Unbounded Cylindrical Domains Research Articles

Stable recovery of a non-compactly supported coefficient of a Schrödinger equation on an infinite waveguide

<p style='text-indent:20px;'>We study the stability issue for the inverse problem of determining a coefficient appearing in a Schrödinger equation defined on an infinite cylindrical waveguide. More precisely, we prove the stable recovery of some general class of non-compactly and non periodic coefficients appearing in an unbounded cylindrical domain. We consider both results of stability from full and partial boundary measurements associated with the so called Dirichlet-to-Neumann map.</p>

RECOVERY OF NON-COMPACTLY SUPPORTED COEFFICIENTS OF ELLIPTIC EQUATIONS ON AN INFINITE WAVEGUIDE

We consider the unique recovery of a non-compactly supported and non-periodic perturbation of a Schrödinger operator in an unbounded cylindrical domain, also called waveguide, from boundary measurements. More precisely, we prove recovery of a general class of electric potentials from the partial Dirichlet-to-Neumann map, where the Dirichlet data is supported on slightly more than half of the boundary and the Neumann data is taken on the other half of the boundary. We apply this result in different contexts including recovery of some general class of non-compactly supported coefficients from measurements on a bounded subset and recovery of an electric potential, supported on an unbounded cylinder, of a Schrödinger operator in a slab.

Hölder stable determination of a quantum scalar potential in unbounded cylindrical domains

We consider the inverse problem of determining the time independent scalar potential of the dynamic Schrödinger equation in an infinite cylindrical domain from one boundary Neumann observation of the solution. We prove Hölder stability by choosing the Dirichlet boundary condition suitably.

A Carleman estimate for infinite cylindrical quantum domains and the application to inverse problems

We consider the inverse problem of determining the time-independent scalar potential q of the dynamic Schrödinger equation in an infinite cylindrical domain Ω, from one Neumann boundary observation of the solution. Assuming that q is known outside some fixed compact subset of Ω, we prove that q may be Lipschitz stably retrieved by choosing the Dirichlet boundary condition of the system suitably. Since the proof is by means of a global Carleman estimate designed specifically for the Schrödinger operator acting in an unbounded cylindrical domain, the measurement of the Neumann data is performed on an infinitely extended subboundary of the cylinder.

Erratum to: Time-Periodic Solutions of the Navier-Stokes Equations in Unbounded Cylindrical Domains – Leray’s Problem for Periodic Flows

Poiseuille flows in infinite cylindrical pipes, in spite of their enormous simplicity, have a main role in many theoretical and applied problems. As is well known, the Poiseuille flow is a stationary solution of the Stokes and the Navier-Stokes equations with a given constant flux. Time-periodic flows in channels and pipes have a comparable importance. However, the problem of the existence of time-periodic flows in correspondence to any given time-periodic total flux, is still an open problem. A solution is known only in some very particular cases, for instance, the Womersley flows. Our aim is to solve this problem in the general case. The above existence result opens the way to further investigations. As an example of this possibility we consider the extension of the classical Leray's problem for Poiseuille flows to arbitrary time-periodic flows.

Mixed problem for the equation governing inertia-gravity waves in the Boussinesq approximation in a unbounded cylindrical domain

The unique solvability of an initial-boundary value problem for the equation governing inertia-gravity waves in the Boussinesq approximation in an unbounded multidimensional cylindrical domain is studied. The existence and uniqueness of a weak solution is proved, and its asymptotic behavior at long times is analyzed. The proofs are based on the Green’s function constructed in explicit form for the corresponding stationary problem.

A comparison principle for quasilinear operators in unbounded domains

We establish a comparison principle for quasilinear elliptic operators in unbounded cylindrical domains by combining the traditional test function method with nonlinear capacity techniques. Some applications to monotonicity and symmetry of solutions to nonlinear elliptic problems are given.

Phragmén–Lindelöf theorems in cylinders

In unbounded cylindrical domains, new Phragmén–Lindelöf theorems ‘at infinity’ are established for solutions of Dirichlet problems for elliptic equations whose operators belong to a large class to which the existing literature does not apply.

Phragmén–Lindelöf theorems in cylinders

In unbounded cylindrical domains, new Phragmén–Lindelöf theorems ‘at infinity’ are established for solutions of Dirichlet problems for elliptic equations whose operators belong to a large class to which the existing literature does not apply.

Time-Periodic Solutions of the Navier-Stokes Equations in Unbounded Cylindrical Domains – Leray's Problem for Periodic Flows

Time-Periodic Solutions of the Navier-Stokes Equations in Unbounded Cylindrical Domains – Leray's Problem for Periodic Flows

Bifurcation Theory for Dissipative Systems on Unbounded Cylindrical Domains — An Introduction to the Mathematical Theory of Modulation Equations —

In this introductory text we consider dissipative translational invariant nonlinear partial differential equations posed on spatially unbounded cylindrical domains close to the threshold of instability. We give an overview about the mathematical theory of modulation equations which allows us to analyse bifurcation problems occurring in such systems. We explain this theory by presenting new results concerning the dynamics of interface solutions bifurcating in certain reaction-diffusion systems.

A global existence result for a spatially extended 3D Navier-Stokes problem with non-small initial data

We consider the Navier-Stokes equations in a two- or three-dimensional unbounded cylindrical domain. The existence and uniqueness of solutions is discussed in the space of uniformly local square integrable functions. We show for small initial data and small forcing term that the solutions exist globally in time. This result is extended to a non-small data result in the sense that the high frequency modes of the initial conditions and of the forcing terms are allowed to be large. Moreover, we show the existence of a local attractor for this 3D Navier-Stokes problem in an unbounded domain. In contrast to previous results the spaces used are no Hilbert spaces, and secondly we have a linear operator possessing continuous spectrum without spectral gap.

Global existence results for pattern forming processes in infinite cylindrical domains — Applications to 3D Navier–Stokes problems

We consider translational invariant systems on unbounded cylindrical domains in R 3 which are described by the Navier–Stokes equations. The examples which we have in mind are the Taylor–Couette problem and Bénard's problem. For certain parameter ranges these systems exhibit pattern of almost spatial periodic nature. Although classical energy methods fail on unbounded domains the so called Ginzburg–Landau formalism allows us to show the global existence of strong solutions to all initial conditions in a neighborhood U of the weakly unstable ground state. For all times the bifurcating solutions of the original system can be shadowed by pseudo-orbits of the associated formally derived Ginzburg–Landau equation. This allows us to control the size of the solutions in the original system in terms of the bifurcation parameter for t→∞.

Nonlinear Stability of Taylor Vortices in Infinite Cylinders

We consider the Taylor‐Couette problem in an infinitely extended cylindrical domain in the case when Couette flow is weakly unstable and a family of spatially periodic equilibria, called the Taylor vortices, has bifurcated from this trivial ground state. We show that those Taylor vortices which are not linearly unstable in the sense of Eckhaus are in fact nonlinearly stable with respect to small spatially localized perturbations. The main difficulty in showing this result stems from the fact that on unbounded cylindrical domains the Taylor vortices are only linearly marginally stable with continuous spectrum up to the imaginary axis. Bloch‐wave representations of the solutions and renormalization theory allow us to show that the nonlinear problem behaves asymptotically like the linearized one which is under a diffusive regime.

Attractors for modulation equations on unbounded domains-existence and comparison

We are interested in the long-time behaviour of nonlinear parabolic PDEs defined on unbounded cylindrical domains. For dissipative systems defined on bounded domains, the longtime behaviour can often be described by the dynamics in their finite-dimensional attractors. For systems defined on the infinite line, very little is known at present, since the lack of compactness prevents application of the standard existence theory for attractors. We develop an abstract theorem based on the interaction of a uniform and a localizing (weighted) norm which allows us to define global attractors for some dissipative problems on unbounded domains such as the Swift-Hohenberg and the Ginzburg-Landau equation. The second aim of this paper is the comparison of attractors. The so-called Ginzburg-Landau formalism allows us to approximate solutions of weakly unstable systems which exhibit modulated periodic patterns. Here we show that the attractor of the Swift-Hohenberg equation is upper semicontinuous in a particular limit to the attractor of the associated Ginzburg-Landau equation.

Numerical Calculation of Center Manifolds for a Class of Infinite-Dimensional Systems with Applications

In this paper, we describe a method for numerically constructing the center manifold for a class of infinite-dimensional systems generated by certain semilinear elliptic boundary value problems in an unbounded cylindrical domain. The manifold lives in a neighbourhood of a solution that is independent of axial direction. It contains all bounded solutions of the system. Error bounds are given. We also describe an application of this method to the construction of numerical solutions for a specific semilinear elliptic equation.

A value function and applications to translation-invariant semilinear elliptic equations on unbounded domains

We combine results from nonlinear functional analysis relating nonlinear eigenvalues of the type $ g'(u) = \rho u $ to the derivatives of the critical value function $ \gamma (t) := \sup_{\|u\|^2=t} g(u) $ with concentration compactness techniques to study the Dirichlet boundary value problem on $\Omega$, $$ -\Delta u + u = \lambda f(x,u), \tag 0.1 $$ where $\Omega$ is an unbounded cylindrical domain and the dependence on $x$ in the unbounded direction is periodic. We give sufficient conditions on $f$ to obtain an interval in which the $\lambda$'s for which (0.1) has a weak solution are dense.

Numerical approximation of bounded solutions for semilinear elliptic equations in an unbounded cylindrical domain

AbstractWe consider the approximation of bounded solutions for a class of semilinear elliptic equations in an unbounded cylindrical domain. On certain conditions, the existence of approximate solutions for the semidiscretization of the problem is proved. Error estimates, applications, and a numerical example are also given. © 1993 John Wiley & Sons, Inc.

The Asymptotic Behavior of the Solutions of Some Semilinear Elliptic Equations in Cylindrical Domains

The bounded solutions of some semilinear elliptic equations in unbounded cylindrical domains are considered. Their asymptotic behavior as |x| → ∞ (x = coordinate along the axis of the cylinder) is analyzed and used to establish uniqueness and monotonicity in the x-variable of travelling wavefronts of some parabolic problems