This paper gives a new and direct construction of the multi-prime big de Rham–Witt complex, which is defined for every commutative and unital ring; the original construction by Madsen and myself relied on the adjoint functor theorem and accordingly was very indirect. The construction given here also corrects the 2-torsion which was not quite correct in the original version. The new construction is based on the theory of modules and derivations over a$λ$-ring which is developed first. The main result in this first part of the paper is that the universal derivation of a$λ$-ring is given by the universal derivation of the underlying ring together with an additional structure depending directly on the$λ$-ring structure in question. In the case of the ring of big Witt vectors, this additional structure gives rise to divided Frobenius operators on the module of Kähler differentials. It is the existence of these divided Frobenius operators that makes the new construction of the big de Rham–Witt complex possible. It is further shown that the big de Rham–Witt complex behaves well with respect to étale maps, and finally, the big de Rham–Witt complex of the ring of integers is explicitly evaluated.

In this paper, we propose a new stabilized linear finite element method for solving reaction-convection-diffusion equations with arbitrary magnitudes of reaction and diffusion. The key feature of the new method is that the test function in the stabilization term is taken in the adjoint-operator-like form $-\varepsilon \Delta v- ({\mbf a}\cdot\nabla v)/\gamma+\sigma v$, where the parameter $\gamma$ is appropriately designed to adjust the convection strength to achieve high accuracy and stability. We derive the stability estimates for the finite element solutions and establish the explicit dependence of $L^2$ and $H^1$ error bounds on the diffusivity, modulus of the convection field, reaction coefficient and the mesh size. The analysis shows that the proposed method is suitable for a wide range of mesh P\'{e}clet numbers and mesh Damk\"{o}hler numbers. More specifically, if the diffusivity $\varepsilon$ is sufficiently small with $\varepsilon < \Vert{\mbf a}\Vert h$ and the reaction coefficient $\sigma$ is large enough such that $\Vert{\mbf a}\Vert < \sigma h$, then the method exhibits optimal convergence rates in both $L^2$ and $H^1$ norms. However, for a small reaction coefficient satisfying $\Vert{\mbf a}\Vert \ge \sigma h$, the method behaves like the well-known streamline upwind/Petrov-Galerkin formulation of Brooks and Hughes. Several numerical examples exhibiting boundary or interior layers are given to demonstrate the high performance of the proposed method. Moreover, we apply the developed method to time-dependent reaction-convection-diffusion problems and simulation results show the efficiency of the approach.

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This note considers the hyper-contractivity and the uniqueness of the weak solutions to the two dimensional Keller–Segel equations. We prove a refined hyper-contractive property and consequently obtain the uniqueness of global weak solutions provided that the initial data satisfy , and . We also extend the result to higher dimensions.