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
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
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
Mathematics, Applied
Xiaona Cui, Wei Shi, Xuezhi Li, Xin-Guang Yang
Summary: This paper investigates the tempered pullback dynamics of 3-D nonautonomous incompressible Navier-Stokes equations with nonlinear damping and delay in a bounded domain. It proves the existence and uniqueness of weak and strong solutions based on delicate priori estimates, and demonstrates the existence of the minimal family of pullback attractors under appropriate hypotheses on external forces.
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2021)
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
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
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
Aaron Brunk, Maria Lukacova-Medvid'ova
Summary: The aim of this paper is to prove the global in time existence of weak solutions for viscoelastic phase separation. The model couples the diffusive interface model with the Peterlin-Navier-Stokes equations and considers appropriate approximations to obtain the desired results.
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
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, 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
Mathematics, Applied
Cecilia Cavaterra, Elisabetta Rocca, Hao Wu
Summary: This research investigates the long-term dynamics and optimal control problem of tumor growth in the presence of nutrients. By analyzing the control variables and objective functions, optimal treatment strategies are proposed for both long-term and finite-time treatments. The goal of the control problem is to achieve a desired distribution of tumor cells for stability after treatment.
APPLIED MATHEMATICS AND OPTIMIZATION
(2021)
Article
Mathematics, Interdisciplinary Applications
Massimo Carraturo, Elisabetta Rocca, Elena Bonetti, Dietmar Hoemberg, Alessandro Reali, Ferdinando Auricchio
COMPUTATIONAL MECHANICS
(2019)
Article
Mathematics
Alain Miranville, Elisabetta Rocca, Giulio Schimperna
JOURNAL OF DIFFERENTIAL EQUATIONS
(2019)
Letter
Mathematics, Applied
Eduard Feireisl, Hana Petzeltova, Elisabetta Rocca
MATHEMATICAL METHODS IN THE APPLIED SCIENCES
(2019)
Article
Automation & Control Systems
Carlo Orrieri, Elisabetta Rocca, Luca Scarpa
ESAIM-CONTROL OPTIMISATION AND CALCULUS OF VARIATIONS
(2020)
Article
Mathematics, Applied
Ferdinando Auricchio, Elena Bonetti, Massimo Carraturo, Dietmar Hoemberg, Alessandro Reali, Elisabetta Rocca
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2020)
Article
Mathematics, Applied
Pierluigi Colli, Hector Gomez, Guillermo Lorenzo, Gabriela Marinoschi, Alessandro Reali, Elisabetta Rocca
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2020)
Article
Biology
Angelique Perrillat-Mercerot, Alain Miranville, Abramo Agosti, Elisabetta Rocca, Pasquale Ciarletta, Remy Guillevin
Summary: Interfaces play a key role in disease development by determining nutrient energy inflow from surrounding tissues. Cells can utilize different resources through nonlinear mechanisms, with lactate being particularly important in brain tumors and neurodegenerative diseases. Analytical proofs and finite element simulations are used to study lactate dynamics at the interface between cells and the vascular network in the brain.
MATHEMATICAL MEDICINE AND BIOLOGY-A JOURNAL OF THE IMA
(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
Computer Science, Interdisciplinary Applications
Michele Marino, Ferdinando Auricchio, Alessandro Reali, Elisabetta Rocca, Ulisse Stefanelli
Summary: The paper introduces a variational principle that combines phase-field and mixed Hu-Washizu functionals to derive traditional and novel topology optimization formulations, where material distribution quantity is either predetermined as a global constraint or minimized without constraint. Numerical solutions are obtained through mixed finite element schemes, avoiding global constraints and providing guidelines for phase-field parameter settings. The monolithic algorithm solution scheme is easily implemented, showcasing advantages in convergence studies and final design discussions in both two-dimensional and three-dimensional applications.
STRUCTURAL AND MULTIDISCIPLINARY OPTIMIZATION
(2021)
Article
Mathematics, Applied
Elisabetta Rocca, Luca Scarpa, Andrea Signori
Summary: This paper addresses the parameter identification problem of a coupled nonlocal Cahn-Hilliard-reaction-diffusion PDE system arising from a tumor growth model, utilizing optimal control theory techniques. Necessary optimality conditions are obtained for fully relaxed systems in both two- and three-dimensional cases, and the problem is further tackled on non-relaxed models using asymptotic arguments to solve the parameter identification problem in the presence of physically relevant double-well potentials.
MATHEMATICAL MODELS & METHODS IN APPLIED SCIENCES
(2021)
Article
Mathematics, Applied
Andrea Aspri, Elena Beretta, Cecilia Cavaterra, Elisabetta Rocca, Marco Verani
Summary: In this paper, we investigate the inverse problem of detecting cavities and inclusions embedded in a linear elastic isotropic medium using boundary displacement measurements. A constrained minimization problem, which includes a boundary quadratic misfit functional and a regularization term, is considered to penalize the perimeter of the identified cavity or inclusion. By employing a phase field approach, a robust algorithm is developed for the reconstruction of elastic inclusions and cavities with very small elasticity tensor.
APPLIED MATHEMATICS AND OPTIMIZATION
(2022)
Article
Mathematics, Applied
Tania Biswas, Elisabetta Rocca
Summary: The study investigates a phase-field model of prostate cancer growth with chemotherapy and antiangiogenic therapy effects. The model consists of phase-field equation, reaction-diffusion type equation, and linear reaction-diffusion equation for describing tumor growth, nutrient for the tumor, and concentration of prostate-specific antigen (PSA) respectively. The long time dynamics of the model is analyzed, proving the generation of a strongly continuous semigroup for the initial-boundary value problem and the existence of a global attractor in a proper phase space.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES B
(2022)
Article
Mathematics, Applied
Eduard Feireisl, Elisabetta Rocca, Giulio Schimperna, Arghir Zarnescu
Summary: This article studies a system of nonlinear PDEs modeling the electrokinetics of a nematic electrolyte material consisting of various ion species in a nematic liquid crystal. It focuses on the two-species case and proves apriori estimates providing weak sequential stability, the main step towards proving the existence of weak solutions.
DISCRETE AND CONTINUOUS DYNAMICAL SYSTEMS-SERIES S
(2021)
Article
Mathematics, Applied
Pierluigi Colli, Gianni Gilardi, Gabriela Marinoschi, Elisabetta Rocca
APPLIED MATHEMATICS AND OPTIMIZATION
(2019)
