Interplay between the Beale-Kato-Majda theorem and the analyticity-strip method to investigate numerically the incompressible Euler singularity problem

Numerical simulations of the incompressible Euler equations are performed using the Taylor-Green vortex initial conditions and resolutions up to $4096^3$. The results are analyzed in terms of the classical analyticity strip method and Beale, Kato and Majda (BKM) theorem. A well-resolved acceleration of the time-decay of the width of the analyticity strip $\delta(t)$ is observed at the highest resolution for $3.7<t<3.85$ while preliminary 3D visualizations show the collision of vortex sheets. The BKM criterium on the power-law growth of supremum of the vorticity, applied on the same time-interval, is not inconsistent with the occurrence of a singularity around $t \simeq 4$. These new findings lead us to investigate how fast the analyticity strip width needs to decrease to zero in order to sustain a finite-time singularity consistent with the BKM theorem. A new simple bound of the supremum norm of vorticity in terms of the energy spectrum is introduced and used to combine the BKM theorem with the analyticity-strip method. It is shown that a finite-time blowup can exist only if $\delta(t)$ vanishes sufficiently fast at the singularity time. In particular, if a power law is assumed for $\delta(t)$ then its exponent must be greater than some critical value, thus providing a new test that is applied to our $4096^3$ Taylor-Green numerical simulation. Our main conclusion is that the numerical results are not inconsistent with a singularity but that higher-resolution studies are needed to extend the time-interval on which a well-resolved power-law behavior of $\delta(t)$ takes place, and check whether the new regime is genuine and not simply a crossover to a faster exponential decay

Recent Advances Concerning Certain Class of Geophysical Flows

This paper is devoted to reviewing several recent developments concerning certain class of geophysical models, including the primitive equations (PEs) of atmospheric and oceanic dynamics and a tropical atmosphere model. The PEs for large-scale oceanic and atmospheric dynamics are derived from the Navier-Stokes equations coupled to the heat convection by adopting the Boussinesq and hydrostatic approximations, while the tropical atmosphere model considered here is a nonlinear interaction system between the barotropic mode and the first baroclinic mode of the tropical atmosphere with moisture. We are mainly concerned with the global well-posedness of strong solutions to these systems, with full or partial viscosity, as well as certain singular perturbation small parameter limits related to these systems, including the small aspect ratio limit from the Navier-Stokes equations to the PEs, and a small relaxation-parameter in the tropical atmosphere model. These limits provide a rigorous justification to the hydrostatic balance in the PEs, and to the relaxation limit of the tropical atmosphere model, respectively. Some conditional uniqueness of weak solutions, and the global well-posedness of weak solutions with certain class of discontinuous initial data, to the PEs are also presented.Comment: arXiv admin note: text overlap with arXiv:1507.0523

Uncertainty estimates and L_2 bounds for the Kuramoto-Sivashinsky equation

We consider the Kuramoto-Sivashinsky (KS) equation in one spatial dimension with periodic boundary conditions. We apply a Lyapunov function argument similar to the one first introduced by Nicolaenko, Scheurer, and Temam, and later improved by Collet, Eckmann, Epstein and Stubbe, and Goodman, to prove that ||u||_2 < C L^1.5. This result is slightly weaker than that recently announced by Giacomelli and Otto, but applies in the presence of an additional linear destabilizing term. We further show that for a large class of Lyapunov functions \phi the exponent 1.5 is the best possible from this line of argument. Further, this result together with a result of Molinet gives an improved estimate for L_2 boundedness of the Kuramoto-Sivashinsky equation in thin rectangular domains in two spatial dimensions.Comment: 17 pages, 1 figure; typos corrected, references added; figure modifie

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

WHEAT STRAW CONVERSION BY ENZYMATIC SYSTEM OF GANODERMA LUCIDUM

The purpose of this study was to resolve the question of whether various nitrogen sources and concentrations affect characteristics of selected G. lucidum ligninolytic enzymes participating in wheat straw fermentation. This is the first study reporting the presence of versatile peroxidase activity in crude extract of G. lucidum culture, as well as isoforms profile of Mn-oxidizing peroxidases. NH4NO3 was the optimum nitrogen source for laccase and Mn-dependent peroxidase activity, while peptone was the optimum one for versatile peroxidase activity. Four bands with laccase activity were obtained by native PAGE and IEF separations from medium enriched with inorganic nitrogen source, and only two bands from medium containing organic source. Medium composition was not shown to affect isoenzyme patterns of Mn-oxidizing peroxidases. Four isoforms of Mn-dependent peroxidase and three of versatile peroxidase were obtained on native PAGE. By IEF separation, five isoforms of Mn-dependent peroxidase and only two of versatile peroxidase were observed. The results demonstrated that G. lucidum has potential for mineralization and transformation of various agricultural residues and should take more significant participation in large-scale biotechnological processes

Existence of global strong solutions to a beam-fluid interaction system

We study an unsteady non linear fluid-structure interaction problem which is a simplified model to describe blood flow through viscoleastic arteries. We consider a Newtonian incompressible two-dimensional flow described by the Navier-Stokes equations set in an unknown domain depending on the displacement of a structure, which itself satisfies a linear viscoelastic beam equation. The fluid and the structure are fully coupled via interface conditions prescribing the continuity of the velocities at the fluid-structure interface and the action-reaction principle. We prove that strong solutions to this problem are global-in-time. We obtain in particular that contact between the viscoleastic wall and the bottom of the fluid cavity does not occur in finite time. To our knowledge, this is the first occurrence of a no-contact result, but also of existence of strong solutions globally in time, in the frame of interactions between a viscous fluid and a deformable structure

Prophage induction reduces Shiga toxin producing \u3ci\u3eEscherichia coli\u3c/i\u3e (STEC) and Salmonella enterica on tomatoes and spinach: A model study

Fresh produce is increasingly implicated in foodborne outbreaks and most fresh produce is consumed raw, emphasizing the need to develop non-thermal methods to control foodborne pathogens. This study investigates bacterial cell lysis through induction of prophages as a novel approach to control foodborne bacterial pathogens on fresh produce. Shiga toxin producing Escherichia coli (STEC) and Salmonella enterica isolates were exposed to different prophage inducers (i.e. mitomycin C or streptonigrin) and growth of the cells was monitored by measuring the optical density (OD600) during incubation at 37C. Beginning at three hours after addition of the inducer, all concentrations (0.5, 1, 2 mg/mL) of mitomycin C, or 2 mg/mL streptonigrin significantly reduced the OD600 in broth cultures, in a concentration dependent manner, relative to cultures where no inducer was added. PCR confirmed bacterial release of induced bacteriophages and demonstrated that a single compound could successfully induce multiple types of prophages. The ability of mitomycin C to induce prophages in STEC O157:H7 and in S. enterica (serovars Typhimurium and Newport) on fresh produce was evaluated by inoculating red greenhouse tomatoes or spinach leaves with 5 x 107 and 5 x 108 colony forming units, respectively. After allowing time for the inoculum to dry on the fresh produce samples, 6 mg/mL mitomycin C was sprayed onto each sample, while control samples were sprayed with water. Following overnight incubation at 4C, the bacterial cells were recovered and plate counts were performed. A 3 log reduction in STEC O157:H7 cells was observed on tomatoes sprayed with mitomycin C compared to those sprayed with water, while a 1 log reduction was obtained on spinach. Similarly, spraying mitomycin C on tomatoes and spinach inoculated with S. enterica isolates resulted in a 1-1.5 log and 2 log reduction, respectively. These findings serve as a proof of concept that prophage induction can effectively control bacterial foodborne pathogens on fresh produce

On formation of a locally self-similar collapse in the incompressible Euler equations

The paper addresses the question of existence of a locally self-similar blow-up for the incompressible Euler equations. Several exclusion results are proved based on the $L^p$-condition for velocity or vorticity and for a range of scaling exponents. In particular, in $N$ dimensions if in self-similar variables $u \in L^p$ and u \sim \frac{1}{t^{\a/(1+\a)}}, then the blow-up does not occur provided \a >N/2 or -1<\a\leq N/p. This includes the $L^3$ case natural for the Navier-Stokes equations. For \a = N/2 we exclude profiles with an asymptotic power bounds of the form |y|^{-N-1+\d} \lesssim |u(y)| \lesssim |y|^{1-\d}. Homogeneous near infinity solutions are eliminated as well except when homogeneity is scaling invariant.Comment: A revised version with improved notation, proofs, etc. 19 page

Viscous-Inviscid Interactions in a Boundary-Layer Flow Induced by a Vortex Array

In this paper we investigate the asymptotic validity of boundary layer theory. For a flow induced by a periodic row of point-vortices, we compare Prandtl's solution to Navier-Stokes solutions at different $Re$ numbers. We show how Prandtl's solution develops a finite time separation singularity. On the other hand Navier-Stokes solution is characterized by the presence of two kinds of viscous-inviscid interactions between the boundary layer and the outer flow. These interactions can be detected by the analysis of the enstrophy and of the pressure gradient on the wall. Moreover we apply the complex singularity tracking method to Prandtl and Navier-Stokes solutions and analyze the previous interactions from a different perspective

The Inviscid Limit and Boundary Layers for Navier-Stokes Flows

The validity of the vanishing viscosity limit, that is, whether solutions of the Navier-Stokes equations modeling viscous incompressible flows converge to solutions of the Euler equations modeling inviscid incompressible flows as viscosity approaches zero, is one of the most fundamental issues in mathematical fluid mechanics. The problem is classified into two categories: the case when the physical boundary is absent, and the case when the physical boundary is present and the effect of the boundary layer becomes significant. The aim of this article is to review recent progress on the mathematical analysis of this problem in each category.Comment: To appear in "Handbook of Mathematical Analysis in Mechanics of Viscous Fluids", Y. Giga and A. Novotn\'y Ed., Springer. The final publication is available at http://www.springerlink.co