Estimating intermittency in three-dimensional Navier-Stokes turbulence

The issue of why computational resolution in Navier-Stokes turbulence is so hard to achieve is addressed. It is shown that Navier-Stokes solutions can potentially behave differently in two distinct regions of space-time $\mathbb{R}^{\pm}$ where $\mathbb{R}^{-}$ is comprised of a union of disjoint space-time `anomalies'. Large values of |\nabla\bom| dominate $\mathbb{R}^{-}$, which is consistent with the formation of vortex sheets or tightly-coiled filaments. The local number of degrees of freedom $\mathcal{N}^{\pm}$ needed to resolve the regions in $\mathbb{R}^{\pm}$ satisfies \mathcal{N}^{\pm}(\bx, t)\lessgtr c_{\pm}\mathcal{R}_{u}^{3} where $\mathcal{R}_{u} = uL/\nu$ is a Reynolds number dependent on the local velocity field u(\bx, t)

Existence of Multiple Vortices in Supersymmetric Gauge Field Theory

Two sharp existence and uniqueness theorems are presented for solutions of multiple vortices arising in a six-dimensional brane-world supersymmetric gauge field theory under the general gauge symmetry group $G=U(1)\times SU(N)$ and with $N$ Higgs scalar fields in the fundamental representation of $G$. Specifically, when the space of extra dimension is compact so that vortices are hosted in a 2-torus of volume |\Om|, the existence of a unique multiple vortex solution representing $n_1,...,n_N$ respectively prescribed vortices arising in the $N$ species of the Higgs fields is established under the explicitly stated necessary and sufficient condition \[ n_i<\frac{g^2v^2}{8\pi N}|\Om|+\frac{1}{N}(1-\frac{1}{N}[\frac{g}{e}]^2)n,\quad i=1,...,N,] where $e$ and $g$ are the U(1) electromagnetic and SU(N) chromatic coupling constants, $v$ measures the energy scale of broken symmetry, and $n=\sum_{i=1}^N n_i$ is the total vortex number; when the space of extra dimension is the full plane, the existence and uniqueness of an arbitrarily prescribed $n$-vortex solution of finite energy is always ensured. These vortices are governed by a system of nonlinear elliptic equations, which may be reformulated to allow a variational structure. Proofs of existence are then developed using the methods of calculus of variations.Comment: 23 page

A Posteriori Error Estimates for Nonconforming Approximations of Evolutionary Convection-Diffusion Problems

We derive computable upper bounds for the difference between an exact solution of the evolutionary convection-diffusion problem and an approximation of this solution. The estimates are obtained by certain transformations of the integral identity that defines the generalized solution. These estimates depend on neither special properties of the exact solution nor its approximation, and involve only global constants coming from embedding inequalities. The estimates are first derived for functions in the corresponding energy space, and then possible extensions to classes of piecewise continuous approximations are discussed.Comment: 10 page

Vestibular reactions in patients during occlusion form of chorioependymitis

A change in intracranial pressure during chorioependymitis is discussed. The inflammatory process in the choroid plexi and walls of the cerebral ventricles is described. The otoneurological method, not having any significant contraindications, can be employed repeatedly, quaranteeing control in dynamics over the course of the disease and effectiveness of treatment

Optimal well-posedness for the inhomogeneous incompressible Navier-Stokes system with general viscosity

In this paper we obtain new well-possedness results concerning a linear inhomogenous Stokes-like system. These results are used to establish local well-posedness in the critical spaces for initial density $\rho_{0}$ and velocity $u_{0}$ such that $\rho_{0}-\rho\in\dot{B}_{p,1}^{\frac{3}{p}}(\mathbb{R}^{3})$, $u_{0}\in\dot{B}_{p,1}^{\frac{3}{p}-1}(\mathbb{R}^{3})$, $p\in\left( \frac{6}{5},4\right)$, for the inhomogeneous incompressible Navier-Stokes system with variable viscosity. To the best of our knowledge, regarding the $3D$ case, this is the first result in a truly critical framework for which one does not assume any smallness condition on the density

Global Dynamics of Cosmological Expansion with Minimally Coupled Scalar Field

We give a complete description of the asymptotic behavior of a Friedmann-Robertson-Walker Universe with ``normal'' matter and a minimally coupled scalar field. We classify the conditions under which the Universe is or is not accelerating. In particular, we show that only two types of large time behavior exist: an exponential regime, and a subexponential expansion with the logarithmic derivative of the scale factor tending to zero. In the case of the subexponetial expansion the Universe accelerates when the scalar field energy density is dominant and the potential behaves in a specified manner, or if matter violates the strong energy conditon $\rho + 3p >0$. When the expansion is exponential the Universe accelerates, and the scalar field energy density is dominant. We also find that the existence of the Big Bang and a never ending expansion of the Universe constrain the equation of state of matter at large and small densities, respectively.Comment: Submitte to Phys. Lett. A. Minor changes were made to clarify some point

Large-time behavior of the weak solution to 3D Navier-Stokes equations

The weak solution to the Navier-Stokes equations in a bounded domain $D \subset \mathbb{R}^3$ with a smooth boundary is proved to be unique provided that it satisfies an additional requirement. This solution exists for all $t \geq 0$. In a bounded domain $D$ the solution decays exponentially fast as $t\to \infty$ if the force term decays at a suitable rate

Global attractors for the one dimensional wave equation with displacement dependent damping

We study the long-time behavior of solutions of the one dimensional wave equation with nonlinear damping coefficient. We prove that if the damping coefficient function is strictly positive near the origin then this equation possesses a global attractor

Note on Prodi-Serrin-Ladyzhenskaya type regularity criteria for the Navier-Stokes equations

X.Y. is partially supported by the Discovery Grant No. RES0020476 from NSERC.In this article we prove new regularity criteria of the Prodi-Serrin-Ladyzhenskaya type for the Cauchy problem of the three-dimensional incompressible Navier-Stokes equations. Our results improve the classical Lr(0,T;Ls) regularity criteria for both velocity and pressure by factors of certain nagative powers of the scaling invariant norms ||u||L3 and ||u||H1/2.PostprintPeer reviewe

Formation of fatty acids in photochemical conversions of saturated hydrocarbons

Abiogenic synthesis of fatty acids was studied in photochemical conversions of saturated hydrocarbons. It was shown that, in a hydrocarbon water CaCO3 suspension, the action of 254 nm UV rays caused the formation of fatty acids with a maximum number of carbon atoms in the chain not exceeding that in the initial hydrocarbon. Synthesis of acetic, propionic, butyric, valeric, caproic, enanthic and caprylic (in the case of octane) acids occurs in heptane water CaCO3 and octane water CaCO3 systems