arXiv subjects

Franziska Weber

Publications and source records attributed to Franziska Weber.

At least 19 recordsLinked to original sources

arXiv2026

Existence of solutions for a model of the Earth's magnetic field

We study a physically realistic, whole-core mathematical model of the dynamics in the Earth's core and we prove existence of Leray-Hopf type weak solutions to the model. Our model combines Magneto-Hydrodynamic equations in the liquid outer core with solid physics for the electrically conducting inner core, and treats everything exterior to the core as a perfect insulator governed by Maxwell's equations. We prove existence of weak solutions using Galerkin approximations. In order to control the nonlinearities, we must define an appropriate function space for the magnetic field and prove a Biot-Savart type result. The main new difficulty here is properly setting up the functional framework to simultaneously deal with the fluid structure interaction with the inner core and the magnetic transmission problem, with both the perfectly conducting inner core and the perfectly insulating mantle/exterior.

math.AP↗

arXiv2026

Mixing and Small-Scale Formation in a Passive Divergence-Free Vector Field

We study mixing for a divergence-free passive vector field $u$ transported by another divergence-free vector field $U$, where $u$ evolves according to $ \partial_t u + (U \cdot \nabla) u + \nabla p = 0.$ In recent years, a lot of attention has been given to the question of optimal mixing in the scalar case, where there is a Sobolev constraint on the advecting velocity. In the vector setting considered here, however, the pressure term introduces substantial difficulties, since the simple Lagrangian perspective available in the scalar case is no longer applicable. In this paper, we investigate mixing on a torus $\mathbb{T}^d$ under the assumption that the field $U$ satisfies $ \|U(t)\|_{W^{1,q}} \leq C $ and we quantify mixing through the decay of the homogeneous $ H^{-\alpha}$ norm of $u$. We start with establishing conditions on $U$ that guarantee existence and uniqueness of solutions. We then derive lower bounds on the mixing rate for various ranges of $q$ and $\alpha$. In addition, we carry out numerical simulations of mixing by choosing, at each time instant, a field $U$ that maximizes the instantaneous decay of the $ H^{-\alpha}$ norm. These simulations provide evidence that the optimal mixing rate is at least exponential in time. More broadly, we view the present model and its diffusive analogue, as a useful framework for probing mechanisms of small-scale formation in divergence-free vector fields and for formulating simplified versions of open questions related to the incompressible Euler and Navier--Stokes equations.

math.AP↗

arXiv2025

Convergence of a Second-Order Projection Method to Leray-Hopf Solutions of the Incompressible Navier-Stokes Equations

We analyze a second-order projection method for the incompressible Navier-Stokes equations on bounded Lipschitz domains. The scheme employs a Backward Differentiation Formula of order two (BDF2) for the time discretization, combined with conforming finite elements in space. Projection methods are widely used to enforce incompressibility, yet rigorous convergence results for possibly non-smooth solutions have so far been restricted to first-order schemes. We establish, for the first time, convergence (up to subsequence) of a second-order projection method to Leray-Hopf weak solutions under minimal assumptions on the data, namely $u_0 \in L^2_{\text{div}}(\Omega)$ and $f \in L^2(0,T;L^2_{\text{div}}(\Omega))$. Our analysis relies on two ingredients: A discrete energy inequality providing uniform $L^{\infty}(0,T;L^2(\Omega))$ and $L^2(0,T;H^1_0(\Omega))$ bounds for suitable interpolants of the discrete velocities, and a compactness argument combining Simon's theorem with refined time-continuity estimates. These tools overcome the difficulty that only the projected velocity satisfies an approximate divergence-free condition, while the intermediate velocity is controlled in space. We conclude that a subsequence of the approximations converges to a Leray-Hopf weak solution. This result provides the first rigorous convergence proof for a higher-order projection method under no additional assumptions on the solution beyond those following from the standard a priori energy estimate.

math.NA↗

arXiv2025

A convergence proof for a finite element discretization of Chorin's projection method of the incompressible Navier-Stokes equations

We study Chorin's projection method combined with a finite element spatial discretization for the time-dependent incompressible Navier-Stokes equations. The scheme advances the solution in two steps: a prediction step, which computes an intermediate velocity field that is generally not divergence-free, and a projection step, which enforces (approximate) incompressibility by projecting this velocity onto the (approximately) divergence-free subspace. We establish convergence, up to a subsequence, of the numerical approximations generated by this scheme to a Leray-Hopf weak solution of the Navier-Stokes equations, without any additional regularity assumptions beyond square-integrable initial data. A discrete energy inequality yields a priori estimates, which we combine with a new compactness result to prove precompactness of the approximations in $L^2([0,T]\times\Omega)$, where $[0,T]$ is the time interval and $\Omega$ is the spatial domain. Passing to the limit as the discretization parameters vanish, we obtain a weak solution of the Navier-Stokes equations. A central difficulty is that different a priori bounds are available for the intermediate and projected velocity fields; our compactness argument carefully integrates these estimates to complete the convergence proof.

math.NA↗

arXiv2025

Global existence of weak solutions to a two-phase diffuse interface model of ferrofluids dynamics

Ferrofluids are a class of materials that exhibit both fluid and magnetic properties. We consider a two-phase diffuse interface model for the dynamics of ferrofluids on a bounded domain. One phase is assumed to be magnetic, the other phase can be magnetic or non-magnetic. We derive a coupled system of partial differential equations consisting of the incompressible Navier-Stokes equations, an evolution equation for the magnetization, the magnetostatics equations for the magnetic field and the Cahn-Hilliard equations for the evolution of the phase field variable, which are all coupled through various source terms and parameters. In contrast to similar models in the literature, the system in this work formally satisfies an energy balance which remains meaningful even in singular limits such as a limit of zero relaxation time. However, the formal derivation of this balance requires a delicate cancellation of several highly non-linear terms, making it challenging to ensure similar cancellations for approximating systems. Our first main result is to prove the existence of global weak solutions for our ferrofluid system based on a carefully constructed sequence of approximation steps. Additionally, we also study the relaxation towards the quasi-equilibrium, in which case the magnetization equation degenerates to a linear relation between the magnetic field and the magnetization. As our second main result, we prove the rigorous convergence to this limiting system.

math.AP↗

arXiv2025

Sufficient Conditions for the Energy Balance for the Stochastic Incompressible Euler Equations with Additive Noise in two Space Dimensions

We consider vanishing viscosity approximations to solutions of the stochastic incompressible Euler equations in two space dimensions with additive noise. We identify sufficient and necessary conditions under which martingale solutions of the stochastic Euler equations satisfy an exact energy balance in mean. We find that the tightness of the laws of the approximating sequence of solutions of the stochastic Navier-Stokes equations in $L^2([0,T]\times D)$ is equivalent to the limiting martingale solution satisfying an energy balance in mean. Numerical simulations illustrate the theoretical findings.

math.PR↗

arXiv2024

The Zero Inertia Limit for the Q-Tensor Model of Liquid Crystals: Analysis and Numerics

The goal of this work is to rigorously study the zero inertia limit for the Q-tensor model of liquid crystals. Though present in the original derivation of the Ericksen-Leslie equations for nematic liquid crystals, the inertia term of the model is often neglected in analysis and applications. We show wellposedness of the model including inertia and then show using the relative entropy method that solutions of the model with inertia converge to solutions of the model without inertia at a rate $σ$ in $L^\infty(0,T;H^1(\dom))$, where $σ$ is the inertial constant. Furthermore, we present an energy stable finite element scheme that is stable and convergent for all $σ$ and study the zero inertia limit numerically. We also present error estimates for the fully discrete scheme with respect to the discretization parameters in time and space.

math.NA↗

arXiv2024

Finite element analysis of a nematic liquid crystal Landau-de Gennes model with quartic elastic terms

In arXiv:1906.09232v2, Golovaty et al. present a $Q$-tensor model for liquid crystal dynamics which reduces to the well-known Oseen-Frank director field model in uniaxial states. We study a closely related model and present an energy stable scheme for the corresponding gradient flow. We prove the convergence of this scheme via fixed-point iteration and rigorously show the $\Gamma$-convergence of discrete minimizers as the mesh size approaches zero. In the numerical experiments, we successfully simulate isotropic-to-nematic phase transitions as expected.

math.NA↗

arXiv2023

A Convergent Finite Element Scheme for the Q-Tensor Model of Liquid Crystals Subjected to an Electric Field

We study the Landau-de Gennes Q-tensor model of liquid crystals subjected to an electric field and develop a fully discrete numerical scheme for its solution. The scheme uses a convex splitting of the bulk potential, and we introduce a truncation operator for the Q-tensors to ensure well-posedness of the problem. We prove the stability and well-posedness of the scheme. Finally, making a restriction on the admissible parameters of the scheme, we show that up to a subsequence, solutions to the fully discrete scheme converge to weak solutions of the Q-tensor model as the time step and mesh are refined. We then present numerical results computed by the numerical scheme, among which we show that it is possible to simulate the Fr\'eedericksz transition with this scheme.

math.NA↗

arXiv2023

On the Convergence of an IEQ-based first-order Numerical Scheme for the Beris-Edwards System

We present a convergence analysis of an unconditionally energy-stable first-order semi-discrete numerical scheme designed for a hydrodynamic Q-tensor model, the so-called Beris-Edwards system, based on the Invariant Energy Quadratization Method (IEQ). The model consists of the Navier-Stokes equations for the fluid flow, coupled to the Q-tensor gradient flow describing the liquid crystal molecule alignment. By using the Invariant Energy Quadratization Method, we obtain a linearly implicit scheme, accelerating the computational speed. However, this introduces an auxiliary variable to replace the bulk potential energy and it is a priori unclear whether the reformulated system is equivalent to the Beris-Edward system. In this work, we prove stability properties of the scheme and show its convergence to a weak solution of the coupled liquid crystal system. We also demonstrate the equivalence of the reformulated and original systems in the weak sense.

math.NA↗

arXiv2022

A new reduced order model of linear parabolic PDEs

How to build an accurate reduced order model (ROM) for multidimensional time dependent partial differential equations (PDEs) is quite open. In this paper, we propose a new ROM for linear parabolic PDEs. We prove that our new method can be orders of magnitude faster than standard solvers, and is also much less memory intensive. Under some assumptions on the problem data, we prove that the convergence rates of the new method is the same with standard solvers. Numerical experiments are presented to confirm our theoretical result.

math.NA↗

arXiv2022

Convergence analysis of a fully discrete energy-stable numerical scheme for the Q-tensor flow of liquid crystals

We present a fully discrete convergent finite difference scheme for the Q-tensor flow of liquid crystals based on the energy-stable semi-discrete scheme by Zhao, Yang, Gong, and Wang (Comput. Methods Appl. Mech. Engrg. 2017). We prove stability properties of the scheme and show convergence to weak solutions of the Q-tensor flow equations. We demonstrate the performance of the scheme in numerical simulations.

math.NA↗

arXiv2022

On the vanishing viscosity limit of statistical solutions of the incompressible Navier-Stokes equations

We study statistical solutions of the incompressible Navier-Stokes equation and their vanishing viscosity limit. We show that a formulation using correlation measures, which are probability measures accounting for spatial correlations, and moment equations is equivalent to statistical solutions in the Foias-Prodi sense. Under the assumption of weak scaling, a weaker version of Kolmogorov's self-similarity at small scales hypothesis that allows for intermittency corrections, we show that the limit is a statistical solution of the incompressible Euler equations. To pass to the limit, we derive a Karman-Howarth-Monin relation for statistical solutions and combine it with the weak scaling assumption and a compactness theorem for correlation measures.

math.AP↗

arXiv2022

On Bayesian data assimilation for PDEs with ill-posed forward problems

We study Bayesian data assimilation (filtering) for time-evolution PDEs, for which the underlying forward problem may be very unstable or ill-posed. Such PDEs, which include the Navier-Stokes equations of fluid dynamics, are characterized by a high sensitivity of solutions to perturbations of the initial data, a lack of rigorous global well-posedness results as well as possible non-convergence of numerical approximations. Under very mild and readily verifiable general hypotheses on the forward solution operator of such PDEs, we prove that the posterior measure expressing the solution of the Bayesian filtering problem is stable with respect to perturbations of the noisy measurements, and we provide quantitative estimates on the convergence of approximate Bayesian filtering distributions computed from numerical approximations. For the Navier-Stokes equations, our results imply uniform stability of the filtering problem even at arbitrarily small viscosity, when the underlying forward problem may become ill-posed, as well as the compactness of numerical approximants in a suitable metric on time-parametrized probability measures.

math.AP↗

arXiv2021

Well-posedness of Bayesian inverse problems for hyperbolic conservation laws

We study the well-posedness of the Bayesian inverse problem for scalar hyperbolic conservation laws where the statistical information about inputs such as the initial datum and (possibly discontinuous) flux function are inferred from noisy measurements. In particular, the Lipschitz continuity of the measurement to posterior map as well as the stability of the posterior to approximations, are established with respect to the Wasserstein distance. Numerical experiments are presented to illustrate the derived estimates.

math.NA↗

arXiv2021

A convergent numerical scheme for a model of liquid crystal dynamics subjected to an electric field

We present a convergent and constraint-preserving numerical discretization of a mathematical model for the dynamics of a liquid crystal subjected to an electric field. This model can be derived from the Oseen-Frank director field theory, assuming that the dynamics of the electric field are governed by the electrostatics equations with a suitable constitutive relation for the electric displacement field that describes the coupling with the liquid crystal director field. The resulting system of partial differential equations consists of an elliptic equation that is coupled to the wave map equations through a quadratic source term. We show that the discretization preserves the unit length constraint of the director field, is energy-stable and convergent. In numerical experiments, we show that the method is stable even when singularities develop. Moreover, predictions about the alignment of the director field with the electric field are confirmed.

math.NA↗

arXiv2019

On the dynamics of ferrofluids: Global weak solutions to the Rosensweig system and rigorous convergence to equilibrium

This article establishes the global existence of weak solutions to a model proposed by Rosensweig (Rosensweig, Ferrohydrodynamics (1985)) for the dynamics of ferrofluids. The system is expressed by the conservation of linear momentum, the incompressibility condition, the conservation of angular momentum, and the evolution of the magnetization. The existence proof is inspired by the DiPerna-Lions theory of renormalized solutions. In addition, the rigorous relaxation limit of the equations of ferrohydrodynamics towards the quasi-equilibrium is investigated. The proof relies on the relative entropy method, which involves constructing a suitable functional, analyzing its time evolution and obtaining convergence results for the sequence of approximating solutions.

math.AP↗

arXiv2019

Statistical solutions of hyperbolic systems of conservation laws: numerical approximation

Statistical solutions are time-parameterized probability measures on spaces of integrable functions, that have been proposed recently as a framework for global solutions and uncertainty quantification for multi-dimensional hyperbolic system of conservation laws. By combining high-resolution finite volume methods with a Monte Carlo sampling procedure, we present a numerical algorithm to approximate statistical solutions. Under verifiable assumptions on the finite volume method, we prove that the approximations, generated by the proposed algorithm, converge in an appropriate topology to a statistical solution. Numerical experiments illustrating the convergence theory and revealing interesting properties of statistical solutions, are also presented.

math.NA↗