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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., Comment: 10 pages, 2 figures

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., Comment: 20 pages, 2 figures

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., Comment: 12 pages, 3 figures

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., Comment: 5 pages, 2 figures

We show that many important natural science models in their mathematical formulation can be reduced to non-strictly hyperbolic systems of the same kind. This allows the same methods to be applied to them so that some essential results concerning a particular model can be obtained as corollaries of general theorems. However, in each case, the models have their own characteristics. The article contains an overview of both potentially applicable methods, as well as known results obtained for a particular model. In addition, we introduce new models and obtain results for them., Comment: 13 pages, 3 figures

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., Comment: 12 pages, 4 figures

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 ${\mathbb V}$., Comment: 7 pages

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., Comment: 15 pages, 8 figures

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., Comment: 16 pages, 2 figures

We study the affine solutions of the equations of plane oscillations of cold plasma, which, under the assumption of electrostaticity, correspond to the Euler-Poisson equations in the repulsive case. It is proved that the zero equilibrium state of the cold plasma equations, both with and without the assumption of electrostaticity, is unstable in the class of all affine solutions. It is also shown that an arbitrary perturbation of an axially symmetric electrostatic solution leads to a finite time blow-up., Comment: 11 pages, 7 figures

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 $t\to\infty$., Comment: 14 pages, 3 figures

We consider the classical Cauchy problem for a system of equations describing 3D arbitrary electrostatic oscillations of the cold plasma and introduce an iteration procedure that allows estimating the blow-up time from below. This procedure is constructive provided one succeed to obtain a two-sided estimate of an additional quantity depending on the solution. We show that this is possible in the case of one and two dimensions, as well as for solutions with zero vorticity. For the particular case of two-dimensional initial data with the radial symmetry, refined sufficient conditions for destruction and preservation of smoothness in the first period of oscillations are obtained. Moreover, we give the example of estimating the blow-up time for the data for which results of numerics exist and discuss a roughness of our estimate., Comment: 16 pages, 6 figures

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., Comment: 13 pages

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 $ \nu $ 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 $ \nu $, we establish a criterion for maintaining the global smoothness of the solution. Namely, for fixed $ B_0 $ and $ \nu \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 $ \nu $., Comment: 10 pages, 3 figures

For a system of equations describing one-dimensional nonlinear oscillations in a magnetoactive plasma, we study the effect of a constant magnetic field on the breaking of oscillations. For the nonrelativistic case, a criterion for the formation of a finite-dimensional singularity is obtained in terms of the initial data. It is shown that the enhancement of the magnetic field basically leads to an expansion of the class of initial data providing the global smoothness of the solution. The nature of the singularities of the solutions is illustrated by numerical examples., Comment: 16 pages, 8 figures. arXiv admin note: text overlap with arXiv:2103.11685

We consider a couple of integrodifferential PDEs arising from a stochastic Markovian control problem subjected to initial-terminal conditions. These equations correspond to the MFG system for a controlled jump-diffusion process. We prove that for a specific choice of the control function the expectation of the jump-diffusion process can be found explicitly. The study is an extension of similar results known for the pure diffusion process. As an example, we show how this can be applied to the problem of investors evaluating the trend of an asset when choosing an optimal portfolio., Comment: 8 pages. arXiv admin note: substantial text overlap with arXiv:1911.09441

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., Comment: 22 pages, 5 figures

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., Comment: 14 pages, 4 figures

We consider the Cauchy problem with smooth data for compressible Euler equations in many dimensions and concentrate on two cases: solutions with finite mass and energy and solutions corresponding to a compact perturbation of a nontrivial stationary state. We prove the blowup results using the characteristics of the propagation of the solution in space and find upper and lower bounds for the density of a smooth solution in a given region of space in terms of the initial data. To solve the problems, we introduce a special family of integral functionals and study their temporal dynamics., Comment: 21 pages, 2 figures

We consider a quasilinear system of hyperbolic equations that describes plane one-dimensional non-relativistic oscillations of electrons in a cold plasma with allowance for electron-ion collisions. Accounting for collisions leads to the appearance of a term analogous to dry friction in a mechanical system, leading to a decrease in the total energy. We obtain a criterion for the existence of a global in time smooth solution to the Cauchy problem. It allows to accurately separate the initial data into two classes: one corresponds to a globally in time smooth solutions, and the other leads to a finite-time blowup. The influence of electron collision frequency $ \nu $ on the solution is investigated. It is shown that there is a threshold value, after exceeding which the regime of damped oscillations is replaced by the regime of monotonic damping. The set of initial data corresponding to a globally in time smooth solution of the Cauchy problem expands with increasing $ \nu $, however, at an arbitrarily large value there are smooth initial data for which the solution forms a singularity in a finite time, and this time tends to zero as $ \nu $ tends to infinity. The character of the emerging singularities is illustrated by numerical examples., Comment: 14 pages, 6 figures

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., Comment: 15 pages, 3 figures

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 \pi $. 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., Comment: 17 pages, 5 figures

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., Comment: 13 pages, 1 figure

The limiting case of the system of equations of two-dimensional gas dynamics in the presence of the Coriolis force, which can be obtained under the assumption of a small pressure, is considered. With this approach, the equation for the velocity vector (transport equation) is split off from the system and can be solved separately. Using the method of stochastic perturbation along characteristics, we obtain an explicit asymptotic representation of a smooth solution of transport equations and analyze the process of formation of singularities of solution using a specific example. It is concluded that the presence of the Coriolis force prevents the singularities formation., Comment: 7 pages, 3 figures

We show that a small correction due to centrifugal force usually neglected in the $l$-plane model of atmosphere drastically influences on the stability of vortices. Namely, in the presence of the Coriolis force only there exists a wide range of parameter ensuring nonlinear stability of a vortex with uniform deformation. Taking into account the centrifugal force results in a disappearance of stable vortices in the above-mentioned class of motions. We also prove that for the heat ratio $\gamma=2$, corresponding to the one-atomic gas, the system of equations, describing the gas on the $l$-plane with the correction due to centrifugal force can be integrated in a special case., Comment: 12 pages

For the 1D non-isentropic polytropic gas dynamics equations we find sufficient and necessary conditions for blow up of derivatives in the terms of Cauchy data. In particular, the method allows to determine exact class of initial data corresponding to globally smooth in time solution., Comment: 13 pages, 1 figure

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., Comment: 24 pages

The paper is devoted to the special classes of solutions to multidimensional balance laws of gas dynamic type. In the velocity field for the solutions of such class the time and space variables are separated. The simplest case is the solution with the linear profile of velocity in the Euclidean space., Comment: 44 pages, 6 figures

We study the influence of linear friction on the vortex motion in a non-viscous stratified compressible rotating media. Our method can be applied to describe the complex behavior of a tropical cyclone approaching land. In particular, we show that several features of the vortex in the atmosphere such as a significant track deflection, sudden decay and intensification, can be explained already by means of the simplest two dimensional barotropic model, which is a result of averaging over the height in the primitive equations of air motion in the atmosphere. Our theoretical considerations are in a good compliance with the experimental data. In contrast to other models, where first the additional physically reasonable simplifications are made, we deal with special solutions of the full system. Our method is able to explain the phenomenon of the cyclone attracting to the land and interaction of the cyclone with an island., Comment: 21 pages, 10 figures

We study a possibility of existence of localized two-dimensional structures, both smooth and non-smooth, that can move without significant change of their shape in a leading stream of compressible barotropic fluid on a rotating plane., Comment: 9 pages, 1 figure

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., Comment: 7 pages

We construct a large family of exact solutions to the hyperbolic system of 3 equations of ideal granular hydrodynamics in several dimensions for arbitrary adiabatic index $\gamma$. In dependence of initial conditions these solutions can keep smoothness for all times or develop singularity. In particular, in the 2D case the singularity can be formed either in a point or along a line. For $\gamma=-1$ the problem is reduced to the system of two equations, related to a special case of the Chaplygin gas. In the 1D case this system can be written in the Riemann invariant and can be treated in a standard way. The solution to the Riemann problem in this case demonstrate an unusual and complicated behavior., Comment: 8 pages, 4 figures

We prove that the smooth solutions to the Cauchy problem for the compressible general three-dimensional Ericksen--Leslie system modeling nematic liquid crystal flow with conserved mass, linear momentum, and dissipating total energy, generally lose classical smoothness within a finite time., Comment: 7 pages. arXiv admin note: text overlap with arXiv:1206.2850, arXiv:1105.2180 by other authors

We consider solutions to the hyperbolic system of equations of ideal granular hydrodynamics with conserved mass, total energy and finite momentum of inertia and prove that these solutions generically lose the initial smoothness within a finite time in any space dimension $n$ for the adiabatic index $\gamma \le 1+\frac{2}{n}.$ Further, in the one-dimensional case we introduce a solution depending only on the spatial coordinate outside of a ball containing the origin and prove that this solution under rather general assumptions on initial data cannot be global in time too. Then we construct an exact axially symmetric solution with separable time and space variables having a strong singularity in the density component beginning from the initial moment of time, whereas other components of solution are initially continuous., Comment: 13 pages, 3 figures

We present an explicit hedging strategy, which enables to prove arbitrageness of market incorporating at least two assets depending on the same random factor. The implied Black-Scholes volatility, computed taking into account the form of the graph of the option price, related to our strategy, demonstrates the "skewness" inherent to the observational data., Comment: 9 pages, 4 figures

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 $\beta$ - 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., Comment: 19 pages, 10 figures

We extend our result of [1] and show that one can associate with the stochastically perturbed non-viscid Burgers equation a system of viscous balance laws. The Cauchy data for the Burgers equation generates the data for this system. Till the moment of the shock formation in the solution to the Burgers equation the above system of viscous balance laws can be reduced to the pressureless gas dynamics system (in a limit as the parameters of perturbation tend to zero). If the solution to the Burgers equation contains shocks, the limit system is equivalent to the system with a specific pressure, in some sense analogous to the pressure of barotropic monoatomic gas., Comment: 8 pages, submitted for Proceedings of International Conference on Hyperbolic Problems, Beijing, 2010

We study the dependence of volatility on the stock price in the stochastic volatility framework on the example of the Heston model. To be more specific, we consider the conditional expectation of variance (square of volatility) under fixed stock price return as a function of the return and time. The behavior of this function depends on the initial stock price return distribution density. In particular, we show that the graph of the conditional expectation of variance is convex downwards near the mean value of the stock price return. For the Gaussian distribution this effect is strong, but it weakens and becomes negligible as the decay of distribution at infinity slows down., Comment: 9 pages, 4 figures, submitted

We consider a generalization of the compressible barotropic Navier-Stokes equations to the case of non-Newtonian fluid in the whole space. The viscosity tensor is assumed to be coercive with an exponent $q>1.$ We prove that if the total mass and momentum of the system are conserved, then one can find a constant $q_0>1$ depending on the dimension of space $n$ and the heat ratio $\gamma$ such that for $q\in [q_0,n)$ there exists no global in time smooth solution to the Cauchy problem. We prove also an analogous result for solutions to equations of magnetohydrodynamic non-Newtonian fluid in 3D space., Comment: 8 pages