arXiv subjects

Christoph Hurm

Publications and source records attributed to Christoph Hurm.

7 recordsLinked to original sources

arXiv2026

Well-posedness and longtime behavior of the conserved Navier--Stokes--Allen--Cahn equations with unmatched viscosities and singular potential

We consider an incompressible Navier--Stokes system nonlinearly coupled with a conserved Allen--Cahn equation with a singular potential (e.g., of Flory--Huggins type). This model describes a mass-conserving two-phase flow with constant density and non-constant viscosity. First, in three spatial dimensions, we prove the existence and uniqueness of local-in-time strong solutions to the associated initial--boundary value problem, subject to no-slip boundary conditions for the velocity field and homogeneous boundary conditions for the phase field. Next, by means of a relative energy approach, we establish a conditional weak--strong uniqueness principle in three dimensions, as well as unconditional uniqueness of weak solutions in two dimensions. Finally, building on recent seminal results by the first and third authors, we prove for the first time that, in both two and three dimensions and for general singular potentials, every global-in-time weak solution asymptotically separates from the pure phases and converges to a unique equilibrium. This result is obtained under minimal assumptions on the viscosity coefficient. Moreover, under additional regularity assumptions on the viscosity, we combine the asymptotic strict separation property with the conditional weak--strong uniqueness principle to show that weak solutions undergo asymptotic regularization. As a consequence, convergence to equilibrium also holds in higher-order norms.

math.AP↗

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.

math.AP↗

arXiv2026

The stochastic nonlocal Cahn-Hilliard equation with regular potential and multiplicative noise

In this work, we deal with the stochastic counterpart of the nonlocal Cahn-Hilliard equation with regular potential in a smooth bounded one-, two- or three-dimensional domain. The problem is endowed with homogeneous Neumann boundary conditions and random initial data. Furthermore, the system is driven by cylindrical noise of multiplicative type. For the resulting system, we are able to show the existence of probabilistically-weak (or martingale) solutions in two and three dimensions, that are unique and probabilistically-strong under suitable assumptions on the stochastic diffusion. Moreover, we investigate the nonlocal-to-local asymptotics toward solutions of the local stochastic Cahn-Hilliard equations, establishing, under regularity conditions, a precise rate of convergence as well.

math.AP↗

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

Nonlocal-to-local convergence rates for strong solutions to a Navier-Stokes-Cahn-Hilliard system with singular potential

The main goal of this paper is to establish the nonlocal-to-local convergence of strong solutions to a Navier--Stokes--Cahn--Hilliard model with singular potential describing immiscible, viscous two-phase flows with matched densities, which is referred to as the Model H. This means that we show that the strong solutions to the nonlocal Model H converge to the strong solution to the local Model H as the weight function in the nonlocal interaction kernel approaches the delta distribution. Compared to previous results in the literature, our main novelty is to further establish corresponding convergence rates. Before investigating the nonlocal-to-local convergence, we first need to ensure the strong well-posedness of the nonlocal Model H. In two dimensions, this result can already be found in the literature, whereas in three dimensions, it will be shown in the present paper. Moreover, in both two and three dimensions, we establish suitable uniform bounds on the strong solutions of the nonlocal Model H, which are essential to prove the nonlocal-to-local convergence results.

math.AP↗

arXiv2024

Nonlocal-to-Local Convergence for a Cahn-Hilliard Tumor Growth Model

We consider a local Cahn-Hilliard-type model for tumor growth as well as a nonlocal model where, compared to the local system, the Laplacian in the equation for the chemical potential is replaced by a nonlocal operator. The latter is defined as a convolution integral with suitable kernels parametrized by a small parameter. For sufficiently smooth bounded domains in three dimensions, we prove convergence of weak solutions of the nonlocal model towards strong solutions of the local model together with convergence rates with respect to the small parameter. The proof is done via a Gronwall-type argument and a convergence result with rates for the nonlocal integral operator towards the Laplacian due to Abels, Hurm arXiv:2307.02264.

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↗