Том 69, № 4 (2023)

For an integral equation whose solutions provide stationary states of a population distributed in an arithmetic space, we nd the conditions for the existence of its solution and conditions under which this equation has no more than one solution.

In this paper, we study the eta-invariant of elliptic parameter-dependent boundary value problems and its main properties. Using Melrose’s approach, we de ne the eta-invariant as a regularization of the winding number of the family. In this case, the regularization of the trace requires obtaining the asymptotics of the trace of compositions of invertible parameter-dependent boundary value problems for large values of the parameter. Obtaining the asymptotics uses the apparatus of pseudodifferential boundary value problems and is based on the reduction of parameter-dependent boundary value problems to boundary value problems with no parameter.

We consider sewing machinery between nite difference and analytical solutions de ned at different scales: far away and near the source of the perturbation of the flow. One of the essences of the approach is that the coarse problem and the boundary-value problem in the proxy of the source model two different flows. In his remarkable paper, Peaceman proposes a framework for dealing with solutions de ned on different scales for linear time independent problems by introducing the famous Peaceman well block radius. In this article, we consider a novel problem: how to solve this issue for transient flow generated by the compressibility of the fluid. We are proposing a method to glue solution via total fluxes, which are prede ned on coarse grid, and changes in pressure, due to compressibility, in the block containing production (injection) well. It is important to mention that the coarse solution “does not see” the boundary. From an industrial point of view, our report provides a mathematical tool for the analytical interpretation of simulated data for compressible fluid flow around a well in a porous medium. It can be considered a mathematical “shirt” on the Peaceman well-block radius formula for linear (Darcy) transient flow but can be applied in much more general scenarios. In the article, we use the Einstein approach to derive the material balance equation, a key instrument to de ne R0. We will expand the Einstein approach for three regimes of Darcy and non-Darcy flows for compressible fluids (time-dependent): 1. stationary; 2. pseudostationary; 3. boundary dominated. Note that in all known authors literature, the rate of production on the well is time-independent.

An explicit form of weak solutions to the Riemann problem for a degenerate nonlinear parabolic equation with a piecewise constant diffusion coe cient is found. It is shown that the lines of phase transitions (free boundaries) correspond to the minimum point of some strictly convex and coercive function of a nite number of variables. A similar result is true for Stefan’s problem. In the limit, when the number of phases tends to in nity, there arises a variational formulation of self-similar solutions to the equation with an arbitrary nonnegative diffusion function.

The paper is devoted to the Dirichlet problem in a flat bounded domain for a linear secondorder functional differential equation in the divergent form with dilation, contraction and rotation of the argument of the higher-order derivatives of the unknown function. We study the existence, the uniqueness and the smoothness of the generalized solution for all possible values of the coefficients and parameters of transformations in the equation.