Searcharxiv⌕ Search

arXiv subjects

Helmut Abels

Publications and source records attributed to Helmut Abels.

At least 37 records · Page 2Linked to original sources
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.

math.AP↗
arXiv2024

Mathematical Analysis of a Diffuse Interface Model for Multi-Phase Flows of Incompressible Viscous Fluids with Different Densities

We analyze a diffuse interface model for multi-phase flows of $N$ incompressible, viscous Newtonian fluids with different densities. In the case of a bounded and sufficiently smooth domain existence of weak solutions in two and three space dimensions and a singular free energy density is shown. Moreover, in two space dimensions global existence for sufficiently regular initial data is proven. In three space dimension, existence of strong solutions locally in time is shown as well as regularization for large times in the absence of exterior forces. Moreover, in both two and three dimensions, convergence to stationary solutions as time tends to infinity is proved.

math.AP↗
arXiv2023

Strong Nonlocal-to-Local Convergence of the Cahn-Hilliard Equation and its Operator

We prove convergence of a sequence of weak solutions of the nonlocal Cahn-Hilliard equation to the strong solution of the corresponding local Cahn-Hilliard equation. The analysis is done in the case of sufficiently smooth bounded domains with Neumann boundary condition and a $W^{1,1}$-kernel. The proof is based on the relative entropy method. Additionally, we prove the strong $L^2$-convergence of the nonlocal operator to the negative Laplacian together with a rate of convergence.

math.AP↗
arXiv2023

Sharp Interface Limit for a Navier-Stokes/Allen-Cahn System in the Case of a Vanishing Mobility

We consider the sharp interface limit of a Navier-Stokes/Allen Cahn equation in a bounded smooth domain in two space dimensions, in the case of vanishing mobility $m_\varepsilon=\sqrt{\varepsilon}$, where the small parameter $\varepsilon>0$ related to the thickness of the diffuse interface is sent to zero. For well-prepared initial data and sufficiently small times, we rigorously prove convergence to the classical two-phase Navier-Stokes system with surface tension. The idea of the proof is to use asymptotic expansions to construct an approximate solution and to estimate the difference of the exact and approximate solutions with a spectral estimate for the (at the approximate solution) linearized Allen-Cahn operator. In the calculations we use a fractional order ansatz and new ansatz terms in higher orders leading to a suitable $\varepsilon$-scaled and coupled model problem. Moreover, we apply the novel idea of introducing $\varepsilon$-dependent coordinates.

math.AP↗
arXiv2023

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

We discuss the sharp interface limit of a coupled Navier-Stokes/Allen-Cahn system in a two dimensional, bounded and smooth domain, when a parameter $\varepsilon>0$ that is proportional to the thickness of the diffuse interface tends to zero rigorously. We prove convergence of the solutions of the Navier-Stokes/Allen-Cahn system to solutions of a sharp interface model, where the interface evolution is given by the mean curvature flow with an additional convection term coupled to a two-phase Navier-Stokes system with surface tension. This is done by constructing an approximate solution from the limiting system via matched asymptotic expansions together with a novel Ansatz for the highest order term, and then estimating its difference with the real solution with the aid of a refined spectral estimate of the linearized Allen-Cahn operator near the approximate solution.

math.AP↗
arXiv2023

Global regularity and asymptotic stabilization for the incompressible Navier-Stokes-Cahn-Hilliard model with unmatched densities

We study an initial-boundary value problem for the incompressible Navier-Stokes-Cahn-Hilliard system with non-constant density proposed by Abels, Garcke and Grün in 2012. 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 towards an equilibrium state as $t \rightarrow \infty$. More precisely, the concentration function $ϕ$ is a strong solution of the Cahn-Hilliard equation for (arbitrary) positive times, whereas the velocity field $\mathbf{u}$ becomes a strong solution of the momentum equation for large times. Our analysis hinges upon the following key points: a novel global regularity result (with explicit bounds) for the Cahn-Hilliard equation with divergence-free velocity belonging only to $L^2(0,\infty; \mathbf{H}^1_{0,σ}(Ω))$, the energy dissipation of the system, the separation property for large times, a weak-strong uniqueness type result, and the Lojasiewicz-Simon inequality. Additionally, in two dimensions, we show the existence and uniqueness of global strong solutions for the full system. Finally, we discuss the existence of global weak solutions for the case of the double obstacle potential.

math.AP↗
arXiv2022

Fractional-Order Operators on Nonsmooth Domains

The fractional Laplacian $(-Δ)^a$, $a\in(0,1)$, and its generalizations to variable-coefficient $2a$-order pseudodifferential operators $P$, are studied in $L_q$-Sobolev spaces of Bessel-potential type $H^s_q$. For a bounded open set $Ω\subset \mathbb R^n$, consider the homogeneous Dirichlet problem: $Pu =f$ in $Ω$, $u=0$ in $ \mathbb R^n\setminusΩ$. We find the regularity of solutions and determine the exact Dirichlet domain $D_{a,s,q}$ (the space of solutions $u$ with $f\in H_q^s(\overlineΩ)$) in cases where $Ω$ has limited smoothness $C^{1+τ}$, for $2a<τ<\infty $, $0\le s<τ-2a$. Earlier, the regularity and Dirichlet domains were determined for smooth $Ω$ by the second author, and the regularity was found in low-order Hölder spaces for $τ=1$ by Ros-Oton and Serra. The $H_q^s$-results obtained now when $τ<\infty $ are new, even for $(-Δ)^a$. In detail, the spaces $D_{a,s,q}$ are identified as $a$-transmission spaces $H_q^{a(s+2a)}(\overlineΩ)$, exhibiting estimates in terms of $\operatorname{dist}(x,\partialΩ)^a$ near the boundary. The result has required a new development of methods to handle nonsmooth coordinate changes for pseudodifferential operators, which have not been available before; this constitutes another main contribution of the paper.

math.AP↗
arXiv2022

Short time existence of a quasi-stationary fluid-structure interaction problem for plaque growth

We address a quasi-stationary fluid-structure interaction problem coupled with cell reactions and growth, which comes from the plaque formation during the stage of the atherosclerotic lesion in human arteries. The blood is modeled by the incompressible Navier--Stokes equation, while the motion of vessels is captured by a quasi-stationary equation of nonlinear elasticity. The growth happens when both cells in fluid and solid react, diffuse and transport across the interface, resulting in the accumulation of foam cells, which are exactly seen as the plaques. Via a fixed-point argument, we derive the local well-posedness of the nonlinear system, which is sustained by the analysis of decoupled linear systems.

math.AP↗
arXiv2022

Qualitative Properties for a System Coupling Scaled Mean Curvature Flow and Diffusion

We consider a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. Several properties of solutions are analyzed. Emphasis is placed on to what extent the surface in our setting qualitatively evolves similar as for the usual mean curvature flow. To this end, we show that the surface area is strictly decreasing but give an example of a surface that exists for infinite times nevertheless. Moreover, mean convexity is conserved whereas convexity is not. Finally, we construct an embedded hypersurface that develops a self-intersection in the course of time. Additionally, a formal explanation of how our equations can be interpreted as a gradient flow is included.

math.AP↗
arXiv2022

Short Time Existence for Coupling of Scaled Mean Curvature Flow and Diffusion

We prove a short time existence result for a system consisting of a geometric evolution equation for a hypersurface and a parabolic equation on this evolving hypersurface. More precisely, we discuss a mean curvature flow scaled with a term that depends on a quantity defined on the surface coupled to a diffusion equation for that quantity. The proof is based on a splitting ansatz, solving both equations separately using linearization and a contraction argument. Our result is formulated for the case of immersed hypersurfaces and yields a uniform lower bound on the existence time that allows for small changes in the initial value of the height function.

math.AP↗
arXiv2022

Convergence of a Nonlocal to a Local Diffuse Interface Model for Two-Phase Flow with Unmatched Densities

We prove convergence of suitable subsequences of weak solutions of a diffuse interface model for the two-phase flow of incompressible fluids with different densities with a nonlocal Cahn-Hilliard equation to weak solutions of the corresponding system with a standard "local" Cahn-Hilliard equation. The analysis is done in the case of a sufficiently smooth bounded domain with no-slip boundary condition for the velocity and Neumann boundary conditions for the Cahn-Hilliard equation. The proof is based on the corresponding result in the case of a single Cahn-Hilliard equation and compactness arguments used in the proof of existence of weak solutions for the diffuse interface model.

math.AP↗
arXiv2022

Convergence of Thin Vibrating Rods to a Linear Beam Equation

We show that solutions for a specifically scaled nonlinear wave equation of nonlinear elasticity converge to solutions of a linear Euler-Bernoulli beam system. We construct an approximation of the solution, using a suitable asymptotic expansion ansatz based upon solutions to the one-dimensional beam equation. Following this, we derive the existence of appropriately scaled initial data and can bound the difference between the analytical solution and the approximating sequence.

math.AP↗
arXiv2021

On a fluid-structure interaction problem for plaque growth: cylindrical domain

This paper concerns a free-boundary fluid-structure interaction problem for plaque growth proposed by Yang et al. [J. Math. Biol., 72(4):973--996, 2016] with additional viscoelastic effects, which was also investigated by the authors [arXiv preprint: 2110.00042, 2021]. Compared to it, the problem is posed in a cylindrical domain with ninety-degree contact angles, which brings additional difficulties when we deal with the linearization of the system By a reflection argument, we obtain the existence and uniqueness of strong solutions to the model problems for the linear systems, which are then shown to be well-posed in a cylindrical (annular) domain via a localization procedure. Finally, we prove that the full nonlinear system admits a unique strong solution locally with the aid of the contraction mapping principle.

math.AP↗
arXiv2021

On a fluid-structure interaction problem for plaque growth

We study a free-boundary fluid-structure interaction problem with growth, which arises from the plaque formation in blood vessels. The fluid is described by the incompressible Navier-Stokes equation, while the structure is considered as a viscoelastic incompressible neo-Hookean material. Moreover, the growth due to the biochemical process is taken into account. Applying the maximal regularity theory to a linearization of the equations, along with a deformation mapping, we prove the well-posedness of the full nonlinear problem via the contraction mapping principle.

math.AP↗
arXiv2021

Large Times Existence for Thin Vibrating Rods

We consider the dynamical evolution of a thin rod described by an appropriately scaled wave equation of nonlinear elasticity. Under the assumption of well-prepared initial data and external forces, we prove that a solution exists for arbitrarily large times, if the diameter of the cross section is chosen sufficiently small. The scaling regime is such that the limiting equations are linear.

math.AP↗
arXiv2021

Well-posedness of the Muskat problem in subcritical $L_p$-Sobolev spaces

We study the Muskat problem describing the vertical motion of two immiscible fluids in a two-dimensional homogeneous porous medium in an $L_p$-setting with $p\in(1,\infty)$. The Sobolev space $W^s_p(\mathbb{R})$ with $s=1+1/p$ is a critical space for this problem. We prove, for $s\in (1+1/p,2),$ that the Rayleigh-Taylor condition identifies an open subset of $W^s_p(\mathbb{R})$ within which the Muskat problem is of parabolic type. This enables us to establish the local well-posedness of the problem in all these subcritical spaces together with a parabolic smoothing property.

math.AP↗
arXiv2021

Convergence of the Allen-Cahn equation with a nonlinear Robin Boundary Condition to Mean Curvature Flow with Contact Angle close to $90$°

This paper is concerned with the sharp interface limit for the Allen-Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain $Ω\subset\mathbb{R}^2$. We assume that a diffuse interface already has developed and that it is in contact with the boundary $\partialΩ$. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant $α$-contact angle. For $α$ close to $90$° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen-Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen-Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.

math.AP↗