This paper reviews the existence, uniqueness and regularity of weak and strong solutions of the Navier-Stokes system. For this purpose we emphasize semigroup theory and the theory of the Stokes operator. We use dimensional analysis to clarify the meaning of the results for the solutions.
The small-scale properties of turbulence are related to the singularity of the velocity field in the inviscid limit. The energy spectrum in the inertial range is expressed in terms of the fractal dimension and Hölder's exponent of the singularity. It is pointed out that the entrainment process is important in the dynamics of active eddies in the β-model theory of turbulence.
We deal with a weakly-coupled system of semilinear parabolic equations, namely a competition-diffusion system, and prove the existence of a stable spatially-inhomogeneous equilibrium solution on the assumption that the spatial domain is far from being convex and that the corresponding system of ODEs in the absence of diffusion posesses at least two distinct asymptotically stable equilibria. We also consider a non-weakly coupled system of competition type involving cross-diffusion terms, which leave the system quasilinear but no longer semilinear. The point of interest is to see how the shape of the spatial domain or the presence of cross-diffusion terms contributes to the occurrence of pattern formation.
A method for evaluation of the error function of real and complex variable by means of a trapezoidal sum with a correction term is proposed. This method gives a result with high relative accuracy with small number of operations. A precise upper bound of the relative error of the approximation for real variable is given.
Recent development of the theory of completely bounded maps between C*-algebras such as those results by Wittstock, Smith, Paulsen and Huruya are reviewed and discussed as well as the author's results.
A finite element step-by-step time integration scheme of a model linear viscous shallow-water system is analyzed by the approximation theory for semi-groups of linear operators. Strong convergence and O(h+τ) error estimate in L2-sense are established.
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