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A RUDN University mathematician has proposed a new criterion for solving the Boussinesq equations. These equations describe the nonlinear propagation of waves in certain media, e.g. plasma, a surface of liquid of shallow depth, and so on. They examined the Boussinesq equation in three-dimensional space and derived a criterion for uniqueness and the existence of important solutions of a special type to the Boussinesq partial differential equation. The proposed criterion has applications in mechanics of continuous media, which studies the motion of liquids and gases. The article was published in Bulletin of the Brazilian Mathematical Society, New Series.
Both the Boussinesq equations and the Navier-Stokes equations are systems of partial differential equations (differentiation is carried out with respect to all independent variables). Partial differential equations play a significant role in mathematical physics and mechanics. Solving equations of this type is often fraught with great difficulties. The problem of existence and uniqueness of a solution to the Boussinesq equations under given initial conditions (the so-called Cauchy problem) had previously been investigated by many scientists, including the authors of the article. With certain values of the parameters, the Boussinesq equations turn into Navier-Stokes equations. The existence and continuous differentiability,...
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