In this thesis, we study some problems on the Navier-Stokes equations and Boltzmann- Enskog equation which are in the active research area of applied mathematics. The system of Navier-Stokes equations is a typical example of the conservation laws. In the first part of the thesis, we study the global existence and convergence rates of solutions to the three-dimensional compressible Navier-Stokes equations without heat conductivity.
Computer-aided Computation of Abelian integrals and Robust Normal Forms
This PhD thesis consists of a summary and seven papers, where various applications of auto-validated computations are studied.In the first paper we describe a rigorous method to determine unknown parameters in a system of ordinary differential equations from measured data with known bounds on the noise of the measurements.Papers II, III, IV, and V are [...]
STOCHASTIC OPTIMIZATION: ALGORITHMS AND CONVERGENCE
Stochastic approximation is one of the oldest approaches for solving stochastic optimization problems. In the first part of the dissertation, we study the convergence and asymptotic normality of a generalized form of stochastic approximation algorithm with deterministic perturbation sequences. Both one-simulation and two-simulation methods are considered. Assuming a special structure on the deterministic sequence, we [...]