We consider the Stokes equations under some nonlinear boundary conditions, which are described in terms of subdifferentials of maximal monotone graphs and are called leak and slip boundary conditions of friction type. The main objective is to show the existence of strong solutions, say u ∈ H 2 and p ∈ H 1 , to these problems. We start with weak solutions to variational inequalities, and then study the regularity of weak solutions. Our main theorems imply the maximality of Stokes operators with such nonlinear boundary conditions in a suitable Hilbert space and they are of use in analysis of time-dependent problems. Linear boundary conditions of Neumann type, such as slip and penetration conditions, are also discussed. §1. IntroductionLet Ω be a bounded domain in R N , N = 2, 3. We suppose that the boundary ∂Ω of Ω is composed of two connected components Γ and Γ D which are assumed to be Lipschitz continuous, unless otherwise stated. Γ is not empty, whereas Γ D may be empty. In the present paper, we shall mainly discuss the existence of a strong solution u ∈ H 2 (Ω) N and p ∈ H 1 (Ω) to a modified Stokes equation λu − ∆u + ∇p = f, div u = 0 in Ω (1.1)

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We are concerned with the finite-element approximation for the Keller-Segel system that describes the aggregation of slime molds resulting from their chemotactic features. The scheme makes use of a semi-implicit time discretization with a time-increment control and Baba-Tabata's conservative upwind finite-element approximation in order to realize the positivity and mass conservation properties. The main aim is to present error analysis that is an application of the discrete version of the analytical semigroup theory.

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