The Navier-Stokes limit of stationary solutions of the nonlinear Boltzmann equation

We consider the flow of a gas in a channel whose walls are kept at fixed (different) temperatures. There is a constant external force parallel to the boundaries which may themselves also be moving. The system is described by the stationary Boltzmann equation to which are added Maxwellian boundary conditions with unit accommodation coefficient. We prove that when the temperature gap, the relative velocity of the planes, and the force are all sufficiently small, there is a solution which converges, in the hydrodynamic limit, to a local Maxwellian with parameters given by the stationary solution of the corresponding compressible Navier-Stokes equations with no-slip voundary conditions. Corrections to this Maxwellian are obtained in powers of the Knudsen number with a controlled remainder.     

Stationary shapes of deformable particles moving at low Reynolds numbers

We introduce an iterative solution scheme in order to calculate stationary shapes of deformable elastic capsules which are steadily moving through a viscous fluid at low Reynolds numbers. The iterative solution scheme couples hydrodynamic boundary integral methods and elastic shape equations to find the stationary axisymmetric shape and the velocity of an elastic capsule moving in a viscous fluid governed by the Stokes equation. We use this approach to systematically study dynamical shape transitions of capsules with Hookean stretching and bending energies and spherical resting shape sedimenting under the influence of gravity or centrifugal forces. We find three types of possible axisymmetric stationary shapes for sedimenting capsules with fixed volume: a pseudospherical state, a pear-shaped state, and buckled shapes. Capsule shapes are controlled by two dimensionless parameters, the Föppl-von-Kármán number characterizing the elastic properties and a Bond number characterizing the driving force. For increasing gravitational force the spherical shape transforms into a pear shape. For very large bending rigidity (very small Föppl-von-Kármán number) this transition is discontinuous with shape hysteresis. The corresponding transition line terminates, however, in a critical point, such that the discontinuous transition is not present at typical Föppl-von-Kármán numbers of synthetic capsules. In an additional bifurcation, buckled shapes occur upon increasing the gravitational force.     

Phase Transitions in Edge-Weighted Exponential Random Graphs: Near-Degeneracy and Universality

We consider the boundary value problem of the stationary transport equation in the slab domain of general dimensions. In this paper, we discuss the relation between discontinuity of the incoming boundary data and that of the solution to the stationary transport equation. We introduce two conditions posed on the boundary data so that discontinuity of the boundary data propagates along positive characteristic lines as that of the solution to the stationary transport equation. Our analysis does not depend on the celebrated velocity averaging lemma, which is different from previous works. We also introduce an example in two dimensional case which shows that piecewise continuity of the boundary data is not a sufficient condition for the main result.     

Stability of nonlinear diffusion

The nonlinear diffusion equation in bounded geometry with time-independent boundary conditions has a uniquely determined stationary solution. We show that this solution is dynamically stable in the sense of Liapunov. Any initial distribution tends to the stationary one as time goes on. It is shown that the application of the Glansdorff-Prigogine stability criterion requires a more elaborate analysis. We develop a variational procedure which has application in a wide range of nonlinear transport problems.     

Using Asymptotic Analysis for Developing a One-Dimensional Substance Transport Model in the Case of Spatial Heterogeneity

We study a solution with an internal transition layer of a one-dimensional boundary value problem for the stationary reaction–advection–diffusion differential equation that arises in mathematical modeling of transport phenomena in the surface layer of the atmosphere in the case of non-uniform vegetation on the assumption of space isotropy along one of the horizontal axes and neutral atmospheric stratification. The parameters of the model at which a boundary value problem has a stable stationary solution with an internal transition layer localized near the boundary between different vegetation types are provided. The existence of such a solution and its local Lyapunov stability and uniqueness are proven. The results can be used for developing multidimensional substance transfer models in the case of a spatial heterogeneity.     

Asymptotic stability of stationary states in the wave equation coupled to a nonrelativistic particle

We consider the Hamiltonian system consisting of a scalar wave field and a single particle coupled in a translation invariant manner. The point particle is subjected to an external potential. The stationary solutions of the system are a Coulomb type wave field centered at those particle positions for which the external force vanishes. It is assumed that the charge density satisfies the Wiener condition, which is a version of the “Fermi Golden Rule.” We prove that in the large time approximation, any finite energy solution, with the initial state close to the some stable stationary solution, is a sum of this stationary solution and a dispersive wave which is a solution of the free wave equation.     

Asymptotic Stability of the Stationary Solution to the Compressible Navier–Stokes Equations in the Half Space

We investigate the existence and the asymptotic stability of a stationary solution to the initial boundary value problem for the compressible Navier–Stokes equation in a half space. The main concern is to analyze the phenomena that happens when the fluid blows out through the boundary. Thus, it is natural to consider the problem in the Eulerian coordinate. We have obtained the two results for this problem. The first result is concerning the existence of the stationary solution. We present the necessary and sufficient condition which ensures the existence of the stationary solution. Then it is shown that the stationary solution is time asymptotically stable if an initial perturbation is small in the suitable Sobolev space. The second result is proved by using an L²-energy method with the aid of the Poincaré type inequality.The second author's work was supported in part by Grant-in-Aid for Scientific Research (C)(2) 14540200 of the Ministry of Education, Culture, Sports, Science and Technology and the third author's work was supported by JSPS postdoctoral fellowship under P99217.     

Initial boundary value problems for the equations of motion of compressible viscous and heat-conductive fluids

The equations of motion of compressible viscous and heat-conductive fluids are investigated for initial boundary value problems on the half space and on the exterior domain of any bounded region. The global solution in time is proved to exist uniquely and approach the stationary state ast→∞, provided the prescribed initial data and the external force are sufficiently small.     

Weakly nonlinear theory of grain boundary motion in patterns with crystalline symmetry

We study the motion of a grain boundary separating two otherwise stationary domains of hexagonal symmetry. Starting from an order parameter equation, a multiple scale analysis leads to an analytical equation of motion for the boundary that shares many properties with that of a crystalline solid. We find that defect motion is generically opposed by a pinning force that arises from nonadiabatic corrections to the standard amplitude equations. The magnitude of this force depends sharply on the misorientation angle between adjacent domains: the most easily pinned grain boundaries are those with a low angle (typically 4 degrees < or =theta;< or =8 degrees ). Although pinning effects may be small, they can be orders of magnitude larger than those present in smectic phases.     

Long-Time Asymptotics for Solutions of the NLS Equation with a Delta Potential and Even Initial Data: Announcement of Results

We consider the one-dimensional focusing nonlinear Schrödinger equation (NLS) with a delta potential and even initial data. The problem is equivalent to the solution of the initial/boundary problem for NLS on a half-line with Robin boundary conditions at the origin. We follow the method of Bikbaev and Tarasov which utilizes a Bäcklund transformation to extend the solution on the half-line to a solution of the NLS equation on the whole line. We study the asymptotic stability of the stationary 1-soliton solution of the equation under perturbation by applying the nonlinear steepest-descent method for Riemann?CHilbert problems introduced by Deift and Zhou. Our work strengthens, and extends, the earlier work on the problem by Holmer and Zworski.     

Effects of boundary conditions on spatially periodic states

We investigate analytically and numerically the influence of linear homogeneous boundary conditions on the stationary solutions of a simple model for cellular pattern formation in one dimension. For all boundary conditions there exists in a reduced wavenumber band at least one static solution where the amplitude falls below its bulk value near the boundary (“Type-I” solution). A linear stability analysis of the uniform state at threshold reveals that Type-I solutions are often unstable. Then there exists in the full Eckhaus-stable band, a static solution where the amplitude rises above its bulk value near the boundary (“Type-II” solution), or a limit-cycle solution where the amplitude near the boundary oscillates. These solutions bifurcate from the homogeneous state below the bulk threshold and therefore remain finite at threshold.     

Half-space problem of the Boltzmann equation for charged particles

For two particular collision kernels, we explicitly solve the one-dimensional stationary half-space boundary value problem of the linear Boltzmann equation including a constant external field via an extension of Case's eigenfunction technique. In the first collision model we reproduce a solution recently obtained by Cercignani; in the second model the solution of the stationary boundary value problem is presented for the first time.     

Stationary Waves for the Discrete Boltzmann Equation in the Half Space with Reflective Boundaries

The present paper is concerned with stationary solutions for discrete velocity models of the Boltzmann equation with reflective boundary condition in the first half space. We obtain a sufficient condition that guarantees the existence and the uniqueness of stationary solutions satisfying the reflective boundary condition as well as the spatially asymptotic condition given by a Maxwellian state. First, the sufficient condition is obtained for the linearized system. Then, this result is applied to prove the existence theorem for the nonlinear equation through the contraction mapping principle. Also, it is shown that the stationary solution approaches the asymptotic Maxwellian state exponentially as the spatial variable tends to infinity. Moreover, we show the time asymptotic stability of the stationary solutions. In the proof, we employ the standard energy method to obtain a priori estimates for nonstationary solutions. The exponential convergence at the spatial asymptotic state of the stationary solutions gives essential information to handle some error terms. Then we discuss some concrete models of the Boltzmann type as an application of our general theory. Received: 7 July 1999 / Accepted: 3 November 1999     

Lattice Gas Model in Random Medium and Open Boundaries: Hydrodynamic and Relaxation to the Steady State

We consider a lattice gas interacting by the exclusion rule in the presence of a random field given by i.i.d. bounded random variables in a bounded domain in contact with particles reservoir at different densities. We show, in dimensions d≥3, that the rescaled empirical density field almost surely, with respect to the random field, converges to the unique weak solution of a quasilinear parabolic equation having the diffusion matrix determined by the statistical properties of the external random field and boundary conditions determined by the density of the reservoir. Further we show that the rescaled empirical density field, in the stationary regime, almost surely with respect to the random field, converges to the solution of the associated stationary transport equation.     

The Kac Equation with a Thermostatted Force Field

We consider the Kac equation with a thermostatted force field and prove the existence of a global in time solution that converges weakly to a stationary state. As there is no an obvious candidate for the entropy functional, in this case, the convergence result is obtained via Fourier transform techniques.     

Navier-stokes equations for compressible fluids: Global existence and qualitative properties of the solutions in the general case

We consider the equations which describe the motion of a viscous compressible fluid, taking into consideration the case of inflow and/or outflow through the boundary. By means of some a priori estimates we prove the existence of a global (in time) solution. Moreover, as a consequence of a stability result, we show that there exist a periodic solution and a stationary solution.     

Brownian motion of a charged particle in a magnetic field

We develop and numerically illustrate an exact solution of the multivariate, stochastic, differential equations that govern the velocity and position of a charged particle in a plane normal to a uniform, stationary, magnetic field. The equations self-consistently incorporate the Lorentz force into an Ornstein-Uhlenbeck collision model. Properties of the solution in the infinite dissipation limit are explored and the spectral energy density function is found     

On the free boundary problem for stationary compressible Navier-Stokes equations

We consider the equations which describe a stationary motion of a viscous compressible barotropic fluid in a bounded domain in ³ with a free boundary determined by the surface tension. By means of some a priori estimates we prove the existence of rotationally symmetric solutions (in reality with some additional symmetry) for a sufficiently small external force and in the case of rotationally symmetric force and domain (where also we need more symmetry, respectively).     

On the structure of exact solutions of discrete masterequations

A concise version of the proof for the graph theoretical representation of the exact solution of the stationary discrete masterequation is given. Further, a new algorithm is developed for the solution of stationary and nonstationary discrete masterequations with next neighbour transition probabilities in the general case without detailed balance. This algorithm reduces the dimension of the system of masterequations to the number of boundary sites and is also appropriate for computer evaluation.     

Newton trajectories for the tilted Frenkel–Kontorova model

ABSTRACT

Newton trajectories are used for the Frenkel–Kontorova model of a finite chain with free-end boundary conditions. We optimise stationary structures, as well as barrier breakdown points for a critical tilting force were depinning of the chain happens. These special points can be obtained straight forwardly by the tool of Newton trajectories. We explain the theory and add examples for a finite-length chain of a fixed number of 2,?3,?4,?5 and 23 particles.