Vol 70, No 1 (2024): Functional spaces. Differential operators. Problems of mathematics education

The known nonlinear kinetic equations, in particular, the wave kinetic equation and the quantum Nordheim–Uehling–Uhlenbeck equations are considered as a natural generalization of the classical spatially homogeneous Boltzmann equation. To this goal we introduce the general Boltzmann-type kinetic equation that depends on a function of four real variables $F(x,y;v,w)$. The function $F$ is assumed to satisfy certain simple relations. The main properties of this kinetic equation are studied. It is shown that the above mentioned specific kinetic equations correspond to different polynomial forms of the function $F$. Then the problem of discretization of the general Boltzmann-type kinetic equation is considered on the basis of ideas similar to those used for construction of discrete velocity models of the Boltzmann equation. The main attention is paid to discrete models of the wave kinetic equation. It is shown that such models have a monotone functional similarly to the Boltzmann $H$-function. The theorem of existence, uniqueness and convergence to equilibrium of solutions to the Cauchy problem with any positive initial conditions is formulated and discussed. The differences in long time behaviour between solutions of the wave kinetic equation and solutions of its discrete models are also briefly discussed.

In this paper, we construct a solution to the Riemann problem for an inhomogeneous nonstrictly hyperbolic system of two equations, which is a corollary of the Euler-Poisson equations without pressure [9]. These equations can be considered for the cases of attractive and repulsive forces as well as for the cases of zero and nonzero underlying density background. The solution to the Riemann problem for each case is nonstandard and contains a delta-shaped singularity in the density component. In [16], solutions were constructed for the combination corresponding to the cold plasma model (repulsive force and nonzero background density). In this paper, we consider the three remaining cases.

We consider the problem of the existence of limit cycles for autonomous systems of differential equations. We present quite elementary considerations that can be useful in discussing qualitative issues that arise in the course of ordinary differential equations. We establish that any simple closed curve defined by the equation $F(x,y)=1$ with a sufficiently general function $F$ is a limit cycle for the corresponding autonomous system on the plane (and even for an infinite number of systems depending on the real parameter). These systems are written out explicitly. We analyze in detail several specific examples. Graphic illustrations are provided.

In this paper, we provide a review of results on a priori estimates for systems of minimal differential operators in the scale of spaces $L^p(\mathrm{\Omega }),$ where $p\in [1,\mathrm{\infty }].$ We present results on the characterization of elliptic and $l$-quasielliptic systems using a priori estimates in isotropic and anisotropic Sobolev spaces $W_{p,0}^l({\mathbb{R}}^n),$ $p\in [1,\mathrm{\infty }].$ For a given set $l=(l_1,\dots ,l_n)\in {\mathbb{N}}^n$ we prove criteria for the existence of $l$-quasielliptic and weakly coercive systems and indicate wide classes of weakly coercive in $W_{p,0}^l({\mathbb{R}}^n),$ $p\in [1,\mathrm{\infty }],$ nonelliptic, and nonquasielliptic systems. In addition, we describe linear spaces of operators that are subordinate in the $L^{\mathrm{\infty }}({\mathbb{R}}^n)$-norm to the tensor product of two elliptic differential polynomials.

The main goal of the work is to construct analogues of Christoffel symbols for infinitedimensional systems and on this basis to obtain geodesic equations for such systems. These analogies are of particular interest in terms of identifying the relationship between the dynamics of systems with an infinite number of degrees of freedom and Riemannian geometry, as well as geometry defined by the pseudo-Riemannian metric.