Existence of Weak Solutions for a Diffuse Interface Model for Two-Phase Flows of Incompressible Fluids with Different Densities

We prove existence of weak solutions for a diffuse interface model for the flow of two viscous incompressible Newtonian fluids in a bounded domain in two and three space dimensions. In contrast to previous works, we study a new model recently developed by Abels, Garcke, and Gr\"un for fluids with different densities, which leads to a solenoidal velocity field. The model is given by a non-homogeneous Navier-Stokes system with a modified convective term coupled to a Cahn-Hilliard system. The density of the mixture depends on an order parameter.Comment: 33 page

Coercivity and stability results for an extended Navier-Stokes system

In this article we study a system of equations that is known to {\em extend} Navier-Stokes dynamics in a well-posed manner to velocity fields that are not necessarily divergence-free. Our aim is to contribute to an understanding of the role of divergence and pressure in developing energy estimates capable of controlling the nonlinear terms. We address questions of global existence and stability in bounded domains with no-slip boundary conditions. Even in two space dimensions, global existence is open in general, and remains so, primarily due to the lack of a self-contained $L^2$ energy estimate. However, through use of new $H^1$ coercivity estimates for the linear equations, we establish a number of global existence and stability results, including results for small divergence and a time-discrete scheme. We also prove global existence in 2D for any initial data, provided sufficient divergence damping is included.Comment: 29 pages, no figure

Boundedness of Pseudodifferential Operators on Banach Function Spaces

We show that if the Hardy-Littlewood maximal operator is bounded on a separable Banach function space $X(\mathbb{R}^n)$ and on its associate space $X'(\mathbb{R}^n)$, then a pseudodifferential operator $\operatorname{Op}(a)$ is bounded on $X(\mathbb{R}^n)$ whenever the symbol $a$ belongs to the H\"ormander class $S_{\rho,\delta}^{n(\rho-1)}$ with $0<\rho\le 1$, $0\le\delta<1$ or to the the Miyachi class $S_{\rho,\delta}^{n(\rho-1)}(\varkappa,n)$ with $0\le\delta\le\rho\le 1$, $0\le\delta0$. This result is applied to the case of variable Lebesgue spaces $L^{p(\cdot)}(\mathbb{R}^n)$.Comment: To appear in a special volume of Operator Theory: Advances and Applications dedicated to Ant\'onio Ferreira dos Santo

Fractional-order operators: Boundary problems, heat equations

The first half of this work gives a survey of the fractional Laplacian (and related operators), its restricted Dirichlet realization on a bounded domain, and its nonhomogeneous local boundary conditions, as treated by pseudodifferential methods. The second half takes up the associated heat equation with homogeneous Dirichlet condition. Here we recall recently shown sharp results on interior regularity and on $L_p$-estimates up to the boundary, as well as recent H\"older estimates. This is supplied with new higher regularity estimates in $L_2$-spaces using a technique of Lions and Magenes, and higher $L_p$-regularity estimates (with arbitrarily high H\"older estimates in the time-parameter) based on a general result of Amann. Moreover, it is shown that an improvement to spatial $C^\infty$-regularity at the boundary is not in general possible.Comment: 29 pages, updated version, to appear in a Springer Proceedings in Mathematics and Statistics: "New Perspectives in Mathematical Analysis - Plenary Lectures, ISAAC 2017, Vaxjo Sweden

Extracellular vesicle-mediated bilateral communication between glioblastoma and astrocytes

Glioblastoma the most aggressive form of brain cancer, comprises a complex mixture of tumor cells and nonmalignant stromal cells, including neurons, astrocytes, microglia, infiltrating monocytes/macrophages, lymphocytes, and other cell types. All nonmalignant cells within and surrounding the tumor are affected by the presence of glioblastoma. Astrocytes use multiple modes of communication to interact with neighboring cells. Extracellular vesicle-directed intercellular communication has been found to be an important component of signaling between astrocytes and glioblastoma in tumor progression. In this review, we focus on recent findings on extracellular vesicle-mediated bilateral crosstalk, between glioblastoma cells and astrocytes, highlighting the protumor and antitumor roles of astrocytes in glioblastoma development.Chemical Immunolog

Anti-leukemic potential of methyl-cobalamin inactivation by nitrous oxide

Myelo‐cytotoxicity of extended nitrous oxide (N2O) inhalation was described almost forty years ago and then incidentally applied already with temporary success for suppressing leukemia. In 1948 the accompanying megaloblastic maturation arrest was explained by inactivation of the methylcobalamin coenzyme and subsequent folate deficiency. We studied the anti‐leukemic effect of N2O on a transplantable acute leukemia in B(rown) N(orway) rats. Progression of this B,N,M(yelocytic)L(eukemia) was measured as spleen and liver weights, and leukemic blood cell counts. The deoxyuridine (dU)‐suppression test provided in vitro indication of the functional folate activity of leukemic cells. Breathing of N2O‐oxygen considerably reduced but did not eradicate, BNML‐proliferation. Addition of anti‐metabolites, interfering with some enzyme in the folate metabolism beyond the methylcobalamin co‐enzyme dependent methionine synthase step, acted at least synergistically. The anti‐leukemic effect of cycloleucine, which reduces S‐adenosyl‐methionine synthesis by inactivation of methionine adenosyltransferase, was moderate but became much stronger with N2O inhalation. Methotrexate, a potent anti‐leukemic agent by inhibiting tetrahydrofolate (THF) generation through inactivation of di‐HF reductase, became highly anti‐BNML, even in low dosage when combined with or preceded by N2O. 5‐Fluorouracil, which inhibits methylene‐THF dependent thymidilate synthase, itself was surprisingly anti‐BNML, but also became much more potent with previous or concomittant N2O exposure. Preliminary dU‐suppression test results with human acute leukemia cells, exposed to N2O and/or folate antagonists in vitro, correlated well with the in vivo BNML‐experiments. Combining the anticobalamin activity of N2O with an anti‐folate therefore seems to be a promising chemotherapeutic approach. Copyrigh

The Baum-Connes Conjecture via Localisation of Categories

We redefine the Baum-Connes assembly map using simplicial approximation in the equivariant Kasparov category. This new interpretation is ideal for studying functorial properties and gives analogues of the assembly maps for all equivariant homology theories, not just for the K-theory of the crossed product. We extend many of the known techniques for proving the Baum-Connes conjecture to this more general setting