Pub Date : 2022-03-31DOI: 10.1017/s0956792522000067
Min Li, Zhaoyin Xiang, Guanyu Zhou
This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.
{"title":"The stability analysis of a 2D Keller–Segel–Navier–Stokes system in fast signal diffusion","authors":"Min Li, Zhaoyin Xiang, Guanyu Zhou","doi":"10.1017/s0956792522000067","DOIUrl":"https://doi.org/10.1017/s0956792522000067","url":null,"abstract":"This paper investigates the stability of a fully parabolic–parabolic-fluid (PP-fluid) system of the Keller–Segel–Navier–Stokes type in a bounded planar domain under the natural volume-filling hypothesis. In the limit of fast signal diffusion, we first show that the global classical solutions of the PP-fluid system will converge to the solution of the corresponding parabolic–elliptic-fluid (PE-fluid) system. As a by-product, we obtain the global well-posedness of the PE-fluid system for general large initial data. We also establish some new exponential time decay estimates for suitable small initial cell mass, which in particular ensure an improvement of convergence rate on time. To further explore the stability property, we carry out three numerical examples of different types: the nontrivial and trivial equilibriums, and the rotating aggregation. The simulation results illustrate the possibility to achieve the optimal convergence and show the vanishment of the deviation between the PP-fluid system and PE-fluid system for the equilibriums and the drastic fluctuation of error for the rotating solution.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49390556","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-30DOI: 10.1017/s0956792522000080
B. Schweizer
The Helmholtz equation $-nablacdot (anabla u) - omega^2 u = f$ is considered in an unbounded wave guide $Omega := mathbb{R} times S subset mathbb{R}^d$ , $Ssubset mathbb{R}^{d-1}$ a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction $x_1 in mathbb{R}$ or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies $omega$ , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.
{"title":"Inhomogeneous Helmholtz equations in wave guides – existence and uniqueness results with energy methods","authors":"B. Schweizer","doi":"10.1017/s0956792522000080","DOIUrl":"https://doi.org/10.1017/s0956792522000080","url":null,"abstract":"The Helmholtz equation \u0000 \u0000 \u0000 \u0000$-nablacdot (anabla u) - omega^2 u = f$\u0000\u0000 \u0000 is considered in an unbounded wave guide \u0000 \u0000 \u0000 \u0000$Omega := mathbb{R} times S subset mathbb{R}^d$\u0000\u0000 \u0000 , \u0000 \u0000 \u0000 \u0000$Ssubset mathbb{R}^{d-1}$\u0000\u0000 \u0000 a bounded domain. The coefficient a is strictly elliptic and either periodic in the unbounded direction \u0000 \u0000 \u0000 \u0000$x_1 in mathbb{R}$\u0000\u0000 \u0000 or periodic outside a compact subset; in the latter case, two different periodic media can be used in the two unbounded directions. For non-singular frequencies \u0000 \u0000 \u0000 \u0000$omega$\u0000\u0000 \u0000 , we show the existence of a solution u. While previous proofs of such results were based on analyticity arguments within operator theory, here, only energy methods are used.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48729030","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-17DOI: 10.1017/s0956792523000153
Zhomart Turarov, C. Totzeck
We propose an extension of the anisotropic interaction model which allows for collision avoidance in pairwise interactions by a rotation of forces (Totzeck (2020) Kinet. Relat. Models13(6), 1219–1242.) by including the agents’ body size. The influence of the body size on the self-organisation of the agents in channel and crossing scenarios as well as the fundamental diagram is studied. Since the model is stated as a coupled system of ordinary differential equations, we are able to give a rigorous well-posedness analysis. Then we state a parameter calibration problem that involves data from real experiments. We prove the existence of a minimiser and derive the corresponding first-order optimality conditions. With the help of these conditions, we propose a gradient descent algorithm based on mini-batches of the data set. We employ the proposed algorithm to fit the parameter of the collision avoidance and the strength parameters of the interaction forces to given real data from experiments. The results underpin the feasibility of the method.
{"title":"Gradient-based parameter calibration of an anisotropic interaction model for pedestrian dynamics","authors":"Zhomart Turarov, C. Totzeck","doi":"10.1017/s0956792523000153","DOIUrl":"https://doi.org/10.1017/s0956792523000153","url":null,"abstract":"\u0000 We propose an extension of the anisotropic interaction model which allows for collision avoidance in pairwise interactions by a rotation of forces (Totzeck (2020) Kinet. Relat. Models13(6), 1219–1242.) by including the agents’ body size. The influence of the body size on the self-organisation of the agents in channel and crossing scenarios as well as the fundamental diagram is studied. Since the model is stated as a coupled system of ordinary differential equations, we are able to give a rigorous well-posedness analysis. Then we state a parameter calibration problem that involves data from real experiments. We prove the existence of a minimiser and derive the corresponding first-order optimality conditions. With the help of these conditions, we propose a gradient descent algorithm based on mini-batches of the data set. We employ the proposed algorithm to fit the parameter of the collision avoidance and the strength parameters of the interaction forces to given real data from experiments. The results underpin the feasibility of the method.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47100079","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-16DOI: 10.1017/s0956792522000055
M. Rodrigo
Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.
{"title":"A Laplace transform approach to direct and inverse problems for multi-compartment models","authors":"M. Rodrigo","doi":"10.1017/s0956792522000055","DOIUrl":"https://doi.org/10.1017/s0956792522000055","url":null,"abstract":"Multi-compartment models described by systems of linear ordinary differential equations are considered. Catenary models are a particular class where the compartments are arranged in a chain. A unified methodology based on the Laplace transform is utilised to solve direct and inverse problems for multi-compartment models. Explicit formulas for the parameters in a catenary model are obtained in terms of the roots of elementary symmetric polynomials. A method to estimate parameters for a general multi-compartment model is also provided. Results of numerical simulations are presented to illustrate the effectiveness of the approach.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42144327","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-03-11DOI: 10.1017/s0956792522000031
C. D'apice, P. Kogut, R. Manzo, M. Uvarov
In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS – Spatial Data Analytics, GIS Software, Satellite Imagery – is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.
{"title":"Variational model with nonstandard growth conditions for restoration of satellite optical images using synthetic aperture radar","authors":"C. D'apice, P. Kogut, R. Manzo, M. Uvarov","doi":"10.1017/s0956792522000031","DOIUrl":"https://doi.org/10.1017/s0956792522000031","url":null,"abstract":"In this paper, the problem of restoration of cloud contaminated optical images is studied in the case when we have no information about brightness of such images in the damage region. We propose a new variational approach for exact restoration of optical multi-band images utilising Synthetic Aperture Radar (EOS – Spatial Data Analytics, GIS Software, Satellite Imagery – is a cloud-based platform to derive remote sensing data and analyse satellite imagery for business and science purposes) images of the same regions. We prove existence of solutions, propose an alternating minimisation method for computing them, prove convergence of this method to weak solutions of the original problem and derive optimality conditions.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-03-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41804445","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-02-03DOI: 10.1017/s0956792522000018
N. Ray, R. Schulz
Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.
{"title":"Existence and uniqueness of solutions to a flow and transport problem with degenerating coefficients","authors":"N. Ray, R. Schulz","doi":"10.1017/s0956792522000018","DOIUrl":"https://doi.org/10.1017/s0956792522000018","url":null,"abstract":"Structural changes of the pore space and clogging phenomena are inherent to many porous media applications. However, related analytical investigations remain challenging due to potentially vanishing coefficients in the respective systems of partial differential equations. In this research, we apply an appropriate scaling of the unknowns and work with porosity-weighted function spaces. This enables us to prove existence, uniqueness and non-negativity of weak solutions to a combined flow and transport problem with vanishing, but prescribed porosity field, permeability and diffusion.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-02-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"49292902","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2022-01-31DOI: 10.1017/s095679252300013x
Stephan Gärttner, P. Knabner, N. Ray
Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro–macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.
{"title":"Local existence of strong solutions to micro–macro models for reactive transport in evolving porous media","authors":"Stephan Gärttner, P. Knabner, N. Ray","doi":"10.1017/s095679252300013x","DOIUrl":"https://doi.org/10.1017/s095679252300013x","url":null,"abstract":"\u0000 Two-scale models pose a promising approach in simulating reactive flow and transport in evolving porous media. Classically, homogenised flow and transport equations are solved on the macroscopic scale, while effective parameters are obtained from auxiliary cell problems on possibly evolving reference geometries (micro-scale). Despite their perspective success in rendering lab/field-scale simulations computationally feasible, analytic results regarding the arising two-scale bilaterally coupled system often restrict to simplified models. In this paper, we first derive smooth dependence results concerning the partial coupling from the underlying geometry to macroscopic quantities. Therefore, alterations of the representative fluid domain are described by smooth paths of diffeomorphisms. Exploiting the gained regularity of the effective space- and time-dependent macroscopic coefficients, we present local-in-time existence results for strong solutions to the partially coupled micro–macro system using fixed-point arguments. What is more, we extend our results to the bilaterally coupled diffusive transport model including a level-set description of the evolving geometry.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2022-01-31","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46114763","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-12-01DOI: 10.1017/s0956792521000334
W. Ma, Yehui Huang, Fudong Wang
The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.
{"title":"Inverse scattering transforms for non-local reverse-space matrix non-linear Schrödinger equations","authors":"W. Ma, Yehui Huang, Fudong Wang","doi":"10.1017/s0956792521000334","DOIUrl":"https://doi.org/10.1017/s0956792521000334","url":null,"abstract":"The aim of the paper is to explore non-local reverse-space matrix non-linear Schrödinger equations and their inverse scattering transforms. Riemann–Hilbert problems are formulated to analyse the inverse scattering problems, and the Sokhotski–Plemelj formula is used to determine Gelfand–Levitan–Marchenko-type integral equations for generalised matrix Jost solutions. Soliton solutions are constructed through the reflectionless transforms associated with poles of the Riemann–Hilbert problems.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":"1 1","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-12-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"41870943","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Pub Date : 2021-11-01DOI: 10.1017/s0956792521000310
S. AL-ALI, G. Hocking, D. Farrow, H. Zhang
A spectral method is developed to study the steady and unsteady flow of fluid into a line sink from a horizontally confined aquifer, and the results are compared to solutions obtained implementing the finite element package COMSOLTM. The aquifer or drain is considered to be confined below so that the solutions are fundamentally unsteady. Comparison is made between the two methods in determining the drawdown of the surface.
{"title":"A spectral modelling approach for fluid flow into a line sink in a confined aquifer","authors":"S. AL-ALI, G. Hocking, D. Farrow, H. Zhang","doi":"10.1017/s0956792521000310","DOIUrl":"https://doi.org/10.1017/s0956792521000310","url":null,"abstract":"A spectral method is developed to study the steady and unsteady flow of fluid into a line sink from a horizontally confined aquifer, and the results are compared to solutions obtained implementing the finite element package COMSOLTM. The aquifer or drain is considered to be confined below so that the solutions are fundamentally unsteady. Comparison is made between the two methods in determining the drawdown of the surface.","PeriodicalId":51046,"journal":{"name":"European Journal of Applied Mathematics","volume":" ","pages":""},"PeriodicalIF":1.9,"publicationDate":"2021-11-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"43975569","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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