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Helmut Abels

Publications and source records attributed to Helmut Abels.

At least 19 recordsLinked to original sources
arXiv2026

On a visco-elastic Mullins-Sekerka System

We introduce a novel visco-elastic Mullins-Sekerka system with a prescribed constant contact angle at the boundary. The system is derived as an $H^{-1}$-$H^1$-type gradient flow of an energy consisting of the perimeter together with capillary, elastic, and second-gradient contributions. Building on the framework of Hensel and Stinson (Arch. Ration. Mech. Anal. 248, 2024), we introduce a measure-valued solution concept featuring a sharp De Giorgi-type energy-dissipation inequality. Moreover, we establish existence of solutions via an implicit time discretization scheme, and prove existence of $BV$ solutions under an energy-conservation hypothesis.

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arXiv2026

Local Well-Posedness for a Diffuse Interface Model for Two-Phase Flows from Mixture Theory

Local-in-time well-posedness is established for a recently proposed diffuse interface model describing incompressible two-phase flows. The result constitutes the first analytical study of a model introduced by ten Eikelder et al. for the motion of a binary mixture of macroscopically immiscible, viscous, incompressible fluids with unmatched densities. In contrast to classic diffuse interface models based on a single mean velocity, this model is derived within the framework of mixture theory, assigning each phase its own momentum and mass balance, which results in a system of two coupled Navier--Stokes equations and two mass transport equations. The proof of the well-posedness result uses a fixed-point strategy, where the main difficulty lies in the analysis of the principal part of the associated linearized system.

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arXiv2026

Higher Order Convergence for the Sharp Interface Limit of 3D Navier--Stokes/Allen--Cahn Systems

We show convergence of solutions to a Navier--Stokes/Allen--Cahn system as the interfacial thickness $\varepsilon>0$ tends to zero for well-prepared initial data as long as the limit system possesses a sufficiently smooth solution. The limit system consists of a two-phase Navier--Stokes system separated by a sharp interface in the presence of surface tension coupled to a convective mean curvature flow equation. In comparison to previous results we obtain improved convergence estimates for higher-order norms. These enable us to prove convergence in the case of three space dimensions and non-constant viscosity, which was unknown before. The convergence results relies crucially on uniform higher-order estimates for the associated linearized Navier--Stokes/Allen--Cahn system in suitably weighted $L^2$-Sobolev spaces. Here a novel problem-adapted weight proportional to the sum of $\varepsilon$ and the distance to the sharp interface of the limit, which gives improved and sharp estimates, is an important new ingredient. This approach can be potentially adapted to other sharp interface limits as well.

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arXiv2026

Sharp Interface Limit for a Mass-Conserving Navier-Stokes/Allen-Cahn System with Different Viscosities

We perform a rigorous examination of the sharp interface limit of a coupled Navier-Stokes and mass-conserving Allen-Cahn system in a two-dimensional, bounded, and smooth domain as the parameter $\varepsilon > 0$, representing the thickness of the diffuse interface, tends to zero. We prove the convergence of solutions from the mass-conserving Navier-Stokes/Allen-Cahn system to those of its sharp interface limit. In this limit, the interface evolves according to mass-conserving mean curvature flow with a convection term and is coupled to a two-phase Navier-Stokes system with surface tension. Our approach entails the construction of an approximate solution for the limiting system through the use of matched asymptotic expansions, complemented by a special ansatz for the leading-order term. In order to estimate the error between this approximate solution and the exact solution, we employ a refined spectral estimate for the linearized Allen-Cahn operator near the approximate solution.

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arXiv2026

Local Well-Posedness of a Model for Stress-Driven Growth in the Presence of Nutrients

A model for morphoelastic growth, that is, growth influenced by elastic stress, driven by the absorption of nutrients is considered. The model features a multiplicative decomposition of the deformation gradient into an elastic contribution and a growth tensor. While the evolution of the system is governed by an ordinary differential equation for the growth tensor on a suitable Banach space, which depends on the elastic stresses and the concentration of a nutrient field, the total deformation is given by the solution of a quasi-static equilibrium equation arising from the formal Euler-Lagrange equations of a hyperelastic variational integral. The nutrient concentration is determined by a linear elliptic reaction-diffusion equation which is formulated in Lagrangian coordinates and whose coefficients depend on the growth tensor as well as the deformation gradient accounting for the change of material properties due to elastic deformation and growth. Existence and uniqueness of solutions of this fully coupled system of differential equations is proven via a fixed-point argument.

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arXiv2026

Nonlocal-to-local $L^p$-convergence of convolution operators with singular, anisotropic kernels

We study nonlocal convolution-type operators with singular, possibly anisotropic kernels. Our main objective is to establish and quantify their nonlocal-to-local convergence to a local differential operator with natural boundary conditions, as the kernels concentrate at the origin in a suitable way. Such convergence results provide a useful tool for the physical justification of mathematical models, particularly in situations where the desired local differential operator cannot be directly derived from microscopic laws. The present work substantially extends previous results by allowing kernels with stronger singularities (comparable to those of fractional Laplacians), anisotropic and non-localized kernels, and by proving strong convergence in general $L^p$ spaces together with explicit convergence rates.

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arXiv2025

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

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.

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arXiv2025

Well-posedness and sharp interface limit of a non-isothermal Navier--Stokes/Allen--Cahn model

We propose a thermodynamically consistent phase-field model for the flow of a mixture of two different viscous incompressible fluids of equal density in a bounded domain. We prove the well-posedness of local-in-time strong solutions by means of maximal regularity and contraction mapping arguments. We introduce a suitable entropic weak formulation of the problem, replacing the heat equation by the total energy inequality and an entropy production inequality, and we rigorously prove global-in-time existence of such weak solutions, developing a novel approximation scheme. We also show that an entropic weak solution to this non-isothermal phase-field model converges to a distributional (or $BV$) solution to a non-isothermal Navier--Stokes/mean curvature flow, under an energy convergence assumption.

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arXiv2025

Local Well-Posedness of the Cahn-Hilliard-Biot System

We show short-time well-posedness of a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including phase-field dependent material properties, with the Cahn-Hilliard equation to model the evolution of the solid, where we further distinguish between the absence and presence of a visco-elastic term of Kelvin-Voigt type. While both problems will be reduced to a fixed-point equation that can be solved using maximal regularity theory along with a contraction argument, the first case relies on a semigroup approach over suitable Hilbert spaces, whereas treating the second case under minimal assumptions with respect to spatial regularity necessitates the application of Banach scales.

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arXiv2025

Weak Solutions to a Sharp Interface Model for a Two-Phase Flow of Incompressible Viscous Fluids with Different Densities

In this paper we consider the flow of two incompressible, viscous and immiscible fluids in a bounded domain, with different densities and viscosities. This model consists of a coupled system of Navier-Stokes and Mullins-Sekerka type parts, and can be obtained from the sharp interface limit of the diffuse interface model proposed by the first author, Garcke, and Gr\"{u}n (Math. Models Methods Appl. Sci. 22, 2012). We introduce a new notion of weak solutions and prove its global in time existence, together with a consistency result of smooth weak solutions with the classical Navier-Stokes-Mullins-Sekerka system. Our new notion of solution allows to include the case of different densities of the two fluids, a sharp energy dissipation principle \`a la De Giorgi, together with a weak formulation of the constant contact angle condition at the boundary, which were left open in the previous notion of solution proposed by the first author and R\"{o}ger (Ann. Inst. H. Poincar\'e Anal. Non Lin\'eaire 26, 2009).

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arXiv2025

Diffuse Interface Models for Two-Phase Flows with Phase Transition: Modeling and Existence of Weak Solutions

The flow of two macroscopically immiscible, viscous, incompressible fluids with unmatched densities is studied, where a transfer of mass between the constituents by phase transition is taken into account. To this end, two quasi-incompressible diffuse interface models with singular free energies are analyzed, differing primarily in their velocity averaging. Firstly, to generalize a model by Abels, Garcke, and Gr\"un, a thermodynamically consistent system of Navier--Stokes/Cahn--Hilliard type with source terms is derived in a framework of continuum fluid dynamics, followed by a proof of existence of weak solutions to the latter. Secondly, the quasi-stationary version of a model by Aki, Dreyer, Giesselmann, and Kraus is investigated analytically, with existence of weak solutions being established for the resulting quasi-stationary Stokes system coupled to a Cahn--Hilliard equation with a source term.

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arXiv2024

On a diffuse interface model for diblock copolymers interacting with an electric field

We consider a diffuse interface model describing a ternary system constituted by a conductive diblock copolymer and a homopolymer acting as solvent. The resulting dynamics is modeled by two Cahn--Hilliard--Oono equations for the copolymer blocks, accounting for long-range interactions; a classical Cahn--Hiliard equation for the homopolymer and the Maxwell equation for the electric displacement field. A multiphase singular potential is employed in order to ensure physical consistency. First, we show existence of global weak solutions in two and three dimensions. Uniqueness of weak solutions is established in the constant mobility case, and a conditional result is given in the general case. Instantaneous regularization and long-time behavior are also investigated, the latter in the case of affine-linear electric permittivity, showing in particular that solutions converge to a single stationary state.

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arXiv2024

Convergence of the Nonlocal Allen-Cahn Equation to Mean Curvature Flow

We prove convergence of the nonlocal Allen-Cahn equation to mean curvature flow in the sharp interface limit, in the situation when the parameter corresponding to the kernel goes to zero fast enough with respect to the diffuse interface thickness. The analysis is done in the case of a $W^{1,1}$-kernel, under periodic boundary conditions and in both two and three space dimensions. We use the approximate solution and spectral estimate from the local case, and combine the latter with an $L^2$-estimate for the difference of the nonlocal operator and the negative Laplacian from Abels, Hurm arXiv:2307.02264. To this end, we prove a nonlocal Ehrling-type inequality to show uniform $H^3$-estimates for the nonlocal solutions.

math.AP↗
arXiv2024

Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Global Well-Posedness

We show existence and uniqueness of strong solutions to a Navier-Stokes/Cahn-Hilliard type system on a given two-dimensional evolving surface in the case of different densities and a singular (logarithmic) potential. The system describes a diffuse interface model for a two-phase flow of viscous incompressible fluids on an evolving surface. We also establish the validity of the instantaneous strict separation property from the pure phases. To show these results we use our previous achievements on local well-posedness together with suitable novel regularity results for the convective Cahn-Hilliard equation. The latter allows to obtain higher-order energy estimates to extend the local solution globally in time. To this aim the time evolution of energy type quantities has to be calculated and estimated carefully.

math.AP↗
arXiv2024

Diffuse Interface Model for Two-Phase Flows on Evolving Surfaces with Different Densities: Local Well-Posedness

A Cahn-Hilliard-Navier-Stokes system for two-phase flow on an evolving surface with non-matched densities is derived using methods from rational thermodynamics. For a Cahn-Hilliard energy with a singular (logarithmic) potential short time well-posedness of strong solutions together with a separation property is shown, under the assumption of a priori prescribed surface evolution. The problem is reformulated with the help of a pullback to the initial surface. Then a suitable linearization and a contraction mapping argument for the pullback system are used. In order to deal with the linearized system, it is necessary to show maximal $L^2$-regularity for the surface Stokes operator in the case of variable viscosity and to obtain maximal $L^p$-regularity for the linearized Cahn-Hilliard system.

math.AP↗
arXiv2024

Existence of Weak Solutions to a Cahn-Hilliard-Biot System

We prove existence of weak solutions to a diffuse interface model describing the flow of a fluid through a deformable porous medium consisting of two phases. The system non-linearly couples Biot's equations for poroelasticity, including phase-field dependent material properties, with the Cahn-Hilliard equation to model the evolution of the solid, and is further augmented by a visco-elastic regularization of Kelvin-Voigt type. To obtain this result, we approximate the problem in two steps, where first a semi-Galerkin ansatz is employed to show existence of weak solutions to regularized systems, for which later on compactness arguments allow limit passage. Notably, we also establish a maximal regularity theory for linear visco-elastic problems.

math.AP↗
arXiv2024

Convergence of a Convective Allen-Cahn Equation to a Transport Equation

We show convergence of solutions of a convective Allen-Cahn equation for a given smooth and divergence free velocity field to a transport equation for an evolving interface in the case when the thickness of the diffuse interface tends to zero and the mobility coefficient is proportional to the interfacial thickness. This is done for well-prepared initial data by estimating the difference of the exact and an approximate solution of the convective Allen-Cahn equation with the aid of a uniform lower bound for the linearized Allen-Cahn operator. The approximate solution is constructed with the aid of three terms from formally matched asymptotics calculations close to the interface.

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