Parabolic Systems Research Articles

Null Controllability of Coupled Parabolic Systems with Switching Control

Null Controllability of Coupled Parabolic Systems with Switching Control

On a parabolic p-Laplacian system with a convective term

On a parabolic p-Laplacian system with a convective term

Modification of wind turbine wakes by large-scale, convective atmospheric boundary layer structures

In this study, we consider the impact of large-scale, convective structures in an unstable atmospheric boundary layer on wind turbine wakes. Simulation data from a high-fidelity large-eddy simulation (LES) of the AWAKEN wind farm site matching unstable atmospheric conditions were analyzed, and both turbine performance and wake behavior were affected based on their location relative to the convective structures. Turbines located in updraft regions of the flow experienced lower inflow velocity and generated less power, but their wakes were observed to recover faster and saw greater turbulent kinetic energy mixing higher in the boundary layer. The opposite effect was found for turbines in the downdraft regions of the convective structures. A simplified model of this wake behavior was also developed based on a two-dimensional k–ε Reynolds-Averaged Navier–Stokes formulation. This simplified model included the effects of vertical transport, but could be efficiently solved as a parabolic system, and was found to capture similar wake modifications observed in the high-fidelity LES computations.

Design and Performance Characteristics of Single-Axis Tracking Dual Confocal Low-Magnification Parabolic Trough CPV system

With the rapid increase of PV module utilization, the environmental pollution associated with waste photovoltaic (PV) module and its recycling is of concern. This paper proposes a concentrating photovoltaic (CPV) system to reduce the use of PV module. The cross-confocal method is employed for the concentrator to eliminate central dark streaks. In optical simulation, a theoretical uniformity index of 0.0466 and a theoretical optical efficiency of 89.87% were achieved. When the distance between the rotational axis and the center of gravity is less than 125.8mm, the system can withstand winds of 20.4m/s. The temperature range of the PV, obtained from finite element analysis, is from 295.71 K to 363.13 K. With the requirements of the relative optical efficiency of over 90% and the uniformity index of less 0.5, the system has a dimensional translation tolerance of approximately ±10mm, a dimensional rotation tolerance of about 5°, and a tracking angle tolerance of about 1°. The actual daily average output current of this system has increased by 100.91%. The ratio between the increase ratio of output current and the theoretical concentration ratio is 83.4%. The system demonstrates excellent optical performance and system tolerance.

Lipschitz truncation method for parabolic double-phase systems and applications

We discuss a Lipschitz truncation technique for parabolic double-phase problems of p-Laplace type in order to prove energy estimates and uniqueness results for the Dirichlet problem. Moreover, we show existence for a non-homogeneous double-phase problem. The Lipschitz truncation method is based on a Whitney-type covering result and a related partition of unity in the intrinsic geometry for the double-phase problem.

Global solution for coupled parabolic systems with degenerate coefficients and time-weighted sources

In this article we obtained the so-called Fujita exponent for the degenerate parabolic coupled system $$\displaylines{ u_t- \text{div} ( \omega(x)\nabla u )= t^{r} v^{p} \cr v_t- \text{div} ( \omega(x)\nabla v )= t^{s} u^{p} }$$ in \(\mathbb{R}^{N} \times (0,T)\) with initial data belonging to \([ L^\infty(\mathbb{R}^N)]^2\), where \(p,q > 0\) with \(pq > 1\); \(r,s>-1\), and either \(\omega(x) = | x_1|^a\) or \(\omega(x) = | x |^b\) with \(a,b > 0\). For more information see https://ejde.math.txstate.edu/Volumes/2024/67/abstr.html

Global existence and boundedness in a two‐dimensional parabolic chemotaxis system with competing attraction and repulsion effects

In this paper, we investigate a chemotaxis system under homogeneous Neumann boundary conditions within a bounded domain with a smooth boundary. The system describes the movement of cells in response to two chemical signal substances: one acts as a chemoattractant, while the other serves as a chemorepellent, both produced by the cells. The system takes into account chemotactic sensitivity in the reaction movement when detecting these chemicals. Under certain assumptions, we demonstrate the existence of a unique global bounded classical solution for the proposed problem. To further understand the time evolution of the system's solutions, we conduct numerical experiments and analyze the dynamic properties of the norm of the solutions with respect to variations in chemical production rates.

Optical Fiber Technology for Efficient Daylighting and Thermal Control: A Sustainable Approach for Buildings

Different direct solar harvesting systems for daylighting are being explored to achieve high uniform illumination deep within buildings at minimal cost. A promising solution to make these systems cost-effective is the use of plastic optical fibers (POFs). However, heat-related issues with low-cost POFs need to be addressed for the widespread adoption of efficient daylighting technologies. Previous studies have explored solutions for this overheating problem, but their effectiveness remains uncertain. This study proposes a low-cost fiber optic daylighting system integrated with a newly patented mechanical component designed to secure the fiber optic bundle at the focal point, providing three levels of heat filtration while ensuring uniform illumination. Our methodology involves selecting a small area, installing the setup, and measuring both heat and light readings, followed by validation through software simulations. The operational principle of this technology is explained, and experimental tests using lux meters and infrared thermometers were conducted to investigate the system’s characteristics. The three-level heat filtration device reduces temperature by approximately 35 °C at the surface of the optical fiber, and the average illumination of the room is around 400 lux. These results were further verified using RELUX simulation software. The findings demonstrate the promising potential of this new device in solar heat filtration and achieving uniform illumination. Recommendations for mitigating overheating damage and exploring heat filtering possibilities in new parabolic solar daylighting systems for further research are also provided.

Development of a DNI Forecast Tool Based on Machine Learning for the Smart Operation of a Parabolic Trough Collector System with Thermal Energy Storage

In the research project Smart Solar System (S3), the Solar-Institut Jülich (SIJ) developed forecast tools to predict the direct normal irradiance (DNI) in hourly resolution for the current day. The aim of the daily DNI forecast is to use it as input to enable a smart operation of a parabolic trough collector (PTC) system with a concrete thermal energy storage (C-TES) located at the company KEAN Soft Drinks Ltd in Limassol, Cyprus. The main focus in this work is on the development of a DNI forecast tool based on long short-term memory (LSTM), which is a recurrent neural network (RNN), and its comparison with a DNI forecast tool based on analytical algorithms (non-machine learning). Only the non-machine learning DNI forecast tool with hourly update was tested in real-life PTC plant operation since end of 2022. The comparisons between the three DNI forecast tools show that the potential of using machine learning is very high. Different comparisons were made including an evaluation of the accuracy of the tool (i.e. comparison of the DNI forecast data with DNI measurement data). The DNI forecast based on the LSTM network proved to be more accurate than the non-machine learning DNI forecasts when considering errors greater than ±100 W/m2. The error of the LSTM network compared to DNI measurement data was as follows: 43.6 % of the data was within ±100 W/m2, 74.8 % was within ±200 W/m2, 89.8 % was within ±300 W/m2 and 95.8 % was within ±400 W/m2.

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Boundary null controllability of some multi-dimensional linear parabolic systems by the moment method

Li-Yau type estimation of a semilinear parabolic system along geometric flow

This article provides a Li–Yau-type gradient estimate for a semilinear weighted parabolic system of semilinear equations along an abstract geometric flow on a smooth measure space. A Harnack-type inequality on the system is also derived at the end.

Effects of extra resource and harvesting on the pattern formation for a predation system

We deal with a reaction–diffusion predation system with extra resource provided to predator and harvesting on prey. We first discuss the long-time behaviors of parabolic system, including the dissipativeness and persistence of positive solutions. It is shown that under certain constraints of harvesting, the quality of extra resource, the predation and transition rates of predator, the dissipativeness is compatible with persistence. Secondly, some properties of steady-state system are investigated, mainly including the existence of non-constant positive solutions, Turing and steady-state bifurcation phenomena. It is found that the extra resource, prey harvesting and diffusion have significant impacts on the pattern formations. Furthermore, some numerical simulations on Turing patterns and steady-state bifurcation solutions are performed to illustrate the theoretical analysis. We observe that when the quantity of extra resources is low, changes in the quality of extra resources can lead to significant changes in the spatial distribution of species, which is in sharp contrast to the case of high quantity of extra resource. Additionally, we conclude that different diffusion rates of predator can lead to different spatial patterns for the system.

Improved regularity and analyticity of Cannone-Karch solutions of the three-dimensional Navier–Stokes equations on the torus

AbstractWe consider the three-dimensional Navier–Stokes equations, with initial data having second derivatives in the space of pseudomeasures. Solutions of this system with such data have been shown to exist previously by Cannone and Karch. As the Navier–Stokes equations are a parabolic system, the solutions gain regularity at positive times. We demonstrate an improved gain of regularity at positive times as compared to that demonstrated by Cannone and Karch. We further demonstrate that the solutions are analytic at all positive times, with lower bounds given for the radius of analyticity.

Existence of global weak solution to tumor chemotaxis competition systems with loop and signal dependent sensitivity

This article examines the weak solution of a fully parabolic chemotaxis-competition system with loop and signal-dependent sensitivity. The system is subject to homogeneous Neumann boundary conditions within an open, bounded domain \(\Omega\subset\mathbb{R}^n\), where \(n\geq 1\) and \(\partial\Omega\) is smooth. We assume that the parameters in the system are positive constants. Additionally, the initial data \((u_{10}, u_{20}, v_{10}, v_{20})\in L^2(\Omega)\times L^2(\Omega) \times W^{1,2}(\Omega)\times W^{1,2}(\Omega)\) are non-negative. The existence of a weak solution to the problem is established using energy inequality method. For more information see https://ejde.math.txstate.edu/Volumes/2024/56/abstr.html

Null controllability for a strongly coupled stochastic degenerate reaction-diffusion system

This paper concerns the null controllability for a strongly coupled stochastic degenerate reaction-diffusion system in cardiac electrocardiology, which describes the electrical activity in the cardiac tissue with random effects. To deal with degeneracy of this system, we first consider an approximate problem of this coupled stochastic degenerate system. By a weighted identity method, we then prove a uniform Carleman estimate for the adjoint system of this approximate problem, which is a strongly coupled backward stochastic parabolic system with homogeneous Neumann boundary conditions. Based on this Carleman estimate and a limit process, we finally obtain the null controllability result.

W‐schemes in spectral methods for reaction‐diffusion equations

AbstractWe consider reaction‐diffusion equations, which are parabolic systems of partial differential equations. Assuming periodic boundary conditions in space, a multidimensional Fourier transform can be used. A spectral method yields an initial value problem of a stiff system of ordinary differential equations for time‐dependent Fourier coefficients. We investigate the application of Rosenbrock‐Wanner W‐schemes in the time integration of the ordinary differential equations including step size selection. Results of numerical computations are presented using a Turing system in two space dimensions.

An N-dimensional fully parabolic attraction-repulsion chemotaxis system (with logistic source): Global solutions and boundedness

This paper deals with the coupled attraction-repulsion chemotaxis system of aggregation of Microglia in Alzhemer's disease given by(⋆){ut=Δu−χ∇⋅(u∇v)+ξ∇⋅(u∇w)+f(u),x∈Ω,t>0,vt=Δv−βv+αu,x∈Ω,t>0,wt=Δw−δw+γu,x∈Ω,t>0, is considered in a bounded Ω⊂RN(N≥1) with smooth boundary, where χ,α,ξ,β as well as γ and δ are positive constants. Here f(u)≡0 or f(u)≤a−bur for all u≥0 with some a∈R,b>0 and r≥1. By virtue of a novel approach on the basis of Maximal Sobolev regularity property for the heat equation, it is showed that when the repulsion cancels the attraction (i.e. χα=ξγ), the solution for an associated initial-boundary problem (⋆) is globally bounded ifN≤3andf(u)≤a−burwith somea∈R,b>0andr≥1 orf(u)≤a−burwith somea∈R,b>0andr>2N+4N+4, which extends (or partly improves) the corresponding results obtained in Tao-Wang (2013) [26] as well as Shi-Liu-Jin (2017) [25] and Wang-Zhuang-Zheng (2018) [31].

Existence and asymptotical behavior of solutions of a class of parabolic systems with homogeneous nonlinearity

In this paper we investigate the global existence and asymptotical stability of solutions to a class of parabolic systems with homogeneous nonlinearity for both bounded and unbounded domains. First we prove both global existence and finite time blow-up of solutions of the system for different initial conditions by using the potential well method, and the asymptotic behavior of the solutions are also considered. On the other hand, we also obtain global existence and finite time blow-up of solutions for both Sobolev subcritical and critical cases. We use a method of comparing least energy levels with that of semitrivial solutions to overcome the difficulties here.

Silicon porous disc featured parabolic trough collector functioned with paraffin: Thermal performance study

The advancement of renewable solar energy has potential for industrial heat exchanger applications, and its performance is enhanced by using nanofluid. Besides, the losses of heat during the energy transfer from solar to heat exchanger limit thermal performance and lead to reduced thermal efficiency. The conventional solar system functioning with nanofluid needs to be upgraded with an advanced thermal management system to limit energy loss and enrich the thermal performance of the overall system. The investigation aims to enrich the thermal performance of a parabolic trough solar collector (PTC), which features silicon porous discs and phase change material (PCM-paraffin). During the investigation, the flow of fluid ranged from 100 to 200 LPH with an interval of 50 LPH. Influences of silicon porous disc and phase change material melting phase on fluid temperature, temperature, energy stored, and thermal efficiency of the present system are experimentally investigated, and its values are related to conventional absorber performance without porous material and phase change material. The output results prove the significance of silicon porous and phase change material related to conventional absorber systems. The parabolic solar collector functioned with silicon porous material with phase change material phase on 200LPH spotted superior thermal performance including fluid temperature, PCM temperature, energy stored by phase change material, and average thermal & exergy efficiency of 84.2 °C, 98.5 °C 599.7 kJ, and 73.8 % & 15.1 %. These findings underscore the promising potential of porous medium absorbers in significantly elevating the efficiency of parabolic trough collector systems.

Optimal control problems for a parabolic system inspired by a cancer therapy

In this paper, we consider optimal control problems for a parabolic system modeling a therapy, based on oncolytic viruses, for the glioma brain cancer. Using several techniques typical of functional analysis, we prove the global in time well posedness of the control model, the existence of optimal controls for specific objective functionals, which are natural for cancer therapies, and we derive necessary conditions for optimality.