Global Strong Well-posedness of the Three Dimensional Primitive equations in LpL^p-spaces

In this article, an $L^p$-approach to the primitive equations is developed. In particular, it is shown that the three dimensional primitive equations admit a unique, global strong solution for all initial data $a \in [X_p,D(A_p)]_{1/p}$ provided $p \in [6/5,\infty)$. To this end, the hydrostatic Stokes operator $A_p$ defined on $X_p$, the subspace of $L^p$ associated with the hydrostatic Helmholtz projection, is introduced and investigated. Choosing $p$ large, one obtains global well-posedness of the primitive equations for strong solutions for initial data $a$ having less differentiability properties than $H^1$, hereby generalizing in particular a result by Cao and Titi (Ann. Math. 166 (2007), pp. 245-267) to the case of non-smooth initial data.Comment: 26 page

Splitting Madsen-Tillmann spectra II. The Steinberg idempotents and Whitehead conjecture

We show that, at the prime $p=2$, the spectrum $\Sigma^{-n}D(n)$ splits off the Madsen-Tillmann spectrum $MTO(n)=BO(n)^{-\gamma_n}$ which is compatible with the classic splitting of $M(n)$ off $BO(n)_+$. For $n=2$, together with our previous splitting result on Madsen-Tillmann spectra, this shows that $MTO(2)$ is homotopy equivalent to $BSO(3)_+\vee\Sigma^{-2}D(2)$.Comment: Comments are welcome

Obstructions to shellability, partitionability, and sequential Cohen-Macaulayness

For a property $\cal P$ of simplicial complexes, a simplicial complex $\Gamma$ is an obstruction to $\cal P$ if $\Gamma$ itself does not satisfy $\cal P$ but all of its proper restrictions satisfy $\cal P$. In this paper, we determine all obstructions to shellability of dimension $\le 2$, refining the previous work by Wachs. As a consequence we obtain that the set of obstructions to shellability, that to partitionability and that to sequential Cohen-Macaulayness all coincide for dimensions $\le 2$. We also show that these three sets of obstructions coincide in the class of flag complexes. These results show that the three properties, hereditary-shellability, hereditary-partitionability, and hereditary-sequential Cohen-Macaulayness are equivalent for these classes

Penalty method with Crouzeix-Raviart approximation for the Stokes equations under slip boundary condition

The Stokes equations subject to non-homogeneous slip boundary conditions are considered in a smooth domain $\Omega \subset \mathbb R^N \, (N=2,3)$. We propose a finite element scheme based on the nonconforming P1/P0 approximation (Crouzeix-Raviart approximation) combined with a penalty formulation and with reduced-order numerical integration in order to address the essential boundary condition $u \cdot n_{\partial\Omega} = g$ on $\partial\Omega$. Because the original domain $\Omega$ must be approximated by a polygonal (or polyhedral) domain $\Omega_h$ before applying the finite element method, we need to take into account the errors owing to the discrepancy $\Omega \neq \Omega_h$, that is, the issues of domain perturbation. In particular, the approximation of $n_{\partial\Omega}$ by $n_{\partial\Omega_h}$ makes it non-trivial whether we have a discrete counterpart of a lifting theorem, i.e., right-continuous inverse of the normal trace operator $H^1(\Omega)^N \to H^{1/2}(\partial\Omega)$; $u \mapsto u\cdot n_{\partial\Omega}$. In this paper we indeed prove such a discrete lifting theorem, taking advantage of the nonconforming approximation, and consequently we establish the error estimates $O(h^\alpha + \epsilon)$ and $O(h^{2\alpha} + \epsilon)$ for the velocity in the $H^1$- and $L^2$-norms respectively, where $\alpha = 1$ if $N=2$ and $\alpha = 1/2$ if $N=3$. This improves the previous result [T. Kashiwabara et al., Numer. Math. 134 (2016), pp. 705--740] obtained for the conforming approximation in the sense that there appears no reciprocal of the penalty parameter $\epsilon$ in the estimates.Comment: 21 page