On the stability of travelling waves with vorticity obtained by minimisation

We modify the approach of Burton and Toland [Comm. Pure Appl. Math. (2011)] to show the existence of periodic surface water waves with vorticity in order that it becomes suited to a stability analysis. This is achieved by enlarging the function space to a class of stream functions that do not correspond necessarily to travelling profiles. In particular, for smooth profiles and smooth stream functions, the normal component of the velocity field at the free boundary is not required a priori to vanish in some Galilean coordinate system. Travelling periodic waves are obtained by a direct minimisation of a functional that corresponds to the total energy and that is therefore preserved by the time-dependent evolutionary problem (this minimisation appears in Burton and Toland after a first maximisation). In addition, we not only use the circulation along the upper boundary as a constraint, but also the total horizontal impulse (the velocity becoming a Lagrange multiplier). This allows us to preclude parallel flows by choosing appropriately the values of these two constraints and the sign of the vorticity. By stability, we mean conditional energetic stability of the set of minimizers as a whole, the perturbations being spatially periodic of given period.Comment: NoDEA Nonlinear Differential Equations and Applications, to appea

On the existence of an inertial manifold for a deconvolution model of the 2D mean Boussinesq equations

We show the existence of an inertial manifold (i.e. a globally invariant, exponentially attracting, finite-dimensional manifold) for the approximate deconvolution model of the 2D mean Boussinesq equations. This model is obtained by means of the Van Cittern approximate deconvolution operators, which is applied to the 2D filtered Boussinesq equations

A gene regulatory network armature for T lymphocyte specification

Choice of a T lymphoid fate by hematopoietic progenitor cells depends on sustained Notch–Delta signaling combined with tightly regulated activities of multiple transcription factors. To dissect the regulatory network connections that mediate this process, we have used high-resolution analysis of regulatory gene expression trajectories from the beginning to the end of specification, tests of the short-term Notch dependence of these gene expression changes, and analyses of the effects of overexpression of two essential transcription factors, namely PU.1 and GATA-3. Quantitative expression measurements of >50 transcription factor and marker genes have been used to derive the principal components of regulatory change through which T cell precursors progress from primitive multipotency to T lineage commitment. Our analyses reveal separate contributions of Notch signaling, GATA-3 activity, and down-regulation of PU.1. Using BioTapestry (www.BioTapestry.org), the results have been assembled into a draft gene regulatory network for the specification of T cell precursors and the choice of T as opposed to myeloid/dendritic or mast-cell fates. This network also accommodates effects of E proteins and mutual repression circuits of Gfi1 against Egr-2 and of TCF-1 against PU.1 as proposed elsewhere, but requires additional functions that remain unidentified. Distinctive features of this network structure include the intense dose dependence of GATA-3 effects, the gene-specific modulation of PU.1 activity based on Notch activity, the lack of direct opposition between PU.1 and GATA-3, and the need for a distinct, late-acting repressive function or functions to extinguish stem and progenitor-derived regulatory gene expression

On the constants in a Kato inequality for the Euler and Navier-Stokes equations

We continue an analysis, started in [10], of some issues related to the incompressible Euler or Navier-Stokes (NS) equations on a d-dimensional torus T^d. More specifically, we consider the quadratic term in these equations; this arises from the bilinear map (v, w) -> v . D w, where v, w : T^d -> R^d are two velocity fields. We derive upper and lower bounds for the constants in some inequalities related to the above bilinear map; these bounds hold, in particular, for the sharp constants G_{n d} = G_n in the Kato inequality | < v . D w | w >_n | <= G_n || v ||_n || w ||^2_n, where n in (d/2 + 1, + infinity) and v, w are in the Sobolev spaces H^n, H^(n+1) of zero mean, divergence free vector fields of orders n and n+1, respectively. As examples, the numerical values of our upper and lower bounds are reported for d=3 and some values of n. When combined with the results of [10] on another inequality, the results of the present paper can be employed to set up fully quantitative error estimates for the approximate solutions of the Euler/NS equations, or to derive quantitative bounds on the time of existence of the exact solutions with specified initial data; a sketch of this program is given.Comment: LaTeX, 39 pages. arXiv admin note: text overlap with arXiv:1007.4412 by the same authors, not concerning the main result

The Cauchy problem for a class of two-dimensional nonlocal nonlinear wave equations governing anti-plane shear motions in elastic materials

This paper is concerned with the analysis of the Cauchy problem of a general class of two-dimensional nonlinear nonlocal wave equations governing anti-plane shear motions in nonlocal elasticity. The nonlocal nature of the problem is reflected by a convolution integral in the space variables. The Fourier transform of the convolution kernel is nonnegative and satisfies a certain growth condition at infinity. For initial data in $L^{2}$ Sobolev spaces, conditions for global existence or finite time blow-up of the solutions of the Cauchy problem are established.Comment: 15 pages. "Section 6 The Anisotropic Case" added and minor changes. Accepted for publication in Nonlinearit

On the Clark-alpha model of turbulence: global regularity and long--time dynamics

In this paper we study a well-known three--dimensional turbulence model, the filtered Clark model, or Clark-alpha model. This is Large Eddy Simulation (LES) tensor-diffusivity model of turbulent flows with an additional spatial filter of width alpha ($\alpha$). We show the global well-posedness of this model with constant Navier-Stokes (eddy) viscosity. Moreover, we establish the existence of a finite dimensional global attractor for this dissipative evolution system, and we provide an anaytical estimate for its fractal and Hausdorff dimensions. Our estimate is proportional to $(L/l_d)^3$, where $L$ is the integral spatial scale and $l_d$ is the viscous dissipation length scale. This explicit bound is consistent with the physical estimate for the number of degrees of freedom based on heuristic arguments. Using semi-rigorous physical arguments we show that the inertial range of the energy spectrum for the Clark-$\aa$ model has the usual $k^{-5/3}$ Kolmogorov power law for wave numbers $k\aa \ll 1$ and $k^{-3}$ decay power law for $k\aa \gg 1.$ This is evidence that the Clark$-\alpha$ model parameterizes efficiently the large wave numbers within the inertial range, $k\aa \gg 1$, so that they contain much less translational kinetic energy than their counterparts in the Navier-Stokes equations.Comment: 11 pages, no figures, submitted to J of Turbulenc

Global regularity criterion for the 3D Navier-Stokes equations involving one entry of the velocity gradient tensor

In this paper we provide a sufficient condition, in terms of only one of the nine entries of the gradient tensor, i.e., the Jacobian matrix of the velocity vector field, for the global regularity of strong solutions to the three-dimensional Navier-Stokes equations in the whole space, as well as for the case of periodic boundary conditions

Vanishing Viscous Limits for 3D Navier-Stokes Equations with A Navier-Slip Boundary Condition

In this paper, we investigate the vanishing viscosity limit for solutions to the Navier-Stokes equations with a Navier slip boundary condition on general compact and smooth domains in $\mathbf{R}^3$. We first obtain the higher order regularity estimates for the solutions to Prandtl's equation boundary layers. Furthermore, we prove that the strong solution to Navier-Stokes equations converges to the Eulerian one in $C([0,T];H^1(\Omega))$ and L^\infty((0,T)\times\o), where $T$ is independent of the viscosity, provided that initial velocity is regular enough. Furthermore, rates of convergence are obtained also.Comment: 45page

Molecular mechanisms of hematogenous tumor - metastasis

The present manuscript demonstrates that B16F10 melanoma cells activate the enzyme acid sphingomyelinase in thrombocytes via the surface molecule P-selectin, by which ceramide is released. Metastasis of tumor cells in the lung is decreased by up to 95% by genetic deficiency of P-selectin molecule or deficiency of acid sphingomyelinase. After activation of wild type thrombocytes by B16F10 melanoma cells there is a rapid increase in acid sphingomyelinase activity and ceramide production as compared to acid sphingomyelinase-deficient thrombocytes or P-selectin-deficient thrombocytes. A lack of interaction of B16F10 melanoma cells and thrombocytes was excluded by activation of PLCγ, JNK and MAP kinase, indicating that these signaling events are stimulated in both, wild-type and P-selectin-deficient platelets, proving that B16F10 melanoma cells interact with and activate P-selectin-deficient thrombocytes. The molecular mechanisms of tumor metastasis are currently fairly incomplete, though metastasis plays a crucial clinical role in cancer patients. Acid sphingomyelinase is iidentified as a novel target molecule for the inhibition of tumor metastasis. In order to pharmacologically inhibit the thrombocytic P-selectin system, an intravenous injection of fucoidan showed a decrease of tumor metastasis of B16F10 melanoma cells by approximately 75%. This indicates that tumor metastasis can be blocked pharmacologically, which is of great clinical interest

Supermassive Black Holes and Their Environments

We make use of the first high--resolution hydrodynamic simulations of structure formation which self-consistently follows the build up of supermassive black holes introduced in Di Matteo et al. (2007) to investigate the relation between black holes (BH), host halo and large--scale environment. There are well--defined relations between halo and black hole masses and between the activities of galactic nuclei and halo masses at low redshifts. A large fraction of black holes forms anti--hierarchically, with a higher ratio of black hole to halo mass at high than at low redshifts. At $z=1$, we predict group environments (regions of enhanced local density) to contain the highest mass and most active (albeit with a large scatter) BHs while the rest of the BH population to be spread over all densities from groups to filaments and voids. Density dependencies are more pronounced at high rather than low redshift. These results are consistent with the idea that gas rich mergers are likely the main regulator of quasar activity. We find star formation to be a somewhat stronger and tighter function of local density than BH activity, indicating some difference in the triggering of the latter versus the former. There exists a large number of low--mass black holes, growing slowly predominantly through accretion, which extends all the way into the most underdense regions, i.e. in voids.Comment: 18 pages, 15 Figures, accepted for publication in MNRA