Special Session 87: Mathematical Insights into Phase-Field Models

Uniform higher order estimates for the sharp interface limit of a Navier-Stokes/Allen-Cahn system

Helmut Abels University of Regensburg Germany Co-Author(s):    Mingwen Fei, Yadong Liu, Maximilain Moser

Abstract: We consider the linearized system for the sharp interface limit of a Navier-Stokes/Allen-Cahn system around a suitable approximate solution. With the aid of a suitable weight taking the distance to the interface and the interfacial thickness into account we obtain optimal regularity estimates, which are uniformy in the interfacial thickness. This enables to improve previous convergence results in two space dimensions significantly and extend results to three space dimensions.

New results on a variant of the Cahn-Hilliard equation with non-variational structure

Abramo Agosti University of Pavia Italy Co-Author(s):    Andrea Giorgini

Abstract: In this talk I will present a variant of the Cahn-Hilliard equation well-known in the Statistical Physics community, which is not derivable from variational principles and which may describe continuum materials constituted by particles that can convert energy into directed motion. This model has multiple applications in biomedicine and engineering. I will show quantitative results about the phase-separation and coarsening dynamics, and both analytical and numerical results for the model equation.

Optimal Control of the Fidelity Coefficient in Cahn--Hilliard Inpainting Models

Cecilia Cavaterra University of Milan Italy Co-Author(s):    Elena Beretta, Matteo Fornoni, Maurizio Grasselli

Abstract: We study an optimal control problem for a Cahn--Hilliard-type inpainting model, as originally proposed by Bertozzi et al. The model employs a linear reaction term, weighted by a fidelity coefficient, to restore a damaged image within a fixed subdomain. To ensure that the order parameter remains within the physically relevant range $[0,1]$, we employ a singular potential. We formulate a cost functional that accounts for the magnitude of the fidelity coefficient and analyze the resulting control-to-state operator. After proving the existence of optimal controls, we derive first-order optimality conditions and, under suitable assumptions, establish second-order optimality conditions.

A perturbation of the Cahn-Hilliard equation with logarithmic nonlinearity

Monica Conti Politecnico di Milano Italy Co-Author(s):    Pietro Galimberti, Stefania Gatti, Andrea Giorgini

Abstract: We investigate a perturbation of the Cahn-Hilliard equation with non-degenerate mobility and nonlinear terms of logarithmic type. This new model is based on an unconstrained theory recently proposed by F.P. Duda, A.F. Sarmiento and E. Fried. We prove the existence, regularity and uniqueness of weak and strong solutions, as well as separation properties from the pure states, also in three space dimensions. Besides, we prove the convergence to the Cahn-Hilliard equation as the perturbation parameter goes to zero.

On recent analytical results for a thermodynamically consistent diffuse interface model for incompressible two-phase flows with unmatched densities

Harald Garcke Department for Mathematics Germany Co-Author(s):    Helmut Abels, Andrea Giorgini, Maoyin Lv, Hao Wu

Abstract: In the talk, an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard sys- tem with non-constant density, proposed by the speaker together with Abels and Gr\un in 2012, will be analyzed. This model arises in the diffuse interface theory for binary mixtures of viscous incompressible fluids. This system is a generalization of the well- known model H in the case of fluids with unmatched densities. In three dimensions, we prove that any global weak solution (for which uniqueness is not known) exhibits a propagation of regularity in time and stabilizes toward an equilibrium state for large times. The analysis hinges upon the following key points: a global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity, the energy dissipation of the system, the separation property at large times, a weak-strong uniqueness type result, and the Lojasiewicz-Simon inequality. We then discuss mixed regularity conditions on the velocity field and its gradient, under which the global weak solution conserves its energy for all times. Finally, we show Lyapunov stability for each steady state consisting of a zero velocity together with a local energy minimizer of the Ginzburg-Landau functional.

On the Cahn-Hilliard equation with nonlinear diffusion: the non-convex case

Andrea Giorgini Politecnico di Milano Italy Co-Author(s):    Monica Conti, Stefania Gatti, Giulio Schimperna

Abstract: We investigate the Cahn-Hilliard equation with nonlinear diffusion and non-degenerate mobility modeling phase separation phenomena in complex systems (e.g., crystals and polymers). Previous results in the literature on this model relied on the strong convexity assumption of the gradient part of the energy, which excludes relevant cases. In this talk, we remove the convexity condition and establish new qualitative properties of solutions under general assumptions on the diffusion and mobility functions. In two spatial dimensions, we prove uniqueness of weak solutions, their smoothing effect for positive times, and convergence to equilibrium as time tends to infinity. In three dimensions, we show local well-posedness of strong solutions for arbitrary initial data and global existence for data close to energy minimizers, yielding a Lyapunov stability principle. A key ingredient of our analysis is a Lojasiewicz-Simon inequality tailored to the nonlinear diffusion case, which enables us to characterize the longtime dynamics.

Fast reaction limit in a stochastic competition-diffusion system

Danielle Hilhorst CNRS and Universiy Paris-Saclay France Co-Author(s):    Perla El Kettani, Kunwoo Kim

Abstract: We study the fast reaction limit of a stochastic competition-diffusion system and prove that its solution converges to the unique solution of a stochastic Stefan problem with zero latent heat as the reaction term tends to infinity.

Strong nonlocal-to-local convergence of convolution operators with singular, possibly anisotropic kernels

Patrik Knopf University of Regensburg Germany Co-Author(s):    Helmut Abels, Christoph Hurm

Abstract: The goal of nonlocal-to-local convergence is to show that certain singular, nonlocal convolution-type integral operators converge to a local differential operator as the convolution kernel concentrates at zero. This can be a useful tool in the physical justification of mathematical models (e.g., the Cahn--Hilliard equation), especially when a desired local differential operator can not be derived by microscopic laws. The nonlocal-to-local convergence of convolution operators with radially symmetric (i.e., isotropic) kernels having $W^{1,1}$-regularity is already very well understood. However, the assumption of $W^{1,1}$-regularity is too strong for many applications. Also, in some situations (e.g., crystallization phenomena), convolution kernels are not radially symmetric but merely even (i.e., anisotropic). In my talk, I will present our recent results on the strong nonlocal-to-local convergence (with rates) for anisotropic kernels with significantly lower regularity assumption compared to previous contributions. These results can, for example, be applied to nonlocal phase-field models such as some anisotropic variants of the Cahn--Hilliard equation.

Complex pattern formation by Ginzburg-Landau and Swift-Hohenberg dynamics: Analysis and Numerical Simulations

Kei Fong Lam Hong Kong Baptist University Hong Kong Co-Author(s):    Harald Garcke, Robert Nurnberg, Andrea Signori and Ho Hei Tam

Abstract: We discuss pattern formation arising from combinations of Ginzburg-Landau and Swift-Hohenberg dynamics. Individually these are well-studied processes where the former involves separation into phases and the latter involves small scale stripes and dots. Such systems were proposed to generate new patterns by leveraging the competition between Ginzburg-Landau and Swift-Hohenberg dynamics, and were suggested for the development of new porous structures for material sciences. We present a three-species mixture described by a Cahn-Hilliard-Swift-Hohenberg system. Singular potentials are introduced to adhere to essential physical constraints, and well-posedness results will be reported. Numerical simulations demonstrate complex pattern formation as seen in pigment patterns of animals. If time permits, we discuss the numerical analysis of a positive preserving scheme for solving such Cahn-Hilliard-Swift-Hohenberg system with singular potentials.

Global weak solutions to a diffuse interface model for quasi-incompressible two-phase flows with barycentric velocity and singular potential

Yadong Liu Nanjing Normal University Peoples Rep of China Co-Author(s):    Mingwen Fei, Xiang Fei, Yadong Liu and Hao Wu

Abstract: In this talk, I will discuss a thermodynamically consistent diffuse-interface model that describes the motion of two macroscopically immiscible, incompressible, viscous fluids with unmatched densities. This model adopts a mass-averaged velocity so that the fluid mixture is quasi-incompressible: the velocity is no longer divergence-free, and the pressure enters the equation for the chemical potential. For the initial boundary value problem in $\mathbb{T}^3$ with a class of physically relevant and singular free energy densities, we prove the existence of global-in-time weak solutions. The proof relies on a reduction of the model and a two-layer approximation with delicate estimates for the order parameter and the mass density that are inspired by the celebrated BD-entropy introduced by Bresch and Desjardins [Comm. Math. Phys. 2003, 238(1-2): 211--223]. A key observation is that capillarity at the interface provides a damping effect on the density evolution. Moreover, for the limiting passage, delicate tail estimates are derived to exclude possible concentrations of the singular potential, since no integrability of the pressure is available a priori. This work appears to be the first existence result for the Navier--Stokes/Cahn--Hilliard system with general densities and barycentric velocity without spatial regularization.

A high-order spatially accurate structure-preserving scheme for PDEs with dynamic boundary conditions based on a summation-by-parts operator

Makoto Okumura Intelligence and Informatics/Konan University Japan Co-Author(s):

Abstract: The Summation-by-Parts (SBP) method is a spatial discretization technique for partial differential equations (PDEs) that employs the SBP operator to approximate spatial partial derivatives. Moreover, some SBP operators provide highly accurate approximations of spatial partial derivatives, and their use is expected to yield high-precision spatial discretization of PDEs. On the other hand, there is a structure-preserving numerical method called the discrete variational derivative (DVD) method by Furihata and Matsuo (2010). The scheme derived from this method is typically second-order accurate in space. More recently, Umezu et al. (2025) have designed a structure-preserving scheme with high-order spatial accuracy based on the DVD method for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, using the SBP operator and an appropriate projection matrix. However, constructing such a projection matrix is difficult for complex boundary conditions, such as dynamic ones that involve the time derivative of an unknown function. Thus, in this study, instead of using a projection matrix, we incorporated a correction term, the simultaneous approximation (SA) term, corresponding to the residual of the boundary conditions. This enabled us to design a spatially high-accuracy structure-preserving scheme based on the DVD method for PDEs under dynamic boundary conditions.

A stochastic Allen-Cahn equation with jumps for radiotherapy damage

Luca Scarpa Politecnico di Milano Italy Co-Author(s):    Andrea Di Primio, Marvin Fritz, Margherita Zanella

Abstract: We present a stochastic Allen-Cahn equation for the dynamics of biomolecular damage and repair. The jump-type noise models the abrupt, localised damage induced by external radiotherapy shocks. We prove well-posedness of the model in a strong probabilistic sense and present some simulations to illustrate how it captures fundamental biological phenomena, such as damage clustering, stress-induced topology perturbations, and damage dynamics. The works presented in the talk are based on joint collaborations with Andrea Di Primio (Scuola Normale Superiore, Pisa, Italy), Marvin Fritz (University of Vienna, Austria), and Margherita Zanella (Politecnico di Milano, Italy).

Optimal Control of 1D Nonlinear Parabolic System Involving Regularized 1-Harmonic Type Flow

Ken Shirakawa Faculty of Education Japan Co-Author(s):    Salvador Moll and Hiroshi Watanabe

Abstract: This study is based on recent jointwork with S. Moll (University of Valencia, Spain) and H. Watanabe (Oita University, Japan). In this talk, we consider an optimal control problem governed by a one-dimensional nonlinear parabolic system involving a regularized 1-harmonic type flow, which serves as a model of grain boundary motion. The state system is a one-dimensional simplified version of the three-dimensional phase-field model for grain boundary motion studied in [S. Moll et al., J. Nonlinear Sci., 33, 2023]. The optimal control problem is formulated under a state constraint requiring that the range of the state variable is restricted to the unit sphere, together with a control constraint imposed on the domain of the cost functional. In particular, the state constraint introduces essential analytical difficulties in both the mathematical model and the optimal control problem. The aim of this talk is to lay the foundation for the development of an optimal control framework for our state system. The analysis is built upon a recent uniqueness result for the one-dimensional state system. Under suitable assumptions, we focus on the existence of optimal controls and the corresponding necessary optimality conditions.

Existence and Nonlocal-to-Local Convergence for Singular, Anisotropic Nonlocal Cahn-Hilliard Equations

Yutaka Terasawa Nagoya University Japan Co-Author(s):    Helmut Abels

Abstract: We study the nonlocal-to-local convergence for a nonlocal Cahn-Hilliard equation with anisotropic and singular kernels. In particular, we show convergence of weak solutions of the nonlocal Cahn-Hilliard equation to weak solutions of a corresponding anisotropic Cahn-Hilliard equation for suitable subsequences. Moreover, we show existence of weak solutions for the nonlocal equation under a condition, which guarantees existence of weak solutions for suitably localized or singular kernels. We put a focus on the existence part in the talk. This is based on a joint work with Helmut Abels (Regensburg Univ.).

Cahn-Hilliard equation associated with hypergraph Laplacian

Shun Uchida Oita University Japan Co-Author(s):

Abstract: In this talk, we consider the solvability of a multi-valued nonlinear ODE system obtained by replacing the Laplacian of the Cahn-Hilliard equations with a hypergraph Laplacian. The hypergraph Laplacian is a nonlinear multi-valued operator introduce to analyze complex network structures. In previous studies, it is known that solutions to evolution equations with this operator behave similarly to that of the heat equation. By analogy with the Cahn-Hilliard equations, we can expect that the system we consider represents a multi-phase decomposition in a discrete domain. However, due to the nonlinearity and multi-valuedness of the hypergraph Laplacian, standard methods for the Cahn-Hilliard equations can not be applied. In this talk, we will introduce a new proof method by using properties of the hypergraph Laplacian.

Optimal Control of a Navier-Stokes-Cahn-Hilliard System for Membrane-fluid Interaction

HAO WU Fudan University Peoples Rep of China Co-Author(s):    Andrea Signori, Hao Wu

Abstract: We consider an optimal control problem for a two-dimensional Navier-Stokes-Cahn-Hilliard system arising in the modeling of fluid-membrane interaction. The fluid dynamics is governed by the incompressible Navier-Stokes equations, which are nonlinearly coupled with a sixth-order Cahn-Hilliard type equation representing the deformation of a flexible membrane through a phase-field variable. We introduce an external forcing term acting on the fluid as the control variable. Then we seek to minimize a tracking-type cost functional, demonstrating the existence of an optimal control and deriving the associated first-order necessary optimality conditions.