Uniqueness Of Time Periodic Solutions Research Articles

Solving 3D fractional Schrödinger systems on the basis of Phragmén–Lindelöf methods

Our aim in this article is to study the existence and uniqueness of time periodic solutions to 3D fractional Schrödinger systems. The techniques in this work are based on Phragmén–Lindelöf method (Liess, Necessary conditions in Phragmén–Lindelöf type estimates and decomposition for non-divergence type elliptic equations and mixed boundary conditions. Math Nachr 290(8–9):1328–1346, 2017) and Matsaev’s theory (Sukochev, Ferleger, Matsaev’s theorem for symmetric spaces of measurable operators. Math Notes 56(5–6):1185–1189, 1995). Particularly, the fractional maximal effects and the potential nonlinear equation are taken into account which play a key role in the Green–Schrödinger growth properties, especially in the selection of the Schrödinger norm.

Spatial Pointwise Behavior of Time-Periodic Navier–Stokes Flow Induced by Oscillation of a Moving Obstacle

We study the spatial decay of time-periodic Navier–Stokes flow at the rate |x|^{-1} with/without wake structure in 3D exterior domains when a rigid body moves periodically in time. In this regime the existence of time-periodic solutions was established first in the 2006 paper by Galdi and Silvestre, however, with little information about spatial behavior at infinity so that uniqueness of solutions was not available. This latter issue has been addressed by Galdi, who has recently succeeded in construction of a unique time-periodic solution with spatial behavior mentioned above if translational and angular velocities of the body fulfill, besides smallness and regularity, either of the following assumptions: (i) translation or rotation is absent; (ii) both velocities are parallel to the same constant vector. This paper shows the existence of a unique time-periodic Navier–Stokes flow in the small with values in the weak-L^3 space and then deduces the desired pointwise decay of the solution under some condition on the rigid motion of the body, that covers the cases (i), (ii) mentioned above.

Existence and uniqueness of time periodic solutions for quantum versions of three-dimensional Schrödinger equations

Existence and uniqueness of time periodic solutions for quantum versions of three-dimensional Schrödinger equations

Time periodic solutions to the 2D quasi-geostrophic equation with the supercritical dissipation

We consider the 2D dissipative quasi-geostrophic equation with the time periodic external force and prove the existence of a unique time periodic solution in the case of the supercritical dissipation. In this case, the smoothing effect of the semigroup generated by the dissipation term is too weak to control the nonlinearity in the Duhamel term of the corresponding integral equation. In this paper, we give a new approach which does not depend on the contraction mapping principle for the integral equation.

ON TIME PERIODIC SOLUTIONS TO THE GENERALIZED BBM-BURGERS EQUATION WITH TIME-DEPENDENT PERIODIC EXTERNAL FORCE

In this paper, we consider the generalized BBM-Burgers equation with periodic external force in Rn. Existence and uniqueness of time periodic solutions that have the same period as the external force are established in some suitable function space for the space dimension n≥ 3. Moreover, we also discuss the time asymptotic stability of the time periodic solution. The proof is mainly based on the contraction mapping theorem and continuous argument.

Time periodic solutions of one-dimensional forced Kirchhoff equations with x-dependent coefficients under spatial periodic conditions

In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under the spatial periodic conditions, which models the forced vibrations of an inhomogeneous string in presence of a time periodic external forcing with period \(2\pi /\omega \) and amplitude \(\epsilon >0\). By using the Lyapunov–Schmidt reduction and the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period \(2\pi /\omega \) and order \(\epsilon \). Such results hold for parameters \((\omega ,\epsilon )\) in a positive measure Cantor set that has asymptotically full measure as the amplitude \(\epsilon \) goes to zero.

Time Periodic Solutions of One-Dimensional Forced Kirchhoff Equation with Sturm–Liouville Boundary Conditions

This paper is concerned with the time periodic solutions for the Sturm–Liouville boundary value problem of a one-dimensional Kirchhoff equation in presence of a time periodic external forcing with period $$2\pi /\omega $$ and amplitude $$\epsilon $$. Such a model arises from the forced vibrations of a bounded string in which the dependence of the tension on the deformation cannot be neglected. By using the Nash–Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period $$2\pi /\omega $$ and order $$\epsilon $$. Such results hold for parameters $$(\omega ,\epsilon )$$ in a positive measure Cantor set that has asymptotically full measure as $$\epsilon $$ goes to zero.

Unifying averaged dynamics of the Fokker-Planck equation for Paul traps

Collective dynamics of a collisional plasma in a Paul trap is governed by the Fokker-Planck equation, which is usually assumed to lead to a unique asymptotic time-periodic solution irrespective of the initial plasma distribution. This uniqueness is, however, hard to prove in general due to analytical difficulties. For the case of small damping and diffusion coefficients, we apply averaging theory to a special solution to this problem and show that the averaged dynamics can be represented by a remarkably simple 2D phase portrait, which is independent of the applied rf field amplitude. In particular, in the 2D phase portrait, we have two regions of initial conditions. From one region, all solutions are unbounded. From the other region, all solutions go to a stable fixed point, which represents a unique time-periodic solution of the plasma distribution function, and the boundary between these two is a parabola.

Time periodic solutions to the beam equation with weak damping

In this paper, we study the beam equation with weak damping in n-dimensional space. In the case of time-dependent periodic external force imposed, we prove the existence and uniqueness of time periodic solutions that have the same period as the time-dependent periodic external force in some suitable function space for all space dimensions n ≥ 1. The proof is based on the spectral analysis for the solution operators and the contraction mapping theorem. Moreover, we show the time asymptotic stability of time periodic solutions by continuous argument.

Existence and asymptotic stability of time periodic solutions to the generalized double dispersion equation with periodic external force

In this paper, we investigate the generalized double dispersion equation with periodic external force in Rn. We prove the existence and uniqueness of time periodic solutions that have the same period as the external force in some suitable function space for the space dimension n≥3. The proof is based on the spectral analysis for the solution operator and the contraction mapping theorem. In addition, we also discuss the time asymptotic stability of the time periodic solutions by continuous argument.

Time periodic solutions of one-dimensional forced Kirchhoff equations with x-dependent coefficients.

In this paper, we consider the one-dimensional Kirchhoff equation with x-dependent coefficients under Dirichlet boundary conditions, which models the forced vibrations of a clamped inhomogeneous string in the presence of a time periodic external forcing with period 2π/ω and amplitude ϵ. By using the Nash-Moser iteration technique, we obtain the existence, regularity and local uniqueness of time periodic solutions with period 2π/ω and order ϵ. Such results hold for parameters (ω,ϵ) in a positive measure Cantor set that has asymptotically full measure as the amplitude ϵ goes to zero.

Stabilizing the long-time behavior of the forced Navier–Stokes and damped Euler systems by large mean flow

The paper studies the issue of stability of solutions to the forced Navier–Stokes and damped Euler systems in periodic boxes. It is shown that for large, but fixed, Grashoff (Reynolds) number the turbulent behavior of all Leray–Hopf weak solutions of the three-dimensional Navier–Stokes equations, in periodic box, is suppressed, when viewed in the right frame of reference, by large enough average flow of the initial data; a phenomenon that is similar in spirit to the Landau damping. Specifically, we consider an initial data which have large enough spatial average, then by means of the Galilean transformation, and thanks to the periodic boundary conditions, the large time independent forcing term changes into a highly oscillatory force; which then allows us to employ some averaging principles to establish our result. Moreover, we also show that under the action of fast oscillatory-in-time external forces all two-dimensional regular solutions of the Navier–Stokes and the damped Euler equations converge to a unique time-periodic solution.

On the existence and uniqueness of time periodic solutions for a semilinear heat equation in the whole space

This paper is concerned with the existence and uniqueness of time periodic solutions in the whole‐space for a heat equation with nonlinear term. The nonlinear term we considered is of this type, |u|q − 1u + f(x,t), with , N > 2. We show that there exists a unique time periodic solution when the source term f is small. In fact, is a critical exponent; when , there is no time periodic solution. Copyright © 2016 John Wiley & Sons, Ltd.

Existence and uniqueness of periodic solutions for parabolic equation with nonlocal delay

This paper deals with the existence and uniqueness of time periodic solutions for the general periodic parabolic equation boundary problem with nonlocal delay. We apply operator semigroup theory and monotone iterative technique of lower and upper solutions to obtain the existence and uniqueness of ω-periodic mild solutions of some abstract evolution equation under some quasimonotone conditions. In the end, applying our abstract results to parabolic equation with nonlocal delay, we get the existence and uniqueness of ω-periodic solution, which generalize the recent conclusions on this issue.

Asymptotic conditions at infinity for the time-periodic Stokes problem in a system of pipes

A time-periodic Stokes problem set in a domain with cylindrical outlets to infinity is studied. We derive the generalized Green formula. It enables us to impose so-called asymptotic conditions at infinity which allow to determine a unique time-periodic solution. This method was proposed and applied for a steady Stokes problem by Nazarov and co-authors.

Time-periodic solutions of the compressible Navier–Stokes equations in $${\mathbb{R}^{4}}$$ R 4

This paper deals with the existence of time-periodic solutions to the compressible Navier–Stokes equations effected by general form external force in \({\mathbb{R}^{N}}\) with \({N = 4}\). Using a fixed point method, we establish the existence and uniqueness of time-periodic solutions. This paper extends Ma, UKai, Yang’s result [5], in which, the existence is obtained when the space dimension \({N \ge 5}\).

Asymptotic analysis of linear parabolic problems with singularities

Asymptotic methods are used to investigate a second-order linear parabolic problem with lower order coefficients rapidly oscillating in time. The coefficient multiplying the principal stationary operator is assumed to be singular, i.e., it has a simple zero eigenvalue. Under certain additional conditions, the problem is proved to have a unique time-periodic solution and its complete asymptotic expansion is constructed and justified.

Small amplitude periodic solutions for Kirchhoff equations with Neumann conditions

In this paper, we prove existence, regularity and local uniqueness of time-periodic solutions of Kirchhoff type equations with Neumann conditions by means of a Nash–Moser iteration scheme.

Existence and uniqueness of time-periodic solutions with finite kinetic energy for the Navier–Stokes equations in

Our aim is to prove existence and uniqueness of time-periodic strong solutions with finite kinetic energy for the Navier–Stokes equations in . For this, appropriate conditions are imposed on the external force, together with a smallness condition involving the viscosity of the fluid and the period of motion. We extend the method we have recently used to construct steady states with finite kinetic energy to the time-periodic case. First, existence and uniqueness of strong solutions with finite kinetic energy are established for a linearized version of the problem, using the Galerkin method and the Fourier transform in the space variables. Then, a strong solution with finite kinetic energy for the nonlinear problem is obtained by means of the contraction mapping principle. We also show that such a solution satisfies the energy equality and is unique within a class of weak solutions.

Time periodic solutions of compressible Navier–Stokes equations

In this paper, we study the Navier–Stokes equations with a time periodic external force in R n . We show that a time periodic solution exists when the space dimension n ⩾ 5 under some smallness assumption. The main idea is to combine the energy method and the spectral analysis for the optimal decay estimates on the linearized solution operator. With the optimal decay estimates, we prove the existence and uniqueness of time periodic solution in some suitable function space by the contraction mapping theorem. In addition, we also study the time asymptotic stability of the time periodic solution.