A criterion is given for the convergence of numerical solutions of the Navier-Stokes equations in two dimensions under steady conditions. The criterion applies to all cases, of steady viscous flow in two dimensions and shows that if the local ' mesh Reynolds number ', based on the size of the mesh used in the solution, exceeds a certain fixed value, the numerical solution will not converge.

On the Navier-Stokes equations

Preliminaries.- Model Elliptic Problems.- Model Parabolic Problems.- Systems.- Equations with Gradient Terms.- Nonlocal Problems.

Superlinear Parabolic Problems: Blow-up, Global Existence and Steady States

This paper proposes a survey and critical analysis focused on a variety of chemotaxis models in biology, namely the classical Keller–Segel model and its subsequent modifications, which, in several cases, have been developed to obtain models that prevent the non-physical blow up of solutions. The presentation is organized in three parts. The first part focuses on a survey of some sample models, namely the original model and some of its developments, such as flux limited models, or models derived according to similar concepts. The second part is devoted to the qualitative analysis of analytic problems, such as the existence of solutions, blow-up and asymptotic behavior. The third part deals with the derivation of macroscopic models from the underlying description, delivered by means of kinetic theory methods. This approach leads to the derivation of classical models as well as that of new models, which might deserve attention as far as the related analytic problems are concerned. Finally, an overview of the entire contents leads to suggestions for future research activities.

Toward a mathematical theory of Keller–Segel models of pattern formation in biological tissues

Solutions for semilinear parabolic equations in Lp and regularity of weak solutions of the Navier-Stokes system

In Section 21 the basic theory of sectorial operators is developed, including examples and the concept of sectorial approximation In Section 22 we introduce some notation for certain spaces of holomorphic functions on sectors A functional calculus for sectorial operators is constructed in Section 23 along the lines of the abstract framework of Chapter 1 Fundamental properties like the composition rule are proved In Section 25 we give natural extensions of the functional calculus to larger function spaces in the case where the given operator is bounded and/or invertible In this way a panorama of functional calculi is developed In Section 26 some mixed topics are discussed, eg, adjoints and restrictions of sectorial operators and some fundamental boundedness and some first approximation results Section 27 contains a spectral mapping theorem

The Functional Calculus for Sectorial Operators

Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains

A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains

/pdf/generalized-resolvent-estimates-for-the-stokes-system-in-59f5bztj6c.pdf

Generalized resolvent estimates for the Stokes system in bounded and unbounded domains

/pdf/an-lq-approach-to-stokes-and-navier-stokes-equations-in-20vz8k863d.pdf

Hermann Sohr

Papers

Abstract Lp estimates for the Cauchy problem with applications to the Navier-Stokes equations in exterior domains

A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains

Generalized resolvent estimates for the Stokes system in bounded and unbounded domains

An Lq-approach to Stokes and Navier-Stokes equations in general domains

On operators with bounded imaginary powers in banach spaces