Background of computations for mathematical models, based on conservation laws systems and applications to fluid dynamics

The paper is devoted to the background of correctness for systems of nonlinear equations, possessing applied significance ill mathematical physics, particularly, in gas and fluid dynamics (Euler equations), in physical kinetics (Boltzmann and Smoluchowski equations) and in phase transition models. The nonlinear operators in above equations are not continuous in Banach spaces specific for these conservation laws. There are discussed general mathematical structures, connected with approximate methods convergence. The existence and uniqueness theorems for global solutions of Cauchy problem for quasilinear and semilinear systems are proved. The problems of parallel computations in above models are discussed.