form satisfy the equations of the model either exactly or with a discrepancy which is small in a neighborhood of the trajectory of the center of vortex. We perform a numeric study of the change of the shape of the vortex affected by the neglecting the discrepancy term.

math-ph↗
arXiv2010

Typhoon eye trajectory based on a mathematical model: comparing with observational data

We propose a model based on the primitive system of the Navier-Stokes equations in a bidimensional framework as the $l$ - plane approximation, which allows us to explain the variety of tracks of tropical cyclones (typhoons). Our idea is to construct special analytical solutions with a linear velocity profile for the Navier-Stokes systems. The evidence of the structure of linear velocity near the center of vortex can be proven by the observational data. We study solutions with the linear-velocity property for both barotropic and baroclinic cases and show that they follow the same equations in describing the trajectories of the typhoon eye at the equilibrium state (that relates to the conservative phase of the typhoon dynamics). Moreover, at the equilibrium state, the trajectories can be viewed as a superposition of two circular motions: one has period $2π/l,$ the other one has period $2π/b_0,$ where $l$ is the Coriolis parameter and $b_0$ is the height-averaged vorticity at the center of cyclone. Also, we compare our theoretical trajectories based on initial conditions from the flow with tracks obtained from the observational database. It is worth to mention that under certain conditions our results are still compatible with observational data although we did not truly consider the influence of steering effect. %Note that the %motion of the typhoon eye can not be totally determinated by initial %conditions due to the effect of ambient pressure field ("steering" %effect). Finally, we propose the parameter-adopting method so that one could correct the weather prediction in real time. Examples of our analysis and the use of parameter-adopting method for the historic trajectories are provided.

math-ph↗
arXiv2004

Note on the typhoon eye trajectory

We consider a model of typhoon based on the three-dimensional baroclinic compressible equations of atmosphere dynamics averaged over hight and describe a qualitative behavior of the vortex and possible trajectories of the typhoon eye.

physics.ao-ph↗