Generalized solvability of initial-boundary value problems for quasihydrodynamic system of equations in weighted Sobolev spaces

The paper considers the analog of the first initial boundary value problem for a quasihydrodynamic system of equations in the case of a weakly compressible fluid in weighted Sobolev spaces. The system is an elliptic-parabolic system: its first part is an elliptic equation for the pressure gradient, and its second part is a parabolic system for the velocity vector. The unknown variables of the pressure gradient and velocity vector belong to the principal parts of the elliptic equation and the parabolic system. The fixed part of the system is not uniformly elliptic, thus complicating the study of the problem. T.G. Elizarova and B.N. Chetverushkin introduce the system by averaging the known kinetic model. The first versions of the system are the system of quasi-gasodynamic equations. Later, Y.V. Sheretov, based on a more general equation of state, obtains another model, which is called quasihydrodynamic system of equations, and thoroughly analyses its properties. However, the issues of generalized solvability of initial boundary value problems for such systems have not been studied in detail yet. There are only some partial results. The paper aims to fill this gap. We prove generalized solvability of the system in some weight classes characterizing the behavior of solutions at $t_\infty$ according to the Galerkin method and the obtained prior estimates. The decreasing (growing) behavior of the solution depends on the decreasing (growing) right-hand side of the system. The decrease (growth) at $t_\infty$ of the used weight functions can be both exponential and power.