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Controllability Results for a Cross Diffusion System with a Free Boundary by a Flatness Approach 具有自由边界的交叉扩散系统的可控性结果
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-29 DOI: 10.1007/s10440-023-00607-0
Blaise Colle, Jérôme Lohéac, Takéo Takahashi
{"title":"Controllability Results for a Cross Diffusion System with a Free Boundary by a Flatness Approach","authors":"Blaise Colle,&nbsp;Jérôme Lohéac,&nbsp;Takéo Takahashi","doi":"10.1007/s10440-023-00607-0","DOIUrl":"10.1007/s10440-023-00607-0","url":null,"abstract":"<div><p>We study a one-dimensional cross diffusion system with a free boundary modeling the Physical Vapor Deposition. Using the flatness approach, we show several results of boundary controllability for this system in spaces of Gevrey class functions. One of the main difficulties consists in the physical constraints on the state and on the control. More precisely, the state corresponds to volume fractions of the <span>(n+1)</span> chemical species and to the thickness of the film produced in the process, whereas the controls are the fluxes of the chemical species. We obtain the local controllability in the case where we apply <span>(n+1)</span> nonnegative controls and a controllability result for large time in the case where we apply <span>(n)</span> controls without any sign constraints. We also show in this last case that the controllability may fail for small times. We illustrate these results with some numerical simulations.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-29","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50053899","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}
引用次数: 0
New Type of Fractal Functions for the General Data Sets 一般数据集的新型分形函数
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-25 DOI: 10.1007/s10440-023-00604-3
Manuj Verma, Amit Priyadarshi
{"title":"New Type of Fractal Functions for the General Data Sets","authors":"Manuj Verma,&nbsp;Amit Priyadarshi","doi":"10.1007/s10440-023-00604-3","DOIUrl":"10.1007/s10440-023-00604-3","url":null,"abstract":"<div><p>In this paper, we prove the existence of the fractal interpolation function corresponding to a general data set using the Rakotch contraction theory and iterated function system. We also prove the existence of the fractal measure supported on the graph of the fractal interpolation function. We emphasize the fact that our theory covers the fractal interpolation theory for finite cases, countably infinite cases, and many more. We establish dimensional results for the graph of the fractal interpolation function for the general data sets.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50047982","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}
引用次数: 0
A Refinement of the First Eigenvalue and Eigenfunction of the Linearized Moser-Trudinger Problem 线性化Moser-Trudinger问题的第一特征值和特征函数的精化
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-20 DOI: 10.1007/s10440-023-00603-4
Kefan Pan, Jing Yang
{"title":"A Refinement of the First Eigenvalue and Eigenfunction of the Linearized Moser-Trudinger Problem","authors":"Kefan Pan,&nbsp;Jing Yang","doi":"10.1007/s10440-023-00603-4","DOIUrl":"10.1007/s10440-023-00603-4","url":null,"abstract":"<div><p>We revisit the following Moser-Trudinger problem </p><div><div><span>$$ textstylebegin{cases} -Delta u=lambda ue^{u^{2}} &amp;text{in } Omega , u&gt;0&amp;text{in } Omega , u=0 &amp;text{on } partial Omega , end{cases} $$</span></div></div><p> where <span>(Omega subset mathbb{R}^{2})</span> is a smooth bounded domain and <span>(lambda &gt;0)</span> is sufficiently small. Qualitative analysis of peaked solutions for Moser-Trudinger type equation in <span>(mathbb{R}^{2})</span> has been widely studied in recent decades. In this paper, we continue to consider the qualitative properties of the eigenvalues and eigenfunctions for the corresponding linearized Moser-Trudinger problem by using a variety of local Pohozaev identities combined with some elliptic theory in dimension two. Here we give some fine estimates for the first eigenvalue and eigenfunction of the linearized Moser-Trudinger problem. Since this problem is a critical exponent for dimension two and will lose compactness, we have to obtain some new and technical estimates.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50094758","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}
引用次数: 0
On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms 空间相关系数波动方程:奇点和低阶项
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-13 DOI: 10.1007/s10440-023-00601-6
Marco Discacciati, Claudia Garetto, Costas Loizou
{"title":"On the Wave Equation with Space Dependent Coefficients: Singularities and Lower Order Terms","authors":"Marco Discacciati,&nbsp;Claudia Garetto,&nbsp;Costas Loizou","doi":"10.1007/s10440-023-00601-6","DOIUrl":"10.1007/s10440-023-00601-6","url":null,"abstract":"<div><p>This paper complements the study of the wave equation with discontinuous coefficients initiated in (Discacciati et al. in <i>J. Differ. Equ.</i> <b>319</b> (2022) 131–185) in the case of time-dependent coefficients. Here we assume that the equation coefficients are depending on space only and we formulate Levi conditions on the lower order terms to guarantee the existence of a very weak solution as defined in (Garetto and Ruzhansky in <i>Arch. Ration. Mech. Anal.</i> <b>217</b> (2015) 113–154). As a toy model we study the wave equation in conservative form with discontinuous velocity and we provide a qualitative analysis of the corresponding very weak solution via numerical methods.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00601-6.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50025436","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Stabilization Effects of Magnetic Field on a 2D Anisotropic MHD System with Partial Dissipation 磁场对二维各向异性局部耗散MHD系统的稳定效应
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-11 DOI: 10.1007/s10440-023-00602-5
Dongxiang Chen, Fangfang Jian
{"title":"Stabilization Effects of Magnetic Field on a 2D Anisotropic MHD System with Partial Dissipation","authors":"Dongxiang Chen,&nbsp;Fangfang Jian","doi":"10.1007/s10440-023-00602-5","DOIUrl":"10.1007/s10440-023-00602-5","url":null,"abstract":"<div><p>To uncover that the magnetic field mechanism can stabilize electrically conducting turbulent fluids, we investigate the stability of a special two dimensional anisotropic MHD system with vertical dissipation in the horizontal velocity component and partial magnetic damping near a background magnetic field. Since the MHD system has only vertical dissipation in the horizontal velocity and vertical magnetic damping, the stability issue and large time behavior problem of the linearized magneto-hydrodynamic system is not trivial. By performing refined energy estimates on the linear system coupled with a careful analysis of the nonlinearities, the stability of a MHD-type system near a background magnetic field is justified for the initial data belonging to <span>(H^{3}(mathbf{R}^{2}))</span> space. The authors also build the explicit decay rates of the linearized system.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50019991","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}
引用次数: 0
Symmetries and Exact Solutions of the Diffusive Holling–Tanner Prey-Predator Model 扩散Holling-Tanner捕食-捕食模型的对称性和精确解
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-06 DOI: 10.1007/s10440-023-00600-7
Roman Cherniha, Vasyl’ Davydovych
{"title":"Symmetries and Exact Solutions of the Diffusive Holling–Tanner Prey-Predator Model","authors":"Roman Cherniha,&nbsp;Vasyl’ Davydovych","doi":"10.1007/s10440-023-00600-7","DOIUrl":"10.1007/s10440-023-00600-7","url":null,"abstract":"<div><p>We consider the classical Holling–Tanner model extended on 1D space by introducing the diffusion term. Making a reasonable simplification, the diffusive Holling–Tanner system is studied by means of symmetry based methods. Lie and <span>(Q)</span>-conditional (nonclassical) symmetries are identified. The symmetries obtained are applied for finding a wide range of exact solutions, their properties are studied and a possible biological interpretation is proposed. 3D plots of the most interesting solutions are drown as well.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00600-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50012265","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity 具有多孔介质扩散和奇异灵敏度的chemotaxi - stokes模型的全局有界解
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-09-04 DOI: 10.1007/s10440-023-00599-x
Jianping Wang
{"title":"Global Bounded Solution in a Chemotaxis-Stokes Model with Porous Medium Diffusion and Singular Sensitivity","authors":"Jianping Wang","doi":"10.1007/s10440-023-00599-x","DOIUrl":"10.1007/s10440-023-00599-x","url":null,"abstract":"<div><p>This article is concerned with a chemotaxis-Stokes system with porous medium diffusion and singular sensitivity: </p><div><div><span> $$begin{aligned} left { textstylebegin{array}{l@{quad }l} n_{t}+ucdot nabla n=nabla cdot (D(n)nabla n)-nabla cdot (nS(x,n,c)cdot nabla c),&amp;xin Omega , t&gt;0, c_{t}+ucdot nabla c=Delta c-nc,&amp;xin Omega , t&gt;0, u_{t}+nabla P=Delta u+nnabla Phi , nabla cdot u=0,&amp;xin Omega , t&gt;0 end{array}displaystyle right . end{aligned}$$ </span></div></div><p> in a bounded domain <span>(Omega subset mathbb{R}^{N})</span> with <span>(2le Nle 3)</span>, where <span>(Din C^{0}([0,infty ))cap C^{2}((0,infty )))</span> and <span>(Sin C^{2}(bar{Omega }times [0,infty )^{2};mathbb{R}^{Ntimes N}))</span>. The global solvability of the system in a natural weak sense is obtained under the conditions that <span>(D(n)ge k_{D}n^{m-1})</span> and <span>(|S(x,n,c)|le frac{S_{0}(c)}{c^{alpha }})</span> for all <span>((x,n,c)in Omega times (0,infty )^{2})</span> with some <span>(k_{D}&gt;0)</span>, <span>(m&gt;frac{3N-2}{2N})</span>, <span>(alpha in [0,1))</span> and some nondecreasing <span>(S_{0}:(0,infty )rightarrow (0,infty ))</span>. Moreover, in the case that <span>(m=frac{3N-2}{2N})</span> and <span>(alpha in [0,1))</span>, we also get the global weak solutions under smallness assumptions on the initial data <span>(|n_{0}|_{L^{1}(Omega )})</span> and <span>(|c_{0}|_{L^{infty }(Omega )})</span>.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-09-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00599-x.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50015452","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Vanishing Micro-Rotation and Angular Viscosities Limit for the 2D Micropolar Equations in a Bounded Domain 有界域中二维微极方程的消失微旋转和角粘性极限
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-08-30 DOI: 10.1007/s10440-023-00596-0
Yangyang Chu, Yuelong Xiao
{"title":"Vanishing Micro-Rotation and Angular Viscosities Limit for the 2D Micropolar Equations in a Bounded Domain","authors":"Yangyang Chu,&nbsp;Yuelong Xiao","doi":"10.1007/s10440-023-00596-0","DOIUrl":"10.1007/s10440-023-00596-0","url":null,"abstract":"<div><p>In this paper, we investigate the vanishing micro-rotation and angular viscosities limit of solutions to the 2D incompressible micropolar equations in a bounded domain with Navier-type boundary conditions satisfied by the velocity field. In a general bounded smooth domain <span>(Omega )</span>, we establish the uniform <span>(H^{2}(Omega ))</span> estimates (independent of the micro-rotation and angular viscosities) of global strong solutions and prove the rate of convergence of viscosity solutions to the inviscid solutions in <span>(C(0,T;H^{1}(Omega )))</span> for any <span>(T&gt;0)</span>.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50054827","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}
引用次数: 0
On the Homogenization of the Renewal Equation with Heterogeneous External Constraints 关于具有非均匀外部约束的更新方程的均化
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-08-30 DOI: 10.1007/s10440-023-00594-2
Étienne Bernard, Francesco Salvarani
{"title":"On the Homogenization of the Renewal Equation with Heterogeneous External Constraints","authors":"Étienne Bernard,&nbsp;Francesco Salvarani","doi":"10.1007/s10440-023-00594-2","DOIUrl":"10.1007/s10440-023-00594-2","url":null,"abstract":"<div><p>We study the homogenization limit of the renewal equation with heterogeneous external constraints by means of the two-scale convergence theory. We prove that the homogenized limit satisfies an equation involving non-local terms, which are the consequence of the oscillations in the birth and death terms. We have moreover shown that the numerical approximation of the homogenized equation via the two-scale limit gives an alternative way for the numerical study of the solution of the limiting problem.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00594-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50054829","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Exponential Ergodicity of a Degenerate Age-Size Piecewise Deterministic Process 退化年龄大小分段确定性过程的指数遍历性
IF 1.6 4区 数学
Acta Applicandae Mathematicae Pub Date : 2023-08-30 DOI: 10.1007/s10440-023-00597-z
Ignacio Madrid
{"title":"Exponential Ergodicity of a Degenerate Age-Size Piecewise Deterministic Process","authors":"Ignacio Madrid","doi":"10.1007/s10440-023-00597-z","DOIUrl":"10.1007/s10440-023-00597-z","url":null,"abstract":"<div><p>We study the long-time behaviour of a non conservative piecewise deterministic measure-valued Markov process modelling the proliferation of an age-and-size structured population, which generalises the “adder” model of bacterial growth. Firstly, we prove the existence of eigenelements of the associated infinitesimal generator, which are used to bring ourselves back to the study of a conservative Markov process using a Doob <span>(h)</span>-transform. Finally, we obtain the exponential ergodicity of the process via drift-minorisation arguments. Specifically, we show the “petiteness” of the compact sets of the state space. This permits to circumvent the difficulties encountered when trying to construct mixing trajectories at a fixed uniform time on an unbounded two-dimensional space with only advection and degenerate jump terms.</p></div>","PeriodicalId":53132,"journal":{"name":"Acta Applicandae Mathematicae","volume":null,"pages":null},"PeriodicalIF":1.6,"publicationDate":"2023-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s10440-023-00597-z.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"50054828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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