Time-periodic Solutions Research Articles

Existence of time-periodic strong solutions to the Navier-Stokes equation in the whole space

In this paper, the existence of time-periodic strong solutions to the Navier-Stokes equation in Rn is established under a suitable smallness condition on the external force. Our analysis is based on splitting periodic solutions into steady and purely periodic parts. One advantage of this decomposition is the availability of slightly more regularity in time of the purely periodic part. We apply this property to construct time-periodic solutions of the Navier-Stokes equation with information on the classes of their steady and purely periodic parts. It is also shown that the small solution v constructed in our existence theorem is unique within a class of time-periodic, not necessarily small, solutions having the same integrability properties as v.

Time periodic solutions for the 2D Euler equation near Taylor-Couette flow

Time periodic solutions for the 2D Euler equation near Taylor-Couette flow

Rich dynamics of a reaction–diffusion Filippov Leslie–Gower predator–prey model with time delay and discontinuous harvesting

To reflect the harvesting effect, a nonsmooth Filippov Leslie–Gower predator–prey model is proposed. Unlike traditional Filippov models, the time delay and reaction–diffusion under the condition of homogeneous Neumann boundary are considered in our system. The stability of equilibrium and the existence of the spatial Hopf bifurcation of the subsystems at the positive equilibrium are investigated. Furthermore, a comprehensive analysis is conducted on the sliding mode dynamics as well as the regular, virtual, and pseudoequilibria. The findings reveal that our Filippov system exhibits either a globally asymptotically stable regular equilibrium, a globally asymptotically stable time periodic solution, or a globally asymptotically stable pseudoequilibrium, contingent upon the specific values of the time delay and threshold level. A boundary point bifurcation, which transform a stable equilibrium point or periodic solution into a stable pseudoequilibrium, is demonstrated to emphasize the impact of time delay on our Filippov system and the significance of threshold control. Meanwhile, two kinds of global sliding bifurcations are exhibited, which sequentially transform a stable periodic solutions below the threshold into a grazing, sliding switching, and crossing bifurcations, depending on changes in the time delay or threshold level. Our results indicate that bucking bifurcation and crossing bifurcation pose significant challenges to the control of our Filippov system.

Attractivity of Time-periodic Solutions of Ginzburg-Landau Equations of Superconductivity and Numerical Simulations

Attractivity of Time-periodic Solutions of Ginzburg-Landau Equations of Superconductivity and Numerical Simulations

FTSE 100 Index Linked Structured Financial Product Pricing Analysis

Since facing widespread skepticism after the 2008 economic crisis, nowadays structured products have gradually become the preferred investment choice for individual consumers, driven by the development of technology and economy. This study analyzes the price of the two most recent structured products offered by Lowes, which closely resemble the FTSE 100 index. It focuses on analyzing the differences in predicted returns resulting from modest structural changes and explores the best solutions for different time periods. The variability is determined by combining historical variability with the GARCH model, based on the principle of Risk Neutral Pricing Theory. The final result is acquired by the application of the Monte Carlo simulation method. Based on the trial results, product B has a greater yield rate compared to product A, although being more conditional. This compensates for the shortage and resulting in a much higher yield rate for product B. Furthermore, based on the findings on the projected output of various investment goods over different time periods, it can be observed that as the length increases, the anticipated return also increases. Ultimately, the sensitivity analysis reveals that the primary source of product risk stems from the impact of risk-free yields on this particular product.

Time periodic traveling wave solutions of a time-periodic reaction–diffusion SEIR epidemic model with periodic recruitment

This paper focuses on the existence and nonexistence of a time periodic traveling wave solution of a time-periodic reaction–diffusion SEIR epidemic model. The main feature of the model is the possible deficiency of the classical comparison principle such that many known results do not directly work. If the basic reproduction number of the model, denoted by R0, is larger than one, there exists a minimal wave speed c∗>0 satisfying for each c>c∗, the system admits a nontrivial time periodic traveling wave solution with wave speed c and for c<c∗, there exists no nontrivial time periodic traveling waves such that the system; if R0<1, the system admits no nontrivial time periodic traveling waves.

Global weak solutions to a time-periodic body-liquid interaction problem

We prove existence of time-periodic weak solutions to the coupled liquid-structure problem constituted by an incompressible Navier–Stokes fluid interacting with a rigid body of finite size, subject to an undamped linear restoring force. The fluid flow is generated by a uniform, time-periodic velocity field {\boldsymbol V} far from the body. We emphasize that our result is global, in the sense that no restriction is imposed on the magnitude of {\boldsymbol V} and, rather remarkably, the frequency of {\boldsymbol V} is entirely arbitrary. Thus, in particular, it can coincide with any multiple of a natural frequency of vibration of the body so that, with this model, resonance cannot occur. Although based on the classical “invading domains” technique, our approach requires several new ideas. Indeed, due to the lack of sufficient dissipation, it appears quite unfeasible to show the existence of a fixed point of the Poincaré map at the finite-dimensional level along the Galerkin approximant. Therefore, unlike the usual strategy, such a result must be proven directly in a class of weak solutions, and therefore in the infinite-dimensional framework.

Periodic solutions for a coupled system of wave equations with x-dependent coefficients

Abstract This paper is concerned with the periodic solutions for a coupled system of wave equations with x-dependent coefficients. Such a model arises naturally when two waves propagate simultaneously in the nonisotrpic media. In this paper, for the periods having the form T = 2 a − 1 b ( a , b are positive integers ) $T=\frac{2a-1}{b}\left(a,b \text{are\,positive\,integers}\right)$ and some types of boundary conditions, we obtain the existence of the time periodic solutions and analyze the asymptotic behaviors as the coupled parameter goes to zero, when the nonlinearities are superlinear and monotone, by using the variational method. In particular, the condition ess inf η ϱ (x) &gt; 0 is not required.

Projection-based reduced-order modelling of time-periodic problems, with application to flow past flapping hydrofoils

Simulating forced time-periodic flows in industrial applications presents significant computational challenges, partly due to the need to overcome costly transients before achieving time-periodicity. Reduced-order modelling emerges as a promising method to speed-up computations. We extend upon the work of Lotz et al. (2024) where a time-periodic space–time model is introduced. We present a time-periodic reduced-order model that directly finds the time-periodic solution without requiring extensive time integration. The reduced-order model gives a reduction in variables in both space and time. Our approach involves a POD-Galerkin reduced-order model based on a time-periodic full-order model that employs isogeometric analysis, residual-based variational multiscale turbulence modelling and weak boundary conditions. The projection-based reduced-order model inherits these features. We evaluate the reduced-order model with numerical experiments on moving hydrofoils. The motion is known a priori and we restrict ourselves to two spatial dimensions. In these experiments we vary the Strouhal and Reynolds numbers, and the motion profile respectively. Reduced-order model solutions agree well with those of the full-order model. The errors over the entire time period of thrust and lift forces are less than 0.2%. This includes complex scenarios such as the transition from drag to thrust production with increasing Strouhal number. Our time-periodic reduced-order model offers speed-ups ranging from O(102) to O(103) compared to the full-order model, depending upon the basis size. This makes it an appealing solution for prescribed time-periodic problems, with potential for additional speedup through nonlinear reduction techniques such as hyper-reduction.

On periodic motions of a harmonic oscillator interacting with incompressible fluids

We consider a mass–spring system immersed in an incompressible fluid flow governed by the Navier–Stokes equations subject to a prescribed time-periodic flow rate (and possibly external time-periodic body forces on the fluid and the mass). We show that, with no restriction on the period of the flow rate (and of the external forces), when the flow rate is “small”, there exists a weak time-periodic solution to the coupled system. Under some more regularity and “smallness” conditions on the flow rate (and the external forces) we also show that these solutions are, indeed, strong solutions.

Time-periodic solutions of completely resonant Klein–Gordon equations on $\mathbb{S}^{3}$

We prove existence and multiplicity of Cantor families of small-amplitude time-periodic solutions of completely resonant Klein–Gordon equations on the sphere \mathbb{S}^{3} with quadratic, cubic, and quintic nonlinearity, regarded as toy models in general relativity. The solutions are obtained by a variational Lyapunov–Schmidt decomposition, which reduces the problem to the search of mountain pass critical points of a restricted Euler–Lagrange action functional. Compactness properties of its gradient are obtained by Strichartz-type estimates for the solutions of the linear Klein–Gordon equation on \mathbb{S}^{3} .

Coupled approach and its convergence analysis for aggregation and breakage models: Study of extended temporal behaviour

Population balance equations (PBEs) play a significant role in describing the dynamics of star formation in astrophysics, cheese manufacturing in dairy sciences, depolymerization in chemical engineering, and granule preparation in the pharmaceutical industry. However, obtaining analytical solutions for these equations remains a formidable challenge due to complex integral terms. In this work, we propose a generalized approach based on He’s variational iteration method which enables the efficient estimation of series solutions for breakage and aggregation PBEs for shorter time domains. To extend the solutions for longer time periods and larger size domains, the series solutions obtained by the coupled method are further expanded using powerful Padé approximants. Moreover, a thorough convergence analysis is conducted in the Banach space. The testing of the new approach is done by considering several gelling and non gelling kernels, and the results are validated against the existing analytical solutions.

The existence and uniqueness of time-periodic solutions to the non-isothermal model for compressible nematic liquid crystals in a periodic domain

The existence and uniqueness of time-periodic solutions to the non-isothermal model for compressible nematic liquid crystals in a periodic domain

Investigation of the inhibiting effect of environmentally friendly cerium-containing conversion films on the corrosion of zinc coatings

In the present work, the inhibiting effect of environmentally friendly cerium and phosphorus-based conversion films on zinc corrosion in 5% NaCl and 1 N Na2SO4 test media was investigated. The films were prepared by immersion and stirring in the passivating/converting solution for time periods of 3,5,7, 10 min., respectively. The chemical composition of the films was determined by EDX analysis. The results of the electrochemical studies showed that in 5% NaCl medium, cerium- and phosphorus-based films exhibited high inhibition effect only during the first days of polarization resistance (Rp) measurement. In 1 N Na2SO4 medium, the Rp of the conversion films was up to 2.5 times higher than that of pure zinc coatings throughout the test period.

Spectral Galerkin method for Cahn-Hilliard equations with time periodic solution

Spectral Galerkin method for Cahn-Hilliard equations with time periodic solution

A note on 2D Navier-Stokes system in a bounded domain

&lt;p&gt;This paper poses a new question and proves a related result. Particularly, the nonexistence of a nontrivial time-periodic solution to the Navier–Stokes system is proved in a bounded domain in $ \mathbb{R}^2 $.&lt;/p&gt;

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

Time-Periodic Solutions to Quasilinear Hyperbolic Systems on General Networks

On the existence of time-periodic solutions of nonlinear parabolic differential equations with nonlocal boundary conditions of the Bitsadze-Samarskii type

We study a nonlinear parabolic differential equation in a bounded multidimensional domain with nonlocal boundary conditions of the Bitsadze-Samarskii type. We prove existence theorems for a periodic in time generalized solution. Su cient conditions for the existence of generalized solutions contain either an algebraic ellipticity condition or an algebraic strong ellipticity condition for the auxiliary differential-difference operator.

Time Periodic Solutions Close to Localized Radial Monotone Profiles for the 2D Euler Equations

Time Periodic Solutions Close to Localized Radial Monotone Profiles for the 2D Euler Equations

A road map to the blow-up for a Kirchhoff equation with external force

It is well-known that the classical hyperbolic Kirchhoff equation admits infinitely many simple modes, namely time-periodic solutions with only one Fourier component in the space variables. In this paper we assume that, for a suitable choice of the nonlinearity, there exists a heteroclinic connection between two simple modes with different frequencies. Under this assumption, we cook up a forced Kirchhoff equation that admits a solution that blows-up in finite time, despite the regularity and boundedness of the forcing term. The forcing term can be chosen with the maximal regularity that prevents the application of the classical global existence results in analytic and quasi-analytic classes.