Seminar on nonlinear problems of PDE and mathematical physics on topic “The equations of gas dynamics and related systems of nonlinear ODEs”

Speaker:  Doctor of phys.-math.sc., prof. Rozanova O.S.  (Lomonosov Moscow State University)

Title of the talk: The equations of gas dynamics and related systems of nonlinear ODEs.

Abstract. We consider gas dynamics equations in several spatial dimensions and related equations, in particular, those that are used to simulate the processes of air movement in the atmosphere. We show that under some assumptions about the structure of the velocity field, such partial differential equations naturally generate a nonlinear system of  ODEs. This approach was very popular in the years 60-70 and was developed in the works of Ovsyannikov, Dyson, Bogoyavlenskiy, Gaffet, in Lagrangian coordinates and was applied to gas dynamics equations.  However, in the case of equations of dynamics of the atmosphere, there is a very non-trivial problem on equilibria of the ODE system and their stability. Each equilibrium corresponds to a stationary solution of the initial PDE system, prototypic for real gas flows (in particular, a vortex and shear flow).