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Risultati sfruttabili
The research carried out in the frame of this project was focused on mathematical problems in the theory of solids and fluids including non-linear elasticity, plasticity, phase transitions and Newtonian and Non-Newtonian fluids.
A considerable part of this research was devoted to phase transition problems. Necessary restrictions on the energy density function which provide existence of multiphase equilibrium states for problems with zero coefficient of phase tension were derived and (regarding to classical thermoelasticity models) new forms of the equilibrium were obtained.
For the obstacle problem we proved Cl regularity of the boundary of the noncoincidence set in the neighbourhood of the prescribed boundary.
Many boundary value and initial boundary value problems related to the theory of Newtonian and Non-Newtonian fluids were investigated in the project. Partial regularity for 3-D problems and full regularity for 2-D problems in the theory of generalised Newtonian fluids were established. Existence of solutions to some variational problems that arise in the theory of Non-Newtonian fluids and their regularity in Orlicz spaces were proved.
New regularity results were established for solutions to boundary value problems describing quasi-static motions of visco-plastic and pseudo-plastic fluids. Existence of global smooth solutions to initial boundary-value problems for the Kelvin-Voight equations in the theory of visco-elastic fluids was proved. Some problems related to the existence of free boundary in the case of 2-D flows of viscous incompressible fluid were investigated.
For perfectly elasto-plastic problems we obtained a priori estimates of approximation errors. New methods of deriving a posteriori error estimate based on duality theory of the calculus of variations were developed. These methods were used for getting computable upper bounds of approximation errors of various non-linear variational problems.
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