arXiv subjects

Jonas Stange

Publications and source records attributed to Jonas Stange.

8 recordsLinked to original sources

arXiv2026

On a bulk-surface Navier-Stokes-Cahn-Hilliard model: Existence of weak solutions and asymptotic limits

We study a thermodynamically consistent bulk-surface Navier--Stokes--Cahn--Hilliard system describing two-phase flows exhibiting moving contact lines, variable contact angles, and mass transfer. We establish the existence of global weak solutions for non-degenerate mobility functions and singular free-energy potentials. The proof is based on an implicit time-discretization scheme and a subdifferential characterization of the convex part of the bulk-surface free energy. Finally, we study the simultaneous high-friction and decoupling limit, and prove that corresponding weak solutions converge, up to a subsequence, to a weak solution of the Abels--Garcke--Gr\"un model. One key ingredient of the proof is the Mosco convergence of the convex part of the bulk-surface free energy, which might be of independent interest.

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arXiv2026

Long-time dynamics of a bulk-surface convective Cahn--Hilliard system: Pullback attractors and convergence to equilibrium

We study the long-time dynamics of a bulk-surface convective Cahn--Hilliard system describing phase separation processes with bulk-surface interaction. The presence of convection terms leads to a non-autonomous dynamical system and prevents the associated free energy from being a Lyapunov functional, which makes the analysis of the asymptotic behavior considerably more challenging. First, we establish an instantaneous regularization property for weak solutions. Next, interpreting the evolution as a continuous two-parameter process, we prove the existence of a minimal pullback attractor. Finally, under suitable decay assumptions on the velocity fields, we show that every solution converges as $t\to\infty$ to a single steady state. The proof of this convergence relies on the {\L}ojasiewicz--Simon inequality combined with customized decay estimates that compensate for the lack of a monotone energy functional.

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arXiv2025

Global well-posedness of strong solutions to a bulk-surface Navier-Stokes-Cahn-Hilliard model with non-degenerate mobilities in two dimensions

We examine a thermodynamically consistent diffuse interface model for bulk-surface viscous fluid mixtures. This model consists of a Navier--Stokes--Cahn--Hilliard model in the bulk coupled to a surface Navier--Stokes--Cahn--Hilliard system on the boundary. In this paper, we address the global well-posedness of strong solutions in the two-dimensional setting, also covering the physically meaningful case of non-degenerate mobility functions. Lastly, we prove the uniqueness of the corresponding strong solutions and their continuous dependence on the initial data. Our approach hinges upon new well-posedness and regularity theory for a convective bulk-surface Cahn--Hilliard equation with non-degenerate mobilities, as well as a bulk-surface Stokes equation with non-constant coefficients.

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arXiv2025

A thermodynamically consistent model for bulk-surface viscous fluid mixtures: Model derivation and mathematical analysis

We derive and analyze a new diffuse interface model for incompressible, viscous fluid mixtures with bulk-surface interaction. Our system consists of a Navier--Stokes--Cahn--Hilliard model in the bulk that is coupled to a surface Navier--Stokes--Cahn--Hilliard model on the boundary. Compared with previous models, the inclusion of an additional surface Navier--Stokes equation is motivated, for example, by biological applications such as the seminal \textit{fluid mosaic model} (Singer \& Nicolson, \textit{Science}, 1972) in which the surface of biological cells is interpreted as a thin layer of viscous fluids. We derive our new model by means of local mass balance laws, local energy dissipation laws, and the Lagrange multiplier approach. Moreover, we prove the existence of global weak solutions via a semi-Galerkin discretization. The core part of the mathematical analysis is the study of a novel bulk-surface Stokes system and its corresponding bulk-surface Stokes operator. Its eigenfunctions are used as the Galerkin basis to discretize the bulk-surface Navier--Stokes subsystem.

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arXiv2025

Well-posedness and long-time behavior of a bulk-surface Cahn--Hilliard model with non-degenerate mobility

We study a bulk-surface Cahn--Hilliard model with non-degenerate mobility and singular potentials in two dimensions. Following the ideas of the recent work by Conti, Galimberti, Gatti, and Giorgini [Calc. Var. Partial Differential Equations, 64(3):Paper No. 87, 32, 2025] for the Cahn--Hilliard equation with homogeneous Neumann boundary conditions, we show the uniqueness of weak solutions together with a continuous dependence estimate for sufficiently regular mobility functions. Next, under weaker assumptions on the mobility functions, we show the existence of a weak solution that exhibits the propagation of uniform-in-time regularity and satisfies the instantaneous separation property. Lastly, we consider the long-time behavior and prove that the unique weak solution converges to a solution of the stationary bulk-surface Cahn--Hilliard equation. Our approach for the uniqueness proof relies on a new well-posedness and regularity theory for a bulk-surface elliptic system with non-constant coefficients, which may be of independent interest.

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arXiv2025

Regularity of a bulk-surface Cahn-Hilliard model driven by Leray velocity fields

We consider a convective bulk-surface Cahn--Hilliard system with dynamic boundary conditions and singular potentials. For this model, well-posedness results concerning weak and strong solutions have already been established in the literature. However, they require the prescribed velocity fields to belong to function spaces with high time regularity. In this paper, we prove that the well-posedness of weak solutions holds true under more general regularity assumptions on the velocity fields. Next, via an alternative proof for higher regularity, we show the well-posedness of strong solutions for velocity fields of Leray type, which is a more relevant assumption for physical applications. Our approach hinges upon a new well-posedness and regularity theory for a bulk-surface elliptic system with singular nonlinearities, which may be of independent interest.

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arXiv2024

Strong well-posedness and separation properties for a bulk-surface convective Cahn--Hilliard system with singular potentials

This paper addresses the well-posedness of a general class of bulk-surface convective Cahn--Hilliard systems with singular potentials. For this model, we first prove the existence of a global-in-time weak solution by approximating the singular potentials via a Yosida regularization, applying the corresponding results for regular potentials, and eventually passing to the limit in this approximation scheme. Then, we prove the uniqueness of weak solutions and their continuous dependence on the velocity fields and the initial data. Afterwards, assuming additional regularity of the domain as well as the velocity fields, we establish higher regularity properties of weak solutions and eventually the existence of strong solutions. In the end, we discuss strict separation properties for logarithmic type potentials in both two and three dimensions.

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arXiv2024

Well-posedness of a bulk-surface convective Cahn--Hilliard system with dynamic boundary conditions

We consider a general class of bulk-surface convective Cahn--Hilliard systems with dynamic boundary conditions. In contrast to classical Neumann boundary conditions, the dynamic boundary conditions of Cahn--Hilliard type allow for dynamic changes of the contact angle between the diffuse interface and the boundary, a convection-induced motion of the contact line as well as absorption of material by the boundary. The coupling conditions for bulk and surface quantities involve parameters $K,L\in[0,\infty]$, whose choice declares whether these conditions are of Dirichlet, Robin or Neumann type. We first prove the existence of a weak solution to our model in the case $K,L\in (0,\infty)$ by means of a Faedo--Galerkin approach. For all other cases, the existence of a weak solution is then shown by means of the asymptotic limits, where $K$ and $L$ are sent to zero or to infinity, respectively. Eventually, we establish higher regularity for the phase-fields, and we prove the uniqueness of weak solutions given that the mobility functions are constant.

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