Article
Mathematics, Applied
Yanqing Wang, Yulin Ye
Summary: This paper discusses the energy conservation for weak solutions to the 3D incompressible Navier-Stokes-Cahn-Hilliard system and presents corresponding conditions. By analyzing these conditions, the energy equality of weak solutions is proved, improving upon previous research findings.
APPLIED MATHEMATICS LETTERS
(2022)
Article
Mathematics, Applied
Sergio Frigeri
Summary: The study considered a diffuse interface model describing flow and phase separation of a binary isothermal mixture of immiscible viscous fluids with different densities. The model is a nonlocal version consisting of a Navier-Stokes system coupled with a nonlocal Cahn-Hilliard equation. The existence of global weak solutions with degenerate mobility was proven, relying on a regularization technique based on approximation of the singular potential. Additionally, existence and regularity of the pressure field was discussed, along with the establishment of the energy identity in two dimensions for slightly more regular solutions.
ANNALES DE L INSTITUT HENRI POINCARE-ANALYSE NON LINEAIRE
(2021)
Article
Mathematics, Applied
Martin Kalousek, Sourav Mitra, Anja Schloemerkemper
Summary: This article discusses a system of partial differential equations modeling a diffuse interface flow of two Newtonian incompressible magnetic fluids, showing global in time existence of weak solutions to the system using the time discretization method.
NONLINEAR ANALYSIS-REAL WORLD APPLICATIONS
(2021)
Article
Mathematics, Applied
Zhilei Liang, Dehua Wang
Summary: In this study, we investigate the Cahn-Hilliard/Navier-Stokes equations for stationary compressible flows in a three-dimensional bounded domain. We demonstrate the existence of weak solutions when the adiabatic exponent gamma is greater than 4/3, using weighted total energy estimates and new techniques to overcome the challenges posed by capillary stress.
JOURNAL OF NONLINEAR SCIENCE
(2022)
Article
Mathematics, Applied
Juliana Honda Lopes, Gabriela Planas
Summary: This study focuses on the mathematical analysis of a general cell-fluid Navier-Stokes model that incorporates chemotaxis. The model is based on a mixture theory multiphase formulation, consisting of mass balance equations and momentum balance equations for the cell and fluid phase, along with an oxygen convection-diffusion-reaction equation. The existence of weak solutions is investigated in a bounded domain of two or three dimensions, assuming the fluids are incompressible with constant volume fraction.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Mathematics
Qiming Huang, Junxiang Yang
Summary: The present work proposes a linear, decoupled, and energy dissipation-preserving time-marching scheme for simulating the Cahn-Hilliard model in two-phase incompressible fluid flows. By introducing an efficient time-dependent auxiliary variable approach and correcting the modified energy using a relaxation technique, the scheme exhibits desired accuracy, consistency, and energy stability.
Article
Engineering, Multidisciplinary
Shilin Zeng, Ziqing Xie, Xiaofeng Yang, Jiangxing Wang
Summary: In this paper, a fully discrete Fourier-Spectral numerical scheme is introduced to solve the nonlocal Cahn-Hilliard equation coupled with Navier-Stokes/Darcy equations. The proposed scheme achieves full decoupling, linearity, and energy stability through a combination of the Scalar Auxiliary Variable (SAV) and Zero-Energy-Contribution (ZEC) methods. The scheme is highly efficient due to its linear decoupling structure and the need to solve only a few elliptic equations with constant coefficients at each time step.
COMPUTER METHODS IN APPLIED MECHANICS AND ENGINEERING
(2023)
Article
Mathematics, Applied
Aristide Ndongmo Ngana, Gabriel Deugoue, Ttheodore Tachim Medjo
Summary: In this study, we consider the stochastic nonlocal Cahn-Hilliard-Navier-Stokes system with shear-dependent viscosity and investigate the existence of global weak martingale solution and the pathwise uniqueness of the weak solution.
STOCHASTICS-AN INTERNATIONAL JOURNAL OF PROBABILITY AND STOCHASTIC PROCESSES
(2023)
Article
Mathematics, Applied
Chuanjun Chen, Xiaofeng Yang
Summary: This article introduces a novel fully-decoupled numerical technique for the variable-density/viscosity Cahn-Hilliard phase-field model, achieving unconditional energy stability. The scheme only requires solving a series of completely independent linear elliptic equations at each time step, with the Cahn-Hilliard equation and the pressure Poisson equation as constant coefficients.
SCIENCE CHINA-MATHEMATICS
(2022)
Article
Computer Science, Interdisciplinary Applications
Juan Manzanero, Carlos Redondo, Miguel Chavez-Modena, Gonzalo Rubio, Eusebio Valero, Susana Gomez-Alvarez, Angel Rivero-Jimenez
Summary: In this work, a three-phase incompressible Navier-Stokes/Cahn-Hilliard numerical method was developed and applied to simulate three-phase flows present in industrial operations, specifically in the oil and gas industry. The method combines the Cahn-Hilliard diffuse interface model with the kinetic-energy stable incompressible Navier-Stokes equations model, using high-order discontinuous Galerkin spectral element method for spatial discretization. The developed numerical tool was validated and successfully used to simulate multiphase flows in pipes.
COMPUTERS & FLUIDS
(2022)
Article
Computer Science, Interdisciplinary Applications
Xiaoyu Feng, Zhonghua Qiao, Shuyu Sun, Xiuping Wang
Summary: This paper presents a pioneering study on the energy-stable smoothed particle hydrodynamics (SPH) discretization of the Navier-Stokes-Cahn-Hilliard (NSCH) model for incompressible two-phase flows. The proposed numerical scheme inherits mass and momentum conservation and the energy dissipation properties at the fully discrete level, and it satisfies the divergence-free condition through the projection procedure. Numerical experiments are conducted to verify the performance of the energy-stable SPH method for solving the two-phase NSCH model.
JOURNAL OF COMPUTATIONAL PHYSICS
(2023)
Article
Mathematics, Applied
Andrea Di Primio, Maurizio Grasselli, Hao Wu
Summary: We investigate a diffuse-interface model for viscous incompressible two-phase flows with surfactant. The model consists of two coupled Cahn-Hilliard equations and a Navier-Stokes system, describing the concentration differences and fluid velocity. We prove the existence of global and unique weak solutions in two dimensions, as well as the existence of unique strong solutions under stronger regularity assumptions in both two and three dimensions. We also establish continuous dependence estimates and instantaneous regularization properties of the solutions.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2023)
Article
Mathematics, Applied
Haibo Cui, Lei Yao
Summary: This paper considers the initial-boundary value problem of the coupled inhomogeneous incompressible Navier-Stokes equations and Vlasov-Boltzmann equation for the moderately thick spray in three-dimensional space. The global existence of weak solutions is established using an approximation scheme, a fixed point argument, and the weak convergence method.
APPLIED MATHEMATICS LETTERS
(2021)
Article
Mathematics, Applied
Helmut Abels, Josef Weber
Summary: In this study, the well-posedness of a diffuse interface model for a two-phase flow of two viscous incompressible fluids with different densities locally in time is demonstrated. Maximal L-2-regularity for the linearized Stokes part and L-p-regularity for the linearized Cahn-Hilliard system are used to prove the existence of strong solutions.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics, Applied
Chuanjun Chen, Xiaofeng Yang
Summary: In this paper, an efficient numerical scheme with second-order temporal accuracy is developed to solve the Cahn-Hilliard model. The scheme combines the finite element method with the pressure-correction projection method and the explicit-invariant energy quadratization method to decouple and solve linear elliptic equations, resulting in an efficient and stable solution.
SCIENCE CHINA-MATHEMATICS
(2023)
Article
Automation & Control Systems
Pierluigi Colli, Gianni Gilardi, Ionut Munteanu
Summary: This work presents the first contribution to the problem of boundary stabilization for the Cahn-Hilliard type phase field system, with a feedback controller designed to act only on the temperature flow of the system on one part of the boundary. The controller, of proportional type and expressed in terms of Laplace operator eigenfunctions, ensures that the closed loop nonlinear system exponentially reaches the prescribed stationary solution as long as the initial datum is sufficiently close to it.
INTERNATIONAL JOURNAL OF CONTROL
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Davide Manini
Summary: This paper discusses the sliding mode control problem for a second-order generalization of the Caginalp phase-field system, introducing the concept of thermal displacement and two control laws. Existence, uniqueness, and continuous dependence of solutions are proven, along with regularity results and the fact that solutions reach the sliding manifold within finite time under suitable conditions.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Correction
Mathematics, Applied
Pierluigi Colli, Andrea Signori, Jurgen Sprekels
Summary: A correction to this paper has been published.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Jurgen Sprekels
Summary: This study investigates viscous and nonviscous Cahn-Hilliard systems with double-well type nonlinearities and nonsmooth potentials. Existence, uniqueness, and regularity results of the solutions to these systems have been proven. Additionally, the asymptotic behavior of the solutions as the parameter sigma tends to zero in the viscous system is analyzed.
JOURNAL OF EVOLUTION EQUATIONS
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Hector Gomez, Guillermo Lorenzo, Gabriela Marinoschi, Alessandro Reali, Elisabetta Rocca
Summary: Prostate cancer in advanced stages can be lethal and may become resistant to chemotherapy, prompting the need for new therapeutic strategies. Combining cytotoxic and antiangiogenic therapies has shown promising potential, with research indicating that this combination may have advantages in treating advanced prostate cancer.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Elisabetta Rocca, Jurgen Sprekels
Summary: The paper discusses the optimal distributed control of a Cahn-Hilliard-Oono system in R-d, 1 <= d <= 3, with control located in the mass term and including both general and logarithmic potentials. The dependence of the phase variable on the control variable is analyzed, and necessary first-order optimality conditions are studied.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2022)
Article
Operations Research & Management Science
Pierluigi Colli, Andrea Signori, Jurgen Sprekels
Summary: This paper addresses a distributed optimal control problem for a tumor growth model of Cahn-Hilliard type, which involves a double obstacle nonlinearity given by a variational inequality corresponding to the associated potential. Using the deep quench approximation and recent results, the paper derives first-order necessary conditions of optimality and extends them to the double obstacle case by deducing a variational inequality in terms of the associated adjoint state variables. Furthermore, the paper obtains sparsity results for the optimal controls using the resulting variational inequality.
JOURNAL OF OPTIMIZATION THEORY AND APPLICATIONS
(2022)
Article
Mathematics, Applied
Pierluigi Colli, Takeshi Fukao, Luca Scarpa
Summary: This paper presents an asymptotic analysis for a system with a Cahn-Hilliard type equation and a dynamic boundary condition. The study focuses on the case where the coefficient of surface diffusion approaches zero, resulting in a forward-backward dynamic boundary condition at the limit. It is found that the solution loses some regularity during the limiting procedure, but the limit problem remains well-posed. Additionally, it is shown that the solution exhibits more regularity and the boundary condition holds almost everywhere when the two graphs have the same growth.
SIAM JOURNAL ON MATHEMATICAL ANALYSIS
(2022)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Juergen Sprekels
Summary: This paper deals with a general system of equations and conditions arising from a mathematical model of prostate cancer growth with chemotherapy and antiangiogenic therapy. The system includes two nonlinear equations involving evolutionary operators, where one equation describes the dynamics of the tumor phase variable with a double-well potential and an additional nonlinearity depending on the nutrient concentration. The other equation describes the nutrient concentration and also has nonlinear terms coupling both variables.
RENDICONTI LINCEI-MATEMATICA E APPLICAZIONI
(2022)
Article
Mathematics, Applied
Pierluigi Colli, Takeshi Fukao, Luca Scarpa
Summary: This article considers a system with a Cahn-Hilliard-type equation and dynamic boundary condition. By analyzing the problem under consideration, we obtained the asymptotic behavior when the surface diffusion coefficient of the boundary phase variable approaches 0, as well as the dynamic boundary condition at the limit. Furthermore, we proved that the solution of the limit problem is more regular and provided an error estimate for a suitable order of the diffusion parameter.
JOURNAL OF EVOLUTION EQUATIONS
(2022)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Andrea Signori, Juergen Sprekels
Summary: In this paper, optimal control problems for a nonlinear state system modeling nonisothermal phase transitions with a nonconserved order parameter are investigated. The system consists of two nonlinearly coupled partial differential equations governing phase dynamics and internal energy balance law. The study extends recent results for optimal control problems with a differentiable phase transition free energy to the nonsmooth case of a double obstacle potential. By utilizing the deep quench approach, meaningful first-order necessary optimality conditions are established for the double obstacle potential.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2023)
Article
Mathematics, Applied
Ferdinando Auricchio, Pierluigi Colli, Gianni Gilardi, Alessandro Reali, Elisabetta Rocca
Summary: This paper focuses on the well-posedness of a diffusion-reaction system for a susceptible-exposed-infected-recovered (SEIR) mathematical model. The model consists of four nonlinear partial differential equations with nonlinear diffusions dependent on the total SEIR population. It aims to describe the spatio-temporal spread of the COVID-19 pandemic and is a variation of a model introduced, discussed, and tested by Viguerie et al. (2020). The paper presents a mathematical analysis of the resulting Cauchy-Neumann problem, proving the existence of solutions in a general setting and employing a suitable time discretization procedure. It also provides uniqueness theorems for cases with constant diffusion coefficient and more regular data, along with a regularity result for the solutions.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2023)
Article
Automation & Control Systems
Pierluigi Colli, Andrea Signori, Juergen Sprekels
Summary: This paper addresses a distributed optimal control problem for a tumor growth model with chemotaxis, where the control and state variables are nonlinearly coupled. It discusses weak and strong well-posedness of the system under general assumptions for the potentials. The optimization problem is also tackled, with the existence of minimizers and necessary and sufficient optimality conditions established, along with a thorough second-order analysis of the control-to-state operator.
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Juergen Sprekels
Summary: This paper explores well-posedness and regularity results for operator equations with Cahn-Hilliard system structure, investigating distributed optimal control problems with double well and logarithmic potentials and deriving first-order optimal conditions. By using the deep quench method, the optimization problem with non-differentiable double obstacle nonlinearities is effectively addressed.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi
Summary: In this study, a viscous Cahn-Hilliard system with an additional leading term in the chemical potential is explored, showing existence and continuous dependence results under homogeneous Neumann and Dirichlet boundary conditions. By assuming specific conditions for the subdifferential operator, regularity results are demonstrated, along with the sliding mode property under certain conditions for the Dirichlet boundary. The results are applicable to a general and potentially singular multi-well potential acting on the phase variable.
MATHEMATICAL CONTROL AND RELATED FIELDS
(2021)
Article
Mathematics, Applied
Geunsu Choi, Mingu Jung, Sun Kwang Kim, Miguel Martin
Summary: This paper studies weak-star quasi norm attaining operators and proves that the set of such operators is dense in the space of bounded linear operators regardless of the choice of Banach spaces. It is also shown that weak-star quasi norm attaining operators have distinct properties from other types of norm attaining operators, although they may share some equivalent properties under certain conditions.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maria Lorente, Francisco J. Martin-Reyes, Israel P. Rivera-Rios
Summary: In this paper, we provide quantitative one-sided estimates that recover the dependences in the classical setting. We estimate the one-sided maximal function in Lorentz spaces and demonstrate the applicability of the conjugation method for commutators in this context.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Fernando Cobos, Luz M. Fernandez-Cabrera
Summary: We provide a necessary and sufficient condition for the weak compactness of bilinear operators interpolated using the real method. However, this characterization does not hold for interpolated operators using the complex method.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ovgue Gurel Yilmaz, Sofiya Ostrovska, Mehmet Turan
Summary: The Lupas q-analogue Rn,q, the first q-version of the Bernstein polynomials, was originally proposed by A. Lupas in 1987 but gained popularity 20 years later when q-analogues of classical operators in approximation theory became a focus of intensive research. This work investigates the continuity of operators Rn,q with respect to the parameter q in both the strong operator topology and the uniform operator topology, considering both fixed and infinite n.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
M. Agranovsky, A. Koldobsky, D. Ryabogin, V. Yaskin
Summary: This article modifies the concept of polynomial integrability for even dimensions and proves that ellipsoids are the only convex infinitely smooth bodies satisfying this property.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Abel Komalovics, Lajos Molnar
Summary: In this paper, a parametric family of two-variable maps on positive cones of C*-algebras is defined and studied from various perspectives. The square roots of the values of these maps under a faithful tracial positive linear functional are considered as a family of potential distance measures. The study explores the problem of well-definedness and whether these distance measures are true metrics, and also provides some related trace characterizations. Several difficult open questions are formulated.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Frederic Bayart
Summary: The passage describes the construction of an operator on a separable Hilbert space that is 5-hypercyclic for all δ in the range (ε, 1) and is not 5-hypercyclic for all δ in the range (0, ε).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Helene Frankowska, Nikolai P. Osmolovskii
Summary: This paper investigates second-order optimality conditions for the minimization problem of a C2 function f on a general set K in a Banach space X. Both necessary and sufficient conditions are discussed, with the sufficiency condition requiring additional assumptions. The paper demonstrates the validity of these assumptions for the case when the set K is an intersection of sets described by smooth inequalities and equalities, such as in mathematical programming problems. The novelty of the approach lies in the arbitrary nature of the set K and the straightforward proofs.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Ole Fredrik Brevig, Kristian Seip
Summary: This paper studies the Hankel operator on the Hardy space and discusses its minimal and maximal norms, as well as the relationship between the maximal norm and the properties of the function.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Alexander Meskhi
Summary: Rubio de Francia's extrapolation theorem is established for new weighted grand Morrey spaces Mp),lambda,theta w (X) with weights w beyond the Muckenhoupt Ap classes. This result implies the one-weight inequality for operators of Harmonic Analysis in these spaces for appropriate weights. The necessary conditions for the boundedness of the Hardy-Littlewood maximal operator and the Hilbert transform in these spaces are also obtained. Some structural properties of new weighted grand Morrey spaces are investigated. Problems are studied in the case when operators and spaces are defined on spaces of homogeneous type.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Maud Szusterman
Summary: In this work, the necessary conditions on the structure of the boundary of a convex body K to satisfy all inequalities are investigated. A new solution for the 3-dimensional case is obtained in particular.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Rami Ayoush, Michal Wojciechowski
Summary: In this article, lower bounds for the lower Hausdorff dimension of finite measures are provided under certain restrictions on their quaternionic spherical harmonics expansions. This estimate is analogous to a result previously obtained by the authors for complex spheres.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
F. G. Abdullayev, V. V. Savchuk
Summary: This paper investigates the convergence and theorem proof of the Takenaka-Malmquist system and Fejer-type operator on the unit circle, and provides relevant results on the class of holomorphic functions representable by Cauchy-type integrals with bounded densities.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Sofiya Ostrovska, Mikhail I. Ostrovskii
Summary: This work aims to establish new results on the structure of transportation cost spaces. The main outcome of this paper states that if a metric space X contains an isometric copy of L1 in its transportation cost space, then it also contains a 1-complemented isometric copy of $1.
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
Article
Mathematics, Applied
Pilar Rueda, Enrique A. Sanchez Perez
Summary: We prove a factorization theorem for Lipschitz operators acting on certain subsets of metric spaces of measurable functions and with values on general metric spaces. Our results show how a Lipschitz operator can be extended to a subset of other metric space of measurable functions that satisfies the following optimality condition: it is the biggest metric space, formed by measurable functions, to which the operator can be extended preserving the Lipschitz constant. Also, we demonstrate the coarsest metric that can be given for a metric space in which an order bounded lattice-valued-Lipschitz map is defined, and provide concrete examples involving the relevant space L0(mu).
JOURNAL OF MATHEMATICAL ANALYSIS AND APPLICATIONS
(2024)
