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A Nonlocal Cahn–Hilliard–Darcy System with Singular Potential, Degenerate Mobility, and Sources
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-21 DOI: 10.1007/s00245-025-10239-5
Cecilia Cavaterra, Sergio Frigeri, Maurizio Grasselli
{"title":"A Nonlocal Cahn–Hilliard–Darcy System with Singular Potential, Degenerate Mobility, and Sources","authors":"Cecilia Cavaterra,&nbsp;Sergio Frigeri,&nbsp;Maurizio Grasselli","doi":"10.1007/s00245-025-10239-5","DOIUrl":"10.1007/s00245-025-10239-5","url":null,"abstract":"<div><p>We consider a Cahn–Hilliard–Darcy system for an incompressible mixture of two fluids we already analyzed in [9]. In this system, the relative concentration difference <span>(varphi )</span> obeys a convective nonlocal Cahn–Hilliard equation with degenerate mobility and singular (e.g., logarithmic) potential, while the volume averaged fluid velocity <span>(varvec{u})</span> is given by a Darcy’s law subject to the Korteweg force <span>(mu nabla varphi )</span>, where the chemical potential <span>(mu )</span> is defined by means of a nonlocal Helmholtz free energy. The kinematic viscosity <span>(eta )</span> depends on <span>(varphi )</span>. With respect to the quoted contribution, here we assume that the Darcy’s law is subject to gravity and to a given additional source. Moreover, we suppose that the Cahn–Hilliard equation and the chemical potential contain source terms. Our main goal is to establish the existence of two notions of weak solutions. The first, called “generalized” weak solution, is based a convenient splitting of <span>(mu )</span> so that the entropy derivative does not need to be integrable. The second is slightly stronger and allows to reconstruct <span>(mu )</span> and to prove the validity of a canonical energy identity. For this reason, the latter is called “natural” weak solution. The rigorous relation between the two notions of weak solution is also analyzed. The existence of a global attractor for generalized weak solutions and time independent sources is then demonstrated via the theory of generalized semiflows introduced by J.M. Ball.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10239-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143455656","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Mean-Field Partial Information Non-zero Sum Stochastic Differential Games
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-19 DOI: 10.1007/s00245-025-10233-x
Tianyang Nie, Ke Yan
{"title":"Mean-Field Partial Information Non-zero Sum Stochastic Differential Games","authors":"Tianyang Nie,&nbsp;Ke Yan","doi":"10.1007/s00245-025-10233-x","DOIUrl":"10.1007/s00245-025-10233-x","url":null,"abstract":"<div><p>In this paper, we study a general mean-field partial information non-zero sum stochastic differential game, in which the dynamic of state is described by a stochastic differential equation (SDE) depending on the distribution of the state and the control domain of each player can be non-convex. Moreover, the control variables of both players can enter the diffusion coefficients of the state equation. We establish a necessary condition in the form of Pontryagin’s maximum principle for optimality. Then a verification theorem is obtained for optimal control when the control domain is convex. As an application, our results are applied to studying linear–quadratic (LQ) mean-field game in the type of scalar interaction.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143438720","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Correction: Optimality Conditions for Sparse Optimal Control of Viscous Cahn–Hilliard Systems with Logarithmic Potential
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-19 DOI: 10.1007/s00245-025-10234-w
Pierluigi Colli, Jürgen Sprekels, Fredi Tröltzsch
{"title":"Correction: Optimality Conditions for Sparse Optimal Control of Viscous Cahn–Hilliard Systems with Logarithmic Potential","authors":"Pierluigi Colli,&nbsp;Jürgen Sprekels,&nbsp;Fredi Tröltzsch","doi":"10.1007/s00245-025-10234-w","DOIUrl":"10.1007/s00245-025-10234-w","url":null,"abstract":"","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10234-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143446516","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Identification of Active Component Functions in Finite-Max Minimisation via a Smooth Reformulation
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-18 DOI: 10.1007/s00245-025-10229-7
Charl J. Ras, Matthew K. Tam, Daniel J. Uteda
{"title":"Identification of Active Component Functions in Finite-Max Minimisation via a Smooth Reformulation","authors":"Charl J. Ras,&nbsp;Matthew K. Tam,&nbsp;Daniel J. Uteda","doi":"10.1007/s00245-025-10229-7","DOIUrl":"10.1007/s00245-025-10229-7","url":null,"abstract":"<div><p>In this work, we consider a nonsmooth minimisation problem in which the objective function can be represented as the maximum of finitely many smooth “component functions”. First, we study a smooth min–max reformulation of the problem. Due to this smoothness, the model provides enhanced capability of exploiting the structure of the problem, when compared to methods that attempt to tackle the nonsmooth problem directly. Then, we present several approaches to identify the set of active component functions at a minimiser, all within finitely many iterations of a first order method for solving the smooth model. As is well known, the problem can be equivalently rewritten in terms of these component functions, but a key challenge is to identify this set a priori. Such an identification is clearly beneficial in an algorithmic sense, since we can discard those component functions which are not necessary to describe the solution, which in turn can facilitate faster convergence. Finally, numerical results comparing the accuracy of each of these approaches are presented, along with the effect they have on reducing the complexity of the original problem.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10229-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143430941","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Path-Dependent Hamilton–Jacobi Equations with u-Dependence and Time-Measurable Hamiltonians
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-15 DOI: 10.1007/s00245-025-10230-0
Elena Bandini, Christian Keller
{"title":"Path-Dependent Hamilton–Jacobi Equations with u-Dependence and Time-Measurable Hamiltonians","authors":"Elena Bandini,&nbsp;Christian Keller","doi":"10.1007/s00245-025-10230-0","DOIUrl":"10.1007/s00245-025-10230-0","url":null,"abstract":"<div><p>We establish existence and uniqueness of minimax solutions for a fairly general class of path-dependent Hamilton–Jacobi equations. In particular, the relevant Hamiltonians can contain the solution and they only need to be measurable with respect to time. We apply our results to optimal control problems of (delay) functional differential equations with cost functionals that have discount factors and with time-measurable data. Our main results are also crucial for our companion paper Bandini and Keller (Non-local Hamilton–Jacobi–Bellman equations for the stochastic optimal control of path-dependent piecewise deterministic processes, 2024, http://arxiv.org/abs/2408.02147), where non-local path-dependent Hamilton–Jacobi–Bellman equations associated to the stochastic optimal control of non-Markovian piecewise deterministic processes are studied.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143423370","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Non-Autonomous Degenerate Parabolic Equations with Robin Boundary Conditions: Carleman Estimates and Null-Controllability
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-15 DOI: 10.1007/s00245-025-10227-9
Mohammad Akil, Genni Fragnelli, Sarah Ismail
{"title":"Non-Autonomous Degenerate Parabolic Equations with Robin Boundary Conditions: Carleman Estimates and Null-Controllability","authors":"Mohammad Akil,&nbsp;Genni Fragnelli,&nbsp;Sarah Ismail","doi":"10.1007/s00245-025-10227-9","DOIUrl":"10.1007/s00245-025-10227-9","url":null,"abstract":"<div><p>The Earth’s climate system naturally adjusts to maintain a balance between the energy received from the Sun and the energy reflected back into space, a concept known as the “Earth’s radiation budget”. However, this balance has been disrupted by human activities, leading to global warming. Starting from the energy balance model proposed by Budyko and Sellers, and considering a time-dependent diffusion coefficient, we prove the null-controllability of non-autonomous degenerate parabolic problems, in the sense that the Earth achieves a desired temperature, by finding new Carleman estimates for the non-homogeneous adjoint problems. At the degeneracy point, we impose Robin boundary condition which is appropriate for modeling heat transfer at the Earth’s surface. Moreover, we provide the equivalence between null-controllability and observability inequality for the non-autonomous case. At the end, we present some extensions and open problems.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10227-9.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143423369","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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 Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-14 DOI: 10.1007/s00245-025-10232-y
Yue Zhou, Changchun Liu
{"title":"Global Weak Solutions in a Haptotactic Cross-Diffusion System Modeling Oncolytic Virotherapy with Nonlinear Diffusion","authors":"Yue Zhou,&nbsp;Changchun Liu","doi":"10.1007/s00245-025-10232-y","DOIUrl":"10.1007/s00245-025-10232-y","url":null,"abstract":"<div><p>This paper discusses an initial-boundary value problem for a doubly haptotactic cross-diffusion system arising from the oncolytic virotherapy </p><div><div><span>$$begin{aligned} {left{ begin{array}{ll} u_t=Delta u^m-nabla cdot (unabla v) +mu u(1-u)-uz,;;&amp; ;xin Omega ,~t&gt;0, v_t=-(u+w)v,;;&amp; ;xin Omega ,~t&gt;0, w_t=Delta w-nabla cdot (wnabla v)-w+uz,;;&amp; ;xin Omega ,~t&gt;0, z_t=DDelta z-z-uz+beta w,;;&amp; ;xin Omega ,~t&gt;0, frac{partial u^m}{partial nu }-ufrac{partial v}{partial nu }=frac{partial w}{partial nu }-wfrac{partial v}{partial nu }=frac{partial z}{partial nu }=0,;;&amp; ;xin partial Omega ,~t&gt;0, u(x,0)=u_{0},~v(x,0)=v_{0},~w(x,0)=w_{0}, z(x,0)=z_{0},;;&amp; ;xin Omega , end{array}right. } end{aligned}$$</span></div></div><p>in a smooth bounded domain <span>( Omega subset {mathbb {R}}^{N}(N=1,2) )</span> with <span>( m&gt;1, beta&gt;0, mu &gt;0 )</span>, and <span>( D&gt;0)</span>. We prove that for any large initial datum, the problem admits a global ‘very’ weak solution for any <span>(m&gt;1)</span>.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-14","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143404145","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Convergence of the Stochastic Navier–Stokes-(alpha ) Solutions Toward the Stochastic Navier–Stokes Solutions 随机纳维-斯托克斯（Navier-Stokes）-（alpha ）解向随机纳维-斯托克斯解的收敛性
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-13 DOI: 10.1007/s00245-025-10228-8
Jad Doghman, Ludovic Goudenège
{"title":"Convergence of the Stochastic Navier–Stokes-(alpha ) Solutions Toward the Stochastic Navier–Stokes Solutions","authors":"Jad Doghman,&nbsp;Ludovic Goudenège","doi":"10.1007/s00245-025-10228-8","DOIUrl":"10.1007/s00245-025-10228-8","url":null,"abstract":"<div><p>Loosely speaking, the Navier–Stokes-<span>(alpha )</span> model and the Navier–Stokes equations differ by a spatial filtration parametrized by a scale denoted <span>(alpha )</span>. Starting from a strong two-dimensional solution to the Navier–Stokes-<span>(alpha )</span> model driven by a multiplicative noise, we demonstrate that it generates a strong solution to the stochastic Navier–Stokes equations under the condition <span>(alpha rightarrow 0)</span>. The initially introduced probability space and the Wiener process are maintained throughout the investigation, thanks to a local monotonicity property that abolishes the use of Skorokhod’s theorem. High spatial regularity a priori estimates for the fluid velocity vector field are carried out within periodic boundary conditions.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-025-10228-8.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396661","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Before and After Default: Information and Optimal Portfolio via Anticipating Calculus 违约前后：通过预期微积分获得信息和最佳投资组合
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-12 DOI: 10.1007/s00245-024-10210-w
Bernardo D’Auria, Giulia Di Nunno, Jose A. Salmeron
{"title":"Before and After Default: Information and Optimal Portfolio via Anticipating Calculus","authors":"Bernardo D’Auria,&nbsp;Giulia Di Nunno,&nbsp;Jose A. Salmeron","doi":"10.1007/s00245-024-10210-w","DOIUrl":"10.1007/s00245-024-10210-w","url":null,"abstract":"<div><p>Default risk calculus plays a crucial role in portfolio optimization when the risky asset is under threat of bankruptcy. However, traditional stochastic control techniques are not applicable in this scenario, and additional assumptions are required to obtain the optimal solution in a before-and-after default context. We propose an alternative approach using forward integration, which allows to avoid one of the restrictive assumptions, the <i>Jacod density hypothesis</i>. We demonstrate that, in the case of logarithmic utility, the weaker <i>intensity hypothesis</i> is the appropriate condition for optimality. Furthermore, we establish the semimartingale decomposition of the risky asset in the filtration that is progressively enlarged to accommodate the default process, under the assumption of the existence of the optimal portfolio. This work aims to provide valuable insights for developing effective risk management strategies when facing default risk.\u0000</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00245-024-10210-w.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396678","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
Null Controllability for Stochastic Parabolic Equations Coupled by First and Zero Order Terms
IF 1.6 2区 数学
Applied Mathematics and Optimization Pub Date : 2025-02-12 DOI: 10.1007/s00245-025-10231-z
Mahmoud Baroun, Said Boulite, Abdellatif Elgrou, Lahcen Maniar
{"title":"Null Controllability for Stochastic Parabolic Equations Coupled by First and Zero Order Terms","authors":"Mahmoud Baroun,&nbsp;Said Boulite,&nbsp;Abdellatif Elgrou,&nbsp;Lahcen Maniar","doi":"10.1007/s00245-025-10231-z","DOIUrl":"10.1007/s00245-025-10231-z","url":null,"abstract":"<div><p>We prove the null controllability of forward and backward linear stochastic parabolic equations with first and zero order coupling terms, and with bounded coefficients. The null controllability results rely on novel Carleman estimates for the backward and forward adjoint equations, established through the application of the duality technique. In doing so, we resolve an open question (see Remark 2.2 in [Tang and Zhang in SIAM J Control Optim 48:2191–2216, 2009]). Moreover, we provide more accurate estimates for the null-control costs.</p></div>","PeriodicalId":55566,"journal":{"name":"Applied Mathematics and Optimization","volume":"91 2","pages":""},"PeriodicalIF":1.6,"publicationDate":"2025-02-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143396679","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":2,"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
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