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Tim Binz

Publications and source records attributed to Tim Binz.

At least 19 recordsLinked to original sources
arXiv2026

Long-term behaviour of the CAO-system: Optimal polynomial $H^1$-convergence to degenerate equilibria

This article is concerned with a definitive analysis of its long-term dynamics of the CAO-system. We establish the convergence of global strong solutions to constant equilibria in the full $H^1$-topology in the non-perturbative regime. Remarkably, we demonstrate that the CAO-system exhibits a polynomial rate of decay and we determine the optimal rate. This phenomenon stands in stark contrast to the exponential convergence typically expected for uniformly parabolic systems on bounded domains. This algebraic slowing is shown to be a purely nonlinear effect, driven by the emergence of a slow manifold (center manifold) induced by nonlinear coupling conditions. To the best of our knowledge, this provides the first instance of a global polynomial decay result for a non-gradient, non-local dissipative system on a bounded domain.

math.AP↗
arXiv2026

The $\mathrm{L}^1$-Stokes Semigroup

We study the Stokes operator with no-slip boundary conditions on the spaces $\mathrm{L}_{\sigma,n}(\Omega)$ and ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$, where $\Omega\subset\mathbb{R}^d$ is an arbitrary bounded $\mathrm{C}^{1,\alpha}$-domain. We show that the Stokes operator on $\mathrm{L}^1_{\sigma,n}(\Omega)$ does not generate a $\mathrm{C}_0$-semigroup, even though the resolvent problem is uniquely solvable. In stark contrast, its realization on ${\mathrm{L}^1(\Omega,\mathbb{C}^d)}/{\nabla \mathrm{W}^{1,1}(\Omega,\mathbb{C})}$ generates a compact, analytic $\mathrm{C}_0$-semigroup, which leaves $\mathrm{L}^1_{\sigma,n}(\Omega)$ invariant. The key point is that these two realizations, which are canonically identified for $1<p<\infty$ through the Helmholtz decomposition, cease to be equivalent at the endpoint $p=1$. This leads to genuinely different functional analytic properties. Our result provides the first positive generation theorem for the Stokes operator with no-slip boundary conditions in a pure $\mathrm{L}^1$-setting on a bounded domain and settles a problem that had remained open for nearly fifty years; see, e.g., \cite{Koz:01,DHP:01}. In this sense, it completes the theory of the Stokes semigroup across the full scale of solenoidal Lebesgue spaces on (smooth) bounded domains. As an intermediate step, some results on the space of Radon measures are obtained. The proof combines the sun-dual construction with a precise analysis of the failure of the Helmholtz decomposition in $\mathrm{L}^1$, the celebrated result of Abe and Giga \cite{AG:12} on the Stokes semigroup on $\mathrm{C}_{\sigma,0}(\Omega)$ and the regularity theory refinements for the Stokes operator recently developed by Breit and the second author \cite{BG:25}.

math.AP↗
arXiv2026

Homothetic Self-Similar Solutions to the Incompressible Navier-Stokes Equations

We investigate homothetic forward self-similar solutions of the incompressible Navier-Stokes equations: the solutions $\overline{U}$ for which both $\overline{U}$ and $\beta \overline{U}$ are self-similar profiles for some nontrivial $\beta$. Homothetic solutions are, in addition, the only solutions for which a singular limit argument can be used to prove non-uniqueness of Leray-Hopf solutions along the lines of the Jia-\v{S}ver\'ak program. In three dimensions, and for sufficiently regular initial data, we prove a Liouville theorem that rules out the existence of non-trivial homothetic solutions. In two dimensions, our Liouville theorem proves that the only decaying homothetic solution is the Oseen vortex. In addition, we prove that the Euler operator linearized around the Oseen vortex is stable. On the other hand, we also discover a homothetic solution for which an unstable approximate eigenvalue of the linearized Euler operator exists.

math.AP↗
arXiv2026

Strong well-posedness of a fluid--poro-viscoelastic interaction problem: An approach by Spectral analysis

This article investigates a coupled viscoelastic Navier--Stokes--Biot system describing the interaction between an incompressible viscous fluid and a poro--viscoelastic medium in three spatial dimensions. The coupling between the fluid and the porous medium is realized through Beavers--Joseph--Saffman type interface conditions. Using spectral analysis, it is proved that the coupled system admits a unique, strong, global solution for small initial data. In addition, a Serrin--type blow-up criterion is established.

math.AP↗
arXiv2025

Global strong well-posedness of the CAO-problem introduced by Lions, Temam and Wang

Consider the CAO-problem introduced by Lions, Temam and Wang, which concerns a system of two fluids described by two primitive equations coupled by fully nonlinear interface conditions. They proved in their pioneering work the existence of a weak solution to the CAO-system; its uniqueness remained an open problem. In this article, it is shown that this coupled CAO-system is globally strongly well-posed for large data, even in critical Besov spaces. It is furthermore shown that, away from the boundary, the solution is even real analytic. The approach presented relies on an optimal data result for the boundary terms in the linearized system in terms of time-space Triebel-Lizorkin spaces. Boundary terms are then controlled by paraproduct methods in these spaces.

math.AP↗
arXiv2025

Fluid-Structure Interaction with Porous Media: The Beaver-Joseph condition in the strong sense

This article considers fluid structure interaction describing the motion of a fluid contained in a porous medium. The fluid is modelled by Navier-Stokes equations and the coupling between fluid and the porous medium is described by the classical Beaver-Joseph or the Beaver-Joseph-Saffman interface condition. In contrast to previous work these conditions are investigated for the first time in the strong sense and it is shown that the coupled system admits a unique, global strong solution in critical spaces provided the data are small enough. Furthermore, a Serrin-type blow-up criterium is developed and higher regularity estimates at the interface are established, which say that the solution is even analytic provided the forces are so.

math.AP↗
arXiv2024

Stability of the Inviscid Power-Law Vortex

We prove that the power-law vortex $\overline{\omega}(x) = \beta |x|^{-\alpha}$, which explicitly solves the stationary unforced incompressible Euler equations in $\mathbb{R}^2$ in both physical and self-similar coordinates, is exponentially linearly stable in self-similar coordinates with the natural scaling. This result, which is valid for functions in a weighted $L^2$ space and in the un-weighted $L^2$ space with a mild symmetry condition, answers a question from the monograph by Albritton et al. Moreover, we prove that in physical coordinates the linearization around the power law vortex cannot generate an unstable $C_0$-semigroup.

math.AP↗
arXiv2023

Interaction of liquid crystals with a rigid body

This article investigates the interaction of nematic liquid crystals modeled by a simplified Ericksen-Leslie model with a rigid body. It is shown that this problem is locally strongly well-posed, and that it also admits a unique, global strong solution for initial data close to constant equilibria. The proof of the global strong solution relies on a new splitting method for the director in a mean value zero and average part.

math.AP↗
arXiv2023

Uniqueness of weak solutions to the primitive equations in some anisotropic spaces

We consider the 3D or 2D primitive equations for oceans and atmosphere in the isothermal setting. In this paper, we establish a new conditional uniqueness result for weak solutions to the primitive equations, that is, if a weak solution belongs some scaling invariant function spaces, and satisfies some additional assumptions, then the weak solution is unique. In particular, our result can be obtained as different one from $z$-weak solutions framework by adopting some anisotropic approaches with the homogeneous toroidal Besov spaces. As an application of the proof, we establish the energy equality for weak solutions in the uniqueness class given in the main theorem.

math.AP↗
arXiv2022

Interaction of geophysical flows with sea ice dynamics

This article establishes local strong well-posedness and global strong well-posedness close to constant equilibria of a model coupling the primitive equations of ocean and atmospheric dynamics with Hibler's viscous-plastic sea ice model. In order to treat the coupling conditions, an approach involving the hydrostatic Dirichlet and Dirichlet-to-Neumann operator is developed. Mapping properties of the latter operators are investigated for the first time and are of central importance for showing that the operator associated with the linearized coupled system admits a bounded $\mathcal{H}^\infty$-calculus on suitable $\mathrm{L}^q$-spaces. Quasilinear methods allow then to obtain the strong well-posedeness results described above.

math.AP↗
arXiv2021

A convergent finite element algorithm for mean curvature flow in arbitrary codimension

Optimal-order uniform-in-time $H^1$-norm error estimates are given for semi- and full discretizations of mean curvature flow of surfaces in arbitrarily high codimension. The proposed and studied numerical method is based on a parabolic system coupling the surface flow to evolution equations for the mean curvature vector and for the orthogonal projection onto the tangent space. The algorithm uses evolving surface finite elements and linearly implicit backward difference formulae. This numerical method admits a convergence analysis in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. Numerical experiments in codimension 2 illustrate and complement our theoretical results.

math.NA↗
arXiv2021

An abstract framework for interior-boundary conditions

In a configuration space whose boundary can be identified with a subset of its interior, a boundary condition can relate the behaviour of a function on the boundary and in the interior. Additionally, boundary values can appear as additive perturbations. Such boundary conditions have recently provided insight into problems form quantum field theory. We discuss interior-boundary conditions in an abstract setting, with a focus on self-adjoint operators, proving self-adjointness criteria, resolvent formulas, and a classification theorem.

math.SP↗
arXiv2020

A convergent finite element algorithm for generalized mean curvature flows of closed surfaces

An algorithm is proposed for generalized mean curvature flow of closed two-dimensional surfaces, which include inverse mean curvature flow, powers of mean and inverse mean curvature flow, etc. Error estimates are proven for semi- and full discretisations for the generalized flow. The algorithm proposed and studied here combines evolving surface finite elements, whose nodes determine the discrete surface, and linearly implicit backward difference formulae for time integration. The numerical method is based on a system coupling the surface evolution to non-linear second-order parabolic evolution equations for the normal velocity and normal vector. Convergence proofs are presented in the case of finite elements of polynomial degree at least two and backward difference formulae of orders two to five. The error analysis combines stability estimates and consistency estimates to yield optimal-order $H^1$-norm error bounds for the computed surface position, velocity, normal vector, normal velocity, and therefore for the mean curvature. The stability analysis is performed in the matrix-vector formulation, and is independent of geometric arguments, which only enter the consistency analysis. Numerical experiments are presented to illustrate the convergence results, and also to report on monotone quantities, e.g.~Hawking mass for inverse mean curvature flow. Complemented by experiments for non-convex surfaces.

math.NA↗
arXiv2020

The primitive equations with stochastic wind driven boundary conditions

The primitive equations for geophysical flows are studied under the influence of {\em stochastic wind driven boundary conditions} modeled by a cylindrical Wiener process. We adapt an approach by Da Prato and Zabczyk for stochastic boundary value problems to define a notion of solutions. Then a rigorous treatment of these stochastic boundary conditions, which combines stochastic and deterministic methods, yields that these equations admit a unique, local pathwise solution within the anisotropic $L^q_t$-$H^{-1,p}_zL^p_{xy}$-setting. This solution is constructed in critical spaces.

math.PR↗
arXiv2019

Dirichlet-to-Neumann operators on manifolds

We consider the Dirichlet-to-Neumann operator associated to a strictly elliptic operator on the space $\mathrm{C}(\partial M)$ of continuous functions on the boundary $\partial M$ of a compact manifold $\overline{M}$ with boundary. We prove that it generates an analytic semigroup of angle $\frac{\pi}{2}$. This yields that the corresponding strictly elliptic operator with Wentzell boundary conditions generates a compact and analytic semigroups of angle $\frac{\pi}{2}$ on the space $\mathrm{C}(\overline{M})$.

math.FA↗