Basic Scientific Results

1. Fracture mechanics. Fundamental mathematical model of elastic body with cracks based on explicit introduction of cohesion forces, and solutions of basic problems. Introduction of one of the basic characteristics of fracture toughness: cohesion modulus. Basic model of the kinetics of crack propagation. Applications to fracture problems of metals, rocks and polymers. Similarity laws for brittle and quasi-brittle fracture. Scaling laws for fatigue cracks and multiple fracturing, model of small fatigue cracks. Mathematical model of non-local damage accumulation. Mathematical model of self-oscillation, self-similar phenomena in fatigue fracture.

2. Theory of fluid and gas flows in porous media. Fundamental model of flow in fissurized porous rocks and solution of basic problems. Asymptotic solutions of basic problems of unsteady groundwater and gas flows in porous media. Fundamental model of fluid flow in elasto-plastic porous media and solution of basic problems. Non-equilibrium two-phase flow in porous media (capillary imbibition, water-oil displacement, solid phase precipitation): basic mathematical model, fundamental solutions. Mathematical model of gas-condensate flow in fissurized-porous media. Mathematical model of very intense pulse in groundwater flows in porous and fissurized porous rocks.

3. Mechanics of non-classical deformable solids. Mathematical models of neck propagation in polymers (deep analog of flame propagation) and of thermal vibro-creep in polymers. Mathematical model of an impact of viscoplastic body on a rigid obstacle.

4. Turbulence. Turbulence in stratified fluids. Mathematical models of the transport of heavy particles in turbulent flows. Basic model of turbulent patch dynamics in stably stratified fluids, self-similar asymptotic laws; relation to the oceanic microstructure. Mathematical model of non-steady heat and mass transfer in stably stratified turbulent flows. Model of turbulent drag reduction by polymeric additives. Mathematical models of turbulent burst and turbulent shearless wake evolution. Mathematical model of laminar-turbulent transition taking into account the evolution of pre-existing turbulence. Mathematical model of temperature steps formation in stably stratified turbulent flows. Scaling laws for developed turbulent shear flows, in particular for pipe, boundary-layer flows and wall-jets. Mathematical models of dust storms and tropical hurricanes.

5. Self-similarities. Nonlinear waves and intermediate asymptotics. Long-time work performed in general in close collaboration with Ya. B. Zeldovich. Concepts of intermediate asymptotics, self-similar asymptotics of the first and second kinds. Nonlinear eigenvalue problems. Relation between intermediate asymptotics and renormalization groups. Basic model of stability of self-similar solutions and travelling waves. Contributions to the theory of combustion and thermal explosion. New model of surface-tension-driven thin films.