ON THE REGULARITY OF THREE-DIMENSIONAL ROTATING EULER–BOUSSINESQ EQUATIONS

1999 ◽  
Vol 09 (07) ◽  
pp. 1089-1121 ◽  
Author(s):  
A. BABIN ◽  
A. MAHALOV ◽  
B. NICOLAENKO
Keyword(s):  
Initial Data ◽  
Three Dimensional ◽  
Fluid Flows ◽  
Boussinesq Equations ◽  
Stable Stratification ◽  
Primitive Equations ◽  
Time Interval ◽  
Regular Solutions ◽  
Time Intervals ◽  
Long Time

The 3-D rotating Boussinesq equations (the "primitive" equations of geophysical fluid flows) are analyzed in the asymptotic limit of strong stable stratification. The resolution of resonances and a nonstandard small divisor problem are the basis for error estimates for such fast singular oscillating limits. Existence on infinite time intervals of regular solutions to the viscous 3-D "primitive" equations is proven for initial data in Hα, α≥ 3/4. Existence on a long-time interval T*of regular solutions to the 3-D inviscid equations is proven for initial data in Hα, α > 5/2 (T*→∞ as the frequency of gravity waves →∞).

scholarly journals Optimal Perturbations of an Oceanic Vortex Lens

Fluids ◽  
2018 ◽  
Vol 3 (3) ◽  
pp. 63 ◽  
Author(s):  
Thomas Meunier ◽  
Claire Ménesguen ◽  
Xavier Carton ◽  
Sylvie Le Gentil ◽  
Richard Schopp
Keyword(s):  
Normal Mode ◽  
Growth Rates ◽  
Time Interval ◽  
Time Intervals ◽  
Horizontal Structure ◽  
Azimuthal Wave ◽  
Wave Numbers ◽  
Non Linear ◽  
Long Time ◽  
Short Time

The stability properties of a vortex lens are studied in the quasi geostrophic (QG) framework using the generalized stability theory. Optimal perturbations are obtained using a tangent linear QG model and its adjoint. Their fine-scale spatial structures are studied in details. Growth rates of optimal perturbations are shown to be extremely sensitive to the time interval of optimization: The most unstable perturbations are found for time intervals of about 3 days, while the growth rates continuously decrease towards the most unstable normal mode, which is reached after about 170 days. The horizontal structure of the optimal perturbations consists of an intense counter-shear spiralling. It is also extremely sensitive to time interval: for short time intervals, the optimal perturbations are made of a broad spectrum of high azimuthal wave numbers. As the time interval increases, only low azimuthal wave numbers are found. The vertical structures of optimal perturbations exhibit strong layering associated with high vertical wave numbers whatever the time interval. However, the latter parameter plays an important role in the width of the vertical spectrum of the perturbation: short time interval perturbations have a narrow vertical spectrum while long time interval perturbations show a broad range of vertical scales. Optimal perturbations were set as initial perturbations of the vortex lens in a fully non linear QG model. It appears that for short time intervals, the perturbations decay after an initial transient growth, while for longer time intervals, the optimal perturbation keeps on growing, quickly leading to a non-linear regime or exciting lower azimuthal modes, consistent with normal mode instability. Very long time intervals simply behave like the most unstable normal mode. The possible impact of optimal perturbations on layering is also discussed.

scholarly journals Design and interpretation of cell trajectory assays

2013 ◽  
Vol 10 (88) ◽  
pp. 20130630 ◽  
Author(s):  
Lucie G. Bowden ◽  
Matthew J. Simpson ◽  
Ruth E. Baker
Keyword(s):  
Cell Biology ◽  
Time Interval ◽  
Short Time Interval ◽  
Trajectory Data ◽  
Time Intervals ◽  
Different Types ◽  
Long Time ◽  
A Cell ◽  
Cell Trajectory ◽  
Short Time

Cell trajectory data are often reported in the experimental cell biology literature to distinguish between different types of cell migration. Unfortunately, there is no accepted protocol for designing or interpreting such experiments and this makes it difficult to quantitatively compare different published datasets and to understand how changes in experimental design influence our ability to interpret different experiments. Here, we use an individual-based mathematical model to simulate the key features of a cell trajectory experiment. This shows that our ability to correctly interpret trajectory data is extremely sensitive to the geometry and timing of the experiment, the degree of motility bias and the number of experimental replicates. We show that cell trajectory experiments produce data that are most reliable when the experiment is performed in a quasi-one-dimensional geometry with a large number of identically prepared experiments conducted over a relatively short time-interval rather than a few trajectories recorded over particularly long time-intervals.

Long-Time Solutions to Heat-Conduction Transients with Time-Dependent Inputs

1980 ◽  
Vol 102 (1) ◽  
pp. 115-120 ◽  
Author(s):  
H. T. Ceylan ◽  
G. E. Myers
Keyword(s):  
Heat Conduction ◽  
Lower Cost ◽  
Piecewise Linear ◽  
Three Dimensional ◽  
Linear Functions ◽  
Time Dependent ◽  
Time Interval ◽  
Piecewise Linear Functions ◽  
One Dimensional ◽  
Long Time

An economical method for obtaining long-time solutions to one, two, or three-dimensional heat-conduction transients with time-dependent forcing functions is presented. The conduction problem is spatially discretized by finite differences or by finite elements to obtain a system of first-order ordinary differential equations. The time-dependent input functions are each approximated by continuous, piecewise-linear functions each having the same uniform time interval. A set of response coefficients is generated by which a long-time solution can be carried out with a considerably lower cost than for conventional methods. A one-dimensional illustrative example is included.

Renormalization of the transformation operators of quantum electrodynamics

1954 ◽  
Vol 227 (1168) ◽  
pp. 94-102
Keyword(s):  
Quantum Electrodynamics ◽  
Matrix Theory ◽  
Transformation Operator ◽  
Time Interval ◽  
Matrix Elements ◽  
Finite Time Interval ◽  
Time Intervals ◽  
Starting Point ◽  
The Matrix ◽  
Long Time

The transformation operator of electrodynamics for a finite time interval with uncertain boundaries is represented by a continuous switching on and off of the charge. It is shown that its divergencies are the same as those appearing in the S matrix theory, and a covariant procedure is given for isolating their infinite parts. Provided Gupta’s renormalized Lagrangian is used as a starting point all the infinities may be removed. The coefficients of the counter terms are power series in the time-dependent charge with coefficients that are independent of the time interval being considered. The practice of approximating the matrix elements of the transformation operators for long time intervals by matrix elements of the S matrix is discussed and justified. In an appendix the extension of these results to the renormalizable meson theories is discussed.

Long time stability for the dispersive SQG equation and Boussinesq equations in Sobolev space Hs

2018 ◽  
Vol 22 (03) ◽  
pp. 1850063 ◽  
Author(s):  
Renhui Wan
Keyword(s):  
Sobolev Space ◽  
Initial Data ◽  
Global Regularity ◽  
Boussinesq Equations ◽  
Initial Condition ◽  
Decay Estimates ◽  
Time Stability ◽  
Long Time ◽  
Boussinesq Systems ◽  
Long Time Stability

Dispersive SQG equation have been studied by many works (see, e.g., [M. Cannone, C. Miao and L. Xue, Global regularity for the supercritical dissipative quasi-geostrophic equation with large dispersive forcing, Proc. Londen. Math. Soc. 106 (2013) 650–674; T. M. Elgindi and K. Widmayer, Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684; A. Kiselev and F. Nazarov, Global regularity for the critical dispersive dissipative surface quasi-geostrophic equation, Nonlinearity 23 (2010) 549–554; R. Wan and J. Chen, Global well-posedness for the 2D dispersive SQG equation and inviscid Boussinesq equations, Z. Angew. Math. Phys. 67 (2016) 104]), which is very similar to the 3D rotating Euler or Navier–Stokes equations. Long time stability for the dispersive SQG equation without dissipation was obtained by Elgindi–Widmayer [Sharp decay estimates for an anisotropic linear semigroup and applications to the surface quasi-geostrophic and inviscid Boussinesq systems, SIAM J. Math. Anal. 47 (2015) 4672–4684], where the initial condition [Formula: see text] [Formula: see text] plays a important role in their proof. In this paper, by using the Strichartz estimate, we can remove this initial condition. Namely, we only assume the initial data is in the Sobolev space like [Formula: see text]. As an application, we can also obtain similar result for the 2D Boussinesq equations with the initial data near a nontrivial equilibrium.

scholarly journals On the Existence of Long-Time Classical Solutions for the 2D Inviscid Boussinesq Equations

2021 ◽  
Vol 2021 ◽  
pp. 1-7
Author(s):  
Jiafa Xu ◽  
Lishan Liu
Keyword(s):  
Cauchy Problem ◽  
Initial Data ◽  
Life Span ◽  
Boussinesq Equations ◽  
Classical Solutions ◽  
Buoyancy Frequency ◽  
General Initial Data ◽  
The Cauchy Problem ◽  
Long Time

In this paper, we consider the Cauchy problem for the 2D inviscid Boussinesq equations with N being the buoyancy frequency. It is proved that for general initial data u 0 ∈ H s with s > 3 , the life span of the classical solutions satisfies T > C ln     N 3 / 4 .

Stability of Computational Algorithms Used in Molecular Dynamics Simulations

1995 ◽  
Vol 117 (3) ◽  
pp. 531-534 ◽  
Author(s):  
Akira Satoh
Keyword(s):  
Transport Coefficients ◽  
Three Dimensional ◽  
Divergence Time ◽  
Linear Functions ◽  
Time Interval ◽  
Thermodynamic Quantities ◽  
Number Density ◽  
Time Intervals ◽  
Dynamics Simulations ◽  
The Relationship

The present study focuses on a three-dimensional Lennard-Jones system in a thermodynamic equilibrium in order to discuss divergence processes, the relationship between time intervals and divergence times, and the influence of time intervals on thermodynamic quantities and transport coefficients under various number density and temperature. It is found that the velocities of molecules in a system gradually increase with time until the system suddenly diverges exponentially. The time interval-divergence time relationship can be expressed in approximate terms as linear functions if the data are plotted on logarithmic scales, and the system diverges more easily as temperature or number density increases. Thermodynamic quantities show the influence of large time intervals more clearly than do transport coefficients.

scholarly journals Corrosion and Inhibition Effects of Mild Steel in Hydrochloric Acid Solutions Containing Organophosphonic Acid

2013 ◽  
Vol 2013 ◽  
pp. 1-5 ◽  
Author(s):  
Manish Gupta ◽  
Jyotsna Mishra ◽  
K. S. Pitre
Keyword(s):  
Corrosion Rate ◽  
Mild Steel ◽  
Electrochemical Techniques ◽  
Time Interval ◽  
Time Intervals ◽  
Voltammetric Method ◽  
Average Value ◽  
Long Time ◽  
In The Beginning ◽  
Short Time

A study has been made on the mechanism of corrosion of mild steel and the effect of nitrilo trimethylene phosphonic (NTMP) acid as a corrosion inhibitor in acidic medium, that is, 10% HC1 using the weight loss method and electrochemical techniques, that is, potentiodynamic and galvanostatic polarization measurements. Although corrosion is a long-time process, but it takes place at a faster rate in the beginning which goes on decreasing with due course of time. The above-mentioned methods of corrosion rate determination furnish an average value for a long-time interval. Looking at the versatility and minimum detection limit of the voltammetric method, the authors have developed a new voltammetric method for the determination of corrosion rate at short-time intervals. The results of corrosion of mild steel in 10% HC1 solution with and without NTMP inhibitor at short-time intervals have been reported. The corrosion inhibition efficiency of NTMP is 93% after 24 h.

A squeezing property and its applications to a description of long-time behaviour in the three-dimensional viscous primitive equations

2014 ◽  
Vol 144 (4) ◽  
pp. 711-729 ◽  
Author(s):  
Igor Chueshov
Keyword(s):  
Three Dimensional ◽  
Ocean Dynamics ◽  
Primitive Equations ◽  
Time Behaviour ◽  
Time Dynamics ◽  
Squeezing Property ◽  
Long Time Behaviour ◽  
New Information ◽  
Determining Modes ◽  
Long Time

We consider the three-dimensional viscous primitive equations with periodic boundary conditions. These equations arise in the study of ocean dynamics and generate a dynamical system in a Sobolev H1-type space. Our main result establishes the so-called squeezing property in the Ladyzhenskaya form for this system. As a consequence of this property we prove the finiteness of the fractal dimension of the corresponding global attractor, the existence of a finite number of determining modes and the ergodicity of a related random kick model. All these results provide new information concerning the long-time dynamics of oceanic motion.

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