Approximate filter for a two-dimensional nonlinear diffusion observed in a correlated low noise channel

The purpose of this article is to study a nonlinear filtering problem when the signal is a two-dimensional process from which only the second component is noisy and when only its first (and unnoisy) component is observed in a correlated low noise channel. We propose an approximate finite-dimensional filter and we prove that the filtering error converges to zero. The order of magnitude of the error between the approximate filter and the optimal filter, as the observation noise vanishes, is computed.     

Infinite-dimensional Evans function theory for elliptic eigenvalue problems in a channel

An infinite-dimensional Evans function theory is developed for the elliptic eigenvalue problem associated with the stability of travelling solitary waves in a channel. Also, a bundle is constructed over the complex domain, so that its first Chern number gives the number of eigenvalues inside the domain.     

Towards an inverse scattering theory for two-dimensional nondecaying potentials

The inverse scattering method is considered for the nonstationary Schrödinger equation with the potentialu (x ₁,x ₂) nondecaying in a finite number of directions in thex plane. The general resolvent approach, which is particularly convenient for this problem, is tested using a potential that is the Bäcklund transformation of an arbitrary decaying potential and that describes a soliton superimposed on an arbitrary background. In this example, the resolvent, Jost solutions, and spectral data are explicitly constructed, and their properties are analyzed. The characterization equations satisfied by the spectral data are derived, and the unique solution of the inverse problem is obtained. The asymptotic potential behavior at large distances is also studied in detail. The obtained resolvent is used in a dressing procedure to show that with more general nondecaying potentials, the Jost solutions may have an additional cut in the spectral-parameter complex domain. The necessary and sufficient condition for the absence of this additional cut is formulated.     

Effect of constriction height on flow separation in a two-dimensional channel

We studied numerically the effect of the constriction height on viscous flow separation past a two-dimensional channel with locally symmetric constrictions. A numerically stable scheme in primitive variables (velocity and pressure) for the solution of two-dimensional incompressible time-dependent Navier–Stokes equations is employed using finite-difference approximation in staggered grid. The wall shear stresses at different heights of the constriction are computed and presented graphically. It is noticed that the maximum stress and the length of the recirculating region associated with two shear layers of the constriction increase with the increase of the area reduction of the constriction. The critical Reynolds number for symmetry breaking bifurcation for the 50%, 60% and 70% area reduction are obtained numerically. The flow field separates after the symmetry breaking bifurcation and the symmetry of the flow depends on the Reynolds number and the height of the constriction.     

Stability and persistence of two-dimensional patterns

The canonical equations for evolution of the amplitude order parameters order parameters describing the nonlinear development and persistence of two-dimensional three-mode spatial patterns generated by Turing instability in dissipative systems are considered. The stability conditions for persistent hexagonal patterns are generalized, and the conditions under which patterns are either disrupted, exhibit bounded quasiperiodic or chaotic behavior, or decay under nonlinear evolution are derived. These conditions are applied to the specific three-mode amplitude evolution equations derived for the Schnakenberg model and a delay predator system in Chapter 3. Numerical results are presented for the persistence, disruption and decay of patterns in these systems, including fairly detailed comparisons with simulations results for the Snackenberg model.     

Continuous theory of dislocations and disclinations in a two-dimensional medium

A system of equations describing mobile defects in a two-dimensional Cosser at continuum, i.e. in a medium whose motion is determined by the displacement field and rotation field independent of it, is obtained.

The basic equations of the static theory /1–5/ and dynamic continuous theory /6–12/ of defects (dislocations and disclinations) are known for a three-dimensional medium, obtained by a variety of methods. A dislocation model of the misalignment surfaces used in describing the Martensitic transformations /2, 13/ is proposed. The dislocation representations were used in /14–16/ to describe the grain boundaries, and the difference dislocations within the boundaries of separation were studied in /17, 18/. The dislocation structure of internal boundaries of separation was described in /19, 20/ using the differential geometry characteristics (torsion and curvature tensors, non-holonomic object) of three-dimensional media. Surface dislocations and disclinations of the separate Volterra distortions-type were studied in /21/, with liquid crystals and various biological objects indicated as the suitable areas of application of these concepts.     

Stability theory for dissipatively perturbed hamiltonian systems

It is shown that for an appropriate class of dissipatively perturbed Hamiltonian systems, the number of unstable modes of the dynamics linearized at a nondegenerate equilibrium is determined solely by the index of the equilibrium regarded as a critical point of the Hamiltonian. In addition, the movement of the associated eigenvalues in the limit of vanishing dissipation is analyzed. ©1995 John Wiley & Sons, Inc.     

Stability of a two-dimensional stationary rotating flow in an exterior cylinder

We investigate the stability of an exact stationary flow in an exterior cylinder. The horizontal velocity is the two-dimensional rotating flow in an exterior disk with a critical spatial decay, for which the L² stability is known under smallness conditions. We prove its stability property for three-dimensional perturbations although the Hardy type inequalities are absent as in the two-dimensional case. The proof uses a large time estimate for the linearized equations exhibiting different behaviors in the Fourier modes, namely, the standard L²-$L^q$ decay of the two-dimensional mode and an exponential decay of the three-dimensional modes.     

On potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity

The construction of potential theory for two-dimensional quasistatic problems of uncoupled thermoelasticity is carried out by considering the full system of differential equations of the problem as a nonselfadjoint differential operator. Green's second formula for this operator is interpreted as a duality theorem that differs from Mizel's duality theorem. In the case of a homogeneous isotropic medium we construct new integral equations for the basic initial-boundary value problems.Translated fromMatematicheskie Metody i Fiziko-Mekhanicheskie Polya, Issue 33, 1991, pp. 48–52.     

Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle

In this paper, we study the stability of supersonic contact discontinuity for the two-dimensional steady compressible Euler flows in a finitely long nozzle of varying cross-sections. We formulate the problem as an initial–boundary value problem with the contact discontinuity as a free boundary. To deal with the free boundary value problem, we employ the Lagrangian transformation to straighten the contact discontinuity and then the free boundary value problem becomes a fixed boundary value problem. We develop an iteration scheme and establish some novel estimates of solutions for the first order of hyperbolic equations on a cornered domain. Finally, by using the inverse Lagrangian transformation and under the assumption that the incoming flows and the nozzle walls are smooth perturbations of the background state, we prove that the original free boundary problem admits a unique weak solution which is a small perturbation of the background state and the solution consists of two smooth supersonic flows separated by a smooth contact discontinuity.