Rotating Navier-Stokes Equations in $${\mathbb R}^{3}_{+}$$ with Initial Data Nondecreasing at Infinity: The Ekman Boundary Layer Problem

We prove time local existence and uniqueness of solutions to a boundary layer problem in a rotating frame around the stationary solution called the Ekman spiral. We choose initial data in the vector-valued homogeneous Besov space for 2 <  p <  ∞. Here the L ^p -integrability is imposed in the normal direction, while we may have no decay in tangential components, since the Besov space contains nondecaying functions such as almost periodic functions. A crucial ingredient is theory for vector-valued homogeneous Besov spaces. For instance we provide and apply an operator-valued bounded H ^∞-calculus for the Laplacian in for a general Banach space .     

Flow Visualization of Superbursts and of the Log-Layer in a DNS at $$ \operatorname{Re} _{\tau } = 950 $$

This paper uses direct numerical simulations (DNS) of turbulent flow in a channel at (Del álamo, Jiménez, Zandonade, Moser J Fluid Mech 500:135–144, 2004) to provide a picture of the turbulent structures making large contributions to the Reynolds shear stress. Considerable work of this type has been done for the viscous wall region at smaller , for which a log-layer does not exist. Recent PIV measurements of turbulent velocity fluctuations in a plane parallel to the direction of flow have emphasized the dominant contribution of large scale structures in the outer flow. This prompted Hanratty and Papavassiliou (The role of wall vortices in producing turbulence. In: Panton, R.L. (ed) Self-sustaining Mechanism of Wall Turbulence. Computational Mechanics Publications, Southampton, pp. 83–108, 1997) to use DNS at to examine these structures in a plane perpendicular to the direction of flow. They identified plumes which extend from the wall to the center of a channel. The data at are used to explore these results further, to examine the structure of the log-layer, and to test present notions about the viscous wall layer.     

Convergence to Steady States of Solutions of Non-autonomous Heat Equations in $$\mathbb{R}^{N}$$

Under certain assumptions on f and g we prove that positive, global and bounded solutions u of the non-autonomous heat equation

Dobrushin States in the $${\phi^4_1}$$ Model

We consider the van der Waals free energy functional in a bounded interval with inhomogeneous Dirichlet boundary conditions imposing the two stable phases at the endpoints. We compute the asymptotic free energy cost, as the length of the interval diverges, of shifting the interface from the midpoint. We then discuss the effect of thermal fluctuations by analyzing the -measure with Dobrushin boundary conditions. In particular, we obtain a non-trivial limit in a suitable scaling in which the length of the interval diverges and the temperature vanishes. The limiting state is not translation invariant and describes a localized interface. This result can be seen as the probabilistic counterpart of the variational convergence of the associated excess free energy.     

Euler–Poisson Systems as Action-Minimizing Paths in the Wasserstein Space

This paper uses a variational approach to establish existence of solutions (σ _t , v _t ) for the 1-d Euler–Poisson system by minimizing an action. We assume that the initial and terminal points σ ₀, σ _T are prescribed in , the set of Borel probability measures on the real line, of finite second-order moments. We show existence of a unique minimizer of the action when the time interval [0,T] satisfies T < π. These solutions conserve the Hamiltonian and they yield a path t → σ _t in . When σ _t  = δ _y(t) is a Dirac mass, the Euler–Poisson system reduces to . The kinetic version of the Euler–Poisson, that is the Vlasov–Poisson system was studied in Ambrosio and Gangbo (Comm Pure Appl Math, to appear) as a Hamiltonian system. WG gratefully acknowledges the support provided by NSF grants DMS-02-00267, DMS-03-54729 and DMS-06-00791. TN gratefully acknowledges the postdoctoral support provided by NSF grants DMS-03- 54729 and the School of Mathematics. AT gratefully acknowledges the support provided by the School of Mathematics.     

A nonlinear kp-εp particle two-scale turbulence model and its application

A particle nonlinear two-scale turbulence model is proposed for simulating the anisotropic turbulent two-phase flow. The particle kinetic energy equation for two-scale fluctuation, particle energy transfer rate equation for large-scale fluctuation, and particle turbulent kinetic energy dissipation rate equation for small-scale fluctuation are derived and closed. This model is used to simulate gas–particle flows in a sudden-expansion chamber. The simulation is compared with the experiment and with those obtained by using another two kinds of tow-phase turbulence model, such as the single-scale two-phase turbulence model and the particle two-scale second-order moment (USM) two-phase turbulence model. It is shown that the present model gives simulation in much better agreement with the experiment than the single-scale two-phase turbulence model does and is almost as good as the particle two-scale USM turbulence model. The project supported by China Postdoctoral Science Foundation (2004036239).     

Nonparabolic Asymptotic Limits of Solutions of the Heat Equation on $${\mathbb{R}}^N$$

In this paper, we construct solutions u(t,x) of the heat equation on such that has nontrivial limit points in as t → ∞ for certain values of μ > 0 and β > 1/2. We also show the existence of solutions of this type for nonlinear heat equations.      

The Impact of Media on the Control of Infectious Diseases

We develop a three dimensional compartmental model to investigate the impact of media coverage to the spread and control of infectious diseases (such as SARS) in a given region/area. Stability analysis of the model shows that the disease-free equilibrium is globally-asymptotically stable if a certain threshold quantity, the basic reproduction number (), is less than unity. On the other hand, if , it is shown that a unique endemic equilibrium appears and a Hopf bifurcation can occur which causes oscillatory phenomena. The model may have up to three positive equilibria. Numerical simulations suggest that when and the media impact is stronger enough, the model exhibits multiple positive equilibria which poses challenge to the prediction and control of the outbreaks of infectious diseases. Research supported by the NNSF of China (10471066). Research supported by NSERC, MITACS and CFI/OIT of Canada.     

The Stationary Oseen Equations in $${\mathbb{R}}^{3}$$ . An Approach in Weighted Sobolev Spaces

In this paper we solve the stationary Oseen equations in . The behavior of the solutions at infinity is described by setting the problem in weighted Sobolev spaces including anisotropic weights. The study is based on a L^p theory for 1 < p < ∞.     

Vector potential–vorticity relationship for the Stokes flows: application to the Stokes eigenmodes in 2D/3D closed domain

The unsteady dynamics of the Stokes flows, where , is shown to verify the vector potential–vorticity ( ) correlation , where the field is the pressure-gradient vector potential defined by . This correlation is analyzed for the Stokes eigenmodes, , subjected to no-slip boundary conditions on any two-dimensional (2D) closed contour or three-dimensional (3D) surface. It is established that an asymptotic linear relationship appears, verified in the core part of the domain, between the vector potential and vorticity, , where is a constant offset field, possibly zero.     

Invertible Contractions and Asymptotically Stable ODE’S that are not \mathcal{C}^{1}-Linearizable

We present an example of a contraction diffeomorphism in infinite dimensions that is not -linearizable, and we construct a regular ordinary differential equation in a Hilbert space whose time-one map is that diffeomorphism. With this we have an example of an asymptotically stable ODE that is not -conjugate to its linear part.     

Computational modeling of vibration damping in SMA wires

Through a mathematical and computational model of the physical behavior of shape memory alloy wires, this study shows that localized heating and cooling of such materials provide an effective means of damping vibrational energy. The thermally induced pseudo-elastic behavior of a shape memory wire is modeled using a continuum thermodynamic model and solved computationally as described by the authors in [23]. Computational experiments confirm that up to of an initial shock of vibrational energy can be eliminated at the onset of a thermally-induced phase transformation through the use of spatially-distributed transformation regions along the length of a shape memory alloy wire.Received: 30 May 2003, Accepted: 20 December 2003, Published online: 16 April 2004PACS: 02.60.Cb, 68.45.Kg, 64.70.Kb Correspondence to: Petr Klouek.The first two authors were supported in part by the grant NSF DMS-0107539, by the Los Alamos National Laboratory Computer Science Institute (LACSI) through LANL contract number 03891-99-23, as part of the prime contract W-7405-ENG-36 between the Department of Energy and the Regents of the University of California, by the grant NASA SECTP NAG5-8136, and by the grant from Schlumberger Foundation. The work of the second author was performed in part under the auspices of the U.S. Department of Energy by the University of California, Lawrence Livermore National Laboratory under Contract No. W-7405-Eng-48. All three authors were supported in part by a grant from the TRW Foundation. The computations in this paper were performed on a 16 processor SGI Origin 2000, which was partly funded by the NSF SCREMS grant DMS-9872009.     

Solutions for Linear Conservation Laws with Velocity Fields in $$L^{\rm \infty}$$

A Space-Time Integrated Least Squares (STILS) method is derived for solving the linear conservation law with a velocity field in . An existence and uniqueness result is given for the solution of this equation. A maximum principle is established and finally a comparison with a renormalized solution is presented.     

Analysis and control of a class of uncertain linear periodic discrete-time systems

Feedback control problems for linear periodic systems （LPSs） with interval- type parameter uncertainties are studied in the discrete-time domain. First, the stability analysis and stabilization problems are addressed. Conditions based on the linear matrices inequality （LMI） for the asymptotical stability and state feedback stabilization, respec-tively, are given. Problems of L2-gain analysis and control synthesis are studied. For the L2-gain analysis problem, we obtain an LMI-based condition such that the autonomous uncertain LPS is asymptotically stable and has an L2-gain smaller than a positive scalar γ. For the control synthesis problem, we derive an LMI-based condition to build a state feedback controller ensuring that the closed-loop system is asymptotically stable and has an L2-gain smaller than the positive scalar γ. All the conditions are necessary and sufficient.     

A Non-Newtonian Fluid with Navier Boundary Conditions

We consider in this paper the equations of motion of third grade fluids on a bounded domain of or with Navier boundary conditions. Under the assumption that the initial data belong to the Sobolev space H ², we prove the existence of a global weak solution. In dimension two, the uniqueness of such solutions is proven. Additional regularity of bidimensional initial data is shown to imply the same additional regularity for the solution. No smallness condition on the data is assumed.     

Control for going from hovering to small speed flight of a model insect

The longitudinal steady-state control for going from hovering to small speed flight of a model insect is studied, using the method of computational fluid dynamics to compute the aerodynamic derivatives and the techniques based on the linear theories of stability and control for determining the non-zero equilibrium points. Morphological and certain kinematical data of droneflies are used for the model insect. A change in the mean stroke angle （δФ） results in a horizontal forward or backward flight; a change in the stroke amplitude （δФ） or a equal change in the down- and upstroke angles of attack （δα1） results in a vertical climb or decent; a proper combination of δФ and δФ controls （or δФ and δα1 controls） can give a flight of any （small） speed in any desired direction.     

On the dynamic behaviour of an elastically supported beam of infinite length, loaded by a concentrated force

Summary An elastically supported beam of infinite length, initially at rest, carries a variable concentrated force at a prescribed point A. General expressions are given for the deflection and the bending moment at A (6.3 and 6.4). Three special cases are considered; the first one is defined by =0 for and =K=const. for ; the second one by =0 for 0 > > , given function of for 0 ; the third one applies to problems in which, during the period of impact, itself is an unknown. The results given here may be of use in those railway-engineering problems in which a rail can be considered as a beam of infinite length, and in which the supporting ground has the required properties.     

Full Measure Reducibility for Generic One-parameter Family of Quasi-periodic Linear Systems

Let be the set of m × m matrices A(λ) depending analytically on a parameter λ in a closed interval . Consider one-parameter families of quasi-periodic linear differential equations: , where is analytic and sufficiently small. We prove that there is an open and dense set in , such that for each the equation can be reduced to an equation with constant coefficients by a quasi-periodic linear transformation for almost all in Lebesgue measure sense provided that g is sufficiently small. The result gives an affirmative answer to a conjecture of Eliasson (In: Proceeding of Symposia in Pure Mathematics). Dedicated to Professor Zhifen Zhang on the occasion of her 80th birthday     

Asymptotic Behaviour of a Semilinear Elliptic System with a Large Exponent

Consider the problem where Ω is a bounded convex domain in , N > 2, with smooth boundary . We study the asymptotic behaviour of the least energy solutions of this system as . We show that the solution remain bounded for p large. In the limit, we find that the solution develops one or two peaks away from the boundary, and when a single peak occurs, we have a characterization of its location.This research was supported by FONDECYT 1061110 and 3040059.     

The Toda System and Clustering Interfaces in the Allen–Cahn equation

We consider the Allen–Cahn equation in a bounded, smooth domain Ω in , under zero Neumann boundary conditions, where is a small parameter. Let Γ₀ be a segment contained in Ω, connecting orthogonally the boundary. Under certain nondegeneracy and nonminimality assumptions for Γ₀, satisfied for instance by the short axis in an ellipse, we construct, for any given N ≥ 1, a solution exhibiting N transition layers whose mutual distances are and which collapse onto Γ₀ as . Asymptotic location of these interfaces is governed by a Toda-type system and yields in the limit broken lines with an angle at a common height and at main order cutting orthogonally the boundary.