The main focus of the laboratory research is the mathematical analysis of continuum with complex rheology motions. Navier-Stokes equations of viscous heat-conducting gas, mathematical problems in the nonlinear wave theory, equations of the non-Newtonian fluid dynamics, the equations describing the growth of biological materials are among the models under consideration.

Considered models are widely used in astrophysics when describing the evolution of stars and the motion of gas clouds in the universe. The investigation of wave processes in the liquid is necessary for the precise description of such phenomena as tsunamis, sporadic giant waves, processes of energy and mass transportation in the ocean.

Non-Newtonian fluids are widely used in industrial processes. All biological environments are non-Newtonian ones. Knowledge of the motion laws is necessary for the simulation of blood circulation and transplant construction. The theory of the biological material growth is a relatively new discipline that lies at the junction of the theory of elasticity, biochemistry and open systems thermodynamics. Its main goal is to predict the development of cancer and to choose the optimal strategy of treatment.

The results of the laboratory research are the following:

- The correctness of the problem on the plane motion of the isothermal viscous gas has been proved. This problem has a simple formulation, and was the first problem of the theory of viscous gas dynamics equations, for which the global solvability was announced in 1985. The proof was mistaken, and for thirty years the problem remained unsolved;
- The weak solutions existence of the Navier-Stokes stationary equations of viscous gas dynamics for all adiabatic exponents greater than one has been established. Earlier this result was known for adiabatic exponents greater than 3/2 (Lions task). This result is of fundamental importance, as the lack of solutions for small adiabatic exponents indicates the presence of singularities in the form of fibers in the gas clouds. For intergalactic clouds adiabatic index is equal to one. The obtained results make it possible to explore this limiting case;
- The quasi-static model for the growth of biological material has been constructed and justified. This model takes into account the impact of stress and temperature on the growth process. It is shown that at any one time the solution satisfies the principle of the internal energy minimum and an analogue of Prigogine's minimum entropy production.

**International partners of the laboratory:** Center of Mathematics and Fundamental Applications at Lisbon University, Elie Cartan Institute and LEMTA Laboratory, Nancy, France.

**Head of Laboratory:** Doctor in Physics and Mathematics, Pavel Plotnikov,

plotnikov@hydro.nsc.ru
**Section of Applied Mathematics, Department of Mathematics NSU**
** ****Institute of Hydrodynamics, Siberian Branch of th Russina Academy of Sciences**