Dynamics of the Ericksen-Leslie Equations with General Leslie Stress I: The Incompressible Isotropic Case

The Ericksen-Leslie model for nematic liquid crystals in a bounded domain with general Leslie and isotropic Ericksen stress is studied in the case of a non-isothermal and incompressible fluid. This system is shown to be locally well-posed in the $L_p$-setting, and a dynamic theory is developed. The equilibria are identified and shown to be normally stable. In particular, a local solution extends to a unique, global strong solution provided the initial data are close to an equilibrium or the solution is eventually bounded in the topology of the natural state manifold. In this case, the solution converges exponentially to an equilibrium, in the topology of the state manifold. The above results are proven {\em without} any structural assumptions on the Leslie coefficients and in particular {\em without} assuming Parodi's relation

On quasilinear parabolic evolution equations in weighted Lp-spaces II

Our study of abstract quasi-linear parabolic problems in time-weighted L_p-spaces, begun in [17], is extended in this paper to include singular lower order terms, while keeping low initial regularity. The results are applied to reaction-diffusion problems, including Maxwell-Stefan diffusion, and to geometric evolution equations like the surface-diffusion flow or the Willmore flow. The method presented here will be applicable to other parabolic systems, including free boundary problems.Comment: 21 page

On a Class of Energy Preserving Boundary Conditions for Incompressible Newtonian Flows

We derive a class of energy preserving boundary conditions for incompressible Newtonian flows and prove local-in-time well-posedness of the resulting initial boundary value problems, i.e. the Navier-Stokes equations complemented by one of the derived boundary conditions, in an Lp-setting in domains, which are either bounded or unbounded with almost flat, sufficiently smooth boundary. The results are based on maximal regularity properties of the underlying linearisations, which are also established in the above setting.Comment: 53 page

On the well-posedness of a mathematical model describing water-mud interaction

In this paper we consider a mathematical model describing the two-phase interaction between water and mud in a water canal when the width of the canal is small compared to its depth. The mud is treated as a non-Netwonian fluid and the interface between the mud and fluid is allowed to move under the influence of gravity and surface tension. We reduce the mathematical formulation, for small boundary and initial data, to a fully nonlocal and nonlinear problem and prove its local well-posedness by using abstract parabolic theory.Comment: 16 page

Global LpL^{p} estimates for degenerate Ornstein-Uhlenbeck operators with variable coefficients

We consider a class of degenerate Ornstein-Uhlenbeck operators in $\mathbb{R}^{N}$, of the kind [\mathcal{A}\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x) \partial_{x_{i}x_{j}}^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}%] where $(a_{ij})$ is symmetric uniformly positive definite on $\mathbb{R}^{p_{0}}$ ($p_{0}\leq N$), with uniformly continuous and bounded entries, and $(b_{ij})$ is a constant matrix such that the frozen operator $\mathcal{A}_{x_{0}}$ corresponding to $a_{ij}(x_{0})$ is hypoelliptic. For this class of operators we prove global $L^{p}$ estimates ($1<p<\infty$) of the kind:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(\mathbb{R}% ^{N})}\leq c{|\mathcal{A}u|_{L^{p}(\mathbb{R}^{N})}+|u|_{L^{p}(\mathbb{R}% ^{N})}} for i,j=1,2,...,p_{0}.] We obtain the previous estimates as a byproduct of the following one, which is of interest in its own:% [|\partial_{x_{i}x_{j}}^{2}u|_{L^{p}(S_{T})}\leq c{|Lu|_{L^{p}(S_{T})}+|u|_{L^{p}(S_{T})}}] for any $u\in C_{0}^{\infty}(S_{T}),$ where $S_{T}$ is the strip $\mathbb{R}^{N}\times[-T,T]$, $T$ small, and $L$ is the Kolmogorov-Fokker-Planck operator% [L\equiv\sum_{i,j=1}^{p_{0}}a_{ij}(x,t) \partial_{x_{i}x_{j}}% ^{2}+\sum_{i,j=1}^{N}b_{ij}x_{i}\partial_{x_{j}}-\partial_{t}%] with uniformly continuous and bounded $a_{ij}$'s

Anti-ARHGAP26 autoantibodies are associated with isolated cognitive impairment

Autoantibodies against the RhoGTPase-activating protein 26 (ARHGAP26) were originally identified in the context of subacute autoimmune cerebellar ataxia. Further studies identified a wider clinical spectrum including psychotic, affective, and cognitive symptoms. Only a few patients reported so far had evidence of a tumor association. A prospective analysis between January 2015 and December 2017 at the Dept. of Neurology at Charite-Universitatsmedizin Berlin identified 14 patients with ARHGAP26 autoantibodies on a cell-based assay, of which three patients had additional brain immunohistochemistry staining of cerebellar molecular layer and Purkinje cells, who were therefore considered antibody-positive. In all three patients, ARHGAP26 autoantibodies were associated with tumors. In two patients, an isolated cognitive impairment without additional neurological deficits was observed. These cases thus further extend the clinical spectrum associated with ARHGAP26 autoantibodies and strengthen a potential paraneoplastic context