arXiv subjects

Marita Thomas

Publications and source records attributed to Marita Thomas.

12 recordsLinked to original sources

arXiv2026

From damage to delamination via evolutionary Gamma-convergence in a rate-independent quasibrittle regime

We analyze via Evolutionary Gamma-convergence a stratified composite structure consisting of a thin adhesive layer with vanishing thickness and undergoing rate-independent damage, as well as two adjacent elastic adherents. As the width of the intermediate layer tends to zero, we prevent complete degradation of the material by assuming that the damage variable scales minimally like the thickness of the adhesive layer. As a result, we identify a limiting model that combines both a brittle constraint and an adhesive-type energy contribution featuring the jump of the admissible displacements.

math.AP↗

arXiv2026

A mathematical study of an elastic-viscous-plastic sea-ice model with the Kelvin-Voigt rheology

Motivated by the elastic-viscous-plastic (EVP) sea-ice model [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849--1867], which is used in large-scale numerical climate simulations, we proposed in [D. W. Boutros, X. Liu, M. Thomas and E. S. Titi, arXiv:2505.03080 (2025)] the use of the inviscid Voigt regularisation for the constitutive (stress-tensor) relation and proved the global well-posedness of the resulting model. The EVP model treats sea ice as a non-Newtonian fluid. In turn, elastic-viscous-plastic solids often involve a Kelvin-Voigt viscosity in terms of the strain rate. Therefore, in the present work we formulate an elastic-viscous-plastic sea-ice model with a Kelvin-Voigt regularisation in terms of the strain rate. In other words, we introduce the Voigt regularisation in the momentum balance rather than in the constitutive relation (for the stress tensor). We then prove the local well-posedness for the Kelvin-Voigt EVP model with the advection term, in the momentum balance, and the global well-posedness in the absence of the advection term (following a very standard approximation in the latter case). A crucial component of the proof of these results, is a new $L^\infty$-estimate for the stress tensor which relies on the damping structure. Note that, both with and without the advection term, we are able to handle the case of viscosity coefficients without a cutoff from above, which remains a major open problem for the closely related Hibler sea-ice model. We are also able to prove the existence of solutions for much less regular initial data compared to our previous paper on the Voigt-EVP model.

math.AP↗

arXiv2026

Global well-posedness of the elastic-viscous-plastic sea-ice model with the inviscid Voigt-regularisation

In this paper, we initiate the rigorous mathematical analysis of the elastic-viscous-plastic (EVP) sea-ice model, which was introduced in [E. C. Hunke and J. K. Dukowicz, J. Phys. Oceanogr., 27, 9 (1997), 1849-1867]. The EVP model is one of the standard and most commonly used dynamical sea-ice models. We study a regularized version of this model. In particular, we prove the global well-posedness of the EVP model with the inviscid Voigt-regularisation of the evolution equation for the stress tensor. Due to the elastic relaxation and the Voigt regularisation, we are able to handle the case of viscosity coefficients without cutoff, which has been a major issue and a setback in the computational study and analysis of the related Hibler sea-ice model, which was originally introduced in [W. D. Hibler, J. Phys. Oceanogr., 9, 4 (1979), 815-846]. The EVP model shares some structural characteristics with the Oldroyd-B model and related models for viscoelastic non-Newtonian complex fluids.

math.AP↗

arXiv2026

Energy-variational solutions for geodynamical two-phase flows -- From logarithmic to double-obstacle potentials by variational convergence

In [Cheng, Lasarzik, Thomas 2025 ARXIV-Preprint 2509.25508], we studied a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids in the framework of dissipative solutions using a logarithmic potential for the phase-field variable. This choice of potential has the effect that the fluid mixture cannot fully separate into two pure phases. The notion of dissipative solutions is based on a relative energy-dissipation inequality featuring a suitable regularity weight. In this way, this is a very weak solution concept. In the present work, we study the well-posedness of the geodynamical two-phase flow in the notion of energy-variational solutions. They feature an additional scalar energy variable that majorizes the system energy along solutions and they are further characterized by a variational inequality that combines an energy-dissipation estimate with the weak formulation of the system adding an error term that accounts for the mismatch between the energy variable and the system energy multiplied by a suitable regularity weight. We give a comparison of these two concepts. We further study different phase-field potentials for the geodynamical two-phase flow model. In particular, we address the variational limit from a potential with a logarithmic contribution to a double-obstacle potential, then also allowing for the emergence of pure phases. This study underlines that, thanks to its structure, the energy-variational solution is better suited for variational convergence methods than the dissipative solution.

math.AP↗

arXiv2026

Plastic limit of a viscoplastic Burgers equation -- A toy model for sea-ice dynamics

We study the plastic Burgers equation in one space dimension, i.e., the Burgers equation featuring an additional term formally given by the p-Laplacian with p=1, or rather, by the multivalued subdifferential of the total variation functional. Our study highlights that the interplay of the advection term with the stresses given by the multivalued 1-Laplacian is a crucial feature of this model. Eventhough it is an interesting model in itsef, it can also be regarded as a one-dimensional version of the momentum balance of Hibler's model for sea-ice dynamics. Therein, the stress tensor is given by a term with similar properties as the 1-Laplacian in order to account for plastic effects of the ice. For our analysis we start out from a viscoplastic Burgers equation, i.e., a suitably regularized version of the plastic Burgers equation with a small regularization parameter $\varepsilon>0$. For the viscoplastic Burgers equation, we construct a global BV-solution. In the singular limit $\varepsilon\to0$ we deduce the existence of a BV-solution for the plastic Burgers equation. In addition we show that the term arising as the limit of the regularized stresses is indeed related to an element of the subdifferential of the total variation functional.

math.AP↗

arXiv2026

Combined effects of evaporation, sedimentation and solute crystallization on the dynamics of aerosol size distributions on multiple length and time scales

We investigate three aspects of aerosol-mediated air-borne viral infection mechanisms on different length and time scales. First, we address the evolution of the size distribution of a non-interacting ensemble of droplets that are subject to evaporation and sedimentation using a sharp droplet-air interface model. From the exact solution of the evolution equation we derive the viral load in the air and show that it depends sensitively on the relative humidity. Secondly, from Molecular Dynamics simulations we extract the molecular reflection coefficient of single water molecules from the air-water interface. This parameter determines the water condensation and evaporation rate at a liquid droplet surface and therefore the evaporation rate of aqueous droplets. We find the reflection of water to be negligible at room temperature but to rise significantly at elevated temperatures and for grazing incidence angles. Thirdly, we derive a thermodynamically consistent three-dimensional diffuse-interface model for solute-containing droplets that is formulated as a three-phase Cahn-Hilliard/Allen-Cahn system. By numerically solving the coupled system of equations, we explore representative scenarios that show that this model reproduces and generalizes features of the sharp-interface model. These interconnected studies on the dynamics of aerosol droplet evaporation are relevant in order to quantitatively assess the airborne infection risk under varying environmental conditions.

physics.flu-dyn↗

arXiv2025

Analysis of a Cahn--Hilliard model for viscoelastoplastic two-phase flows

We study a Cahn--Hilliard two-phase model describing the flow of two viscoelastoplastic fluids, which arises in geodynamics. A phase-field variable indicates the proportional distribution of the two fluids in the mixture. The motion of the incompressible mixture is described in terms of the volume-averaged velocity. Besides a volume-averaged Stokes-like viscous contribution, the Cauchy stress tensor in the momentum balance contains an additional volume-averaged internal stress tensor to model the elastoplastic behavior. This internal stress has its own evolution law featuring the nonlinear Zaremba-Jaumann time-derivative and the subdifferential of a non-smooth plastic potential. The well-posedness of this system is studied in two cases: Based on a regularization by stress-diffusion we obtain the existence of Leray-Hopf-type weak solutions. In order to deduce existence results also in the absence of the regularization, we introduce the concept of dissipative solutions, which is based on an estimate for the relative energy. We discuss general properties of dissipative solutions and show their existence for the viscoelastoplastic two-phase model in the setting of stress-diffusion. By a limit passage in the relative energy inequality for vanishing stress-diffusion, we conclude an existence result for the non-regularized model.

math.AP↗

arXiv2022

Well-posedness of Hibler's dynamical sea-ice model

This paper establishes the local-in-time well-posedness of solutions to an approximating system constructed by mildly regularizing the dynamical sea-ice model of {\it W.D. Hibler, Journal of Physical Oceanography, 1979}. Our choice of regularization has been carefully designed, prompted by physical considerations, to retain the original coupled hyperbolic-parabolic character of Hibler's model. Various regularized versions of this model have been used widely for the numerical simulation of the circulation and thickness of the Arctic ice cover. However, due to the singularity in the ice rheology, the notion of solutions to the original model is unclear. Instead, an approximating system, which captures current numerical study, is proposed. The well-posedness theory of such a system provides a first-step groundwork in both numerical study and future analytical study.

math.AP↗

arXiv2019

Rate-independent evolution of sets

The goal of this work is to analyze a model for the rate-independent evolution of sets with finite perimeter. The evolution of the admissible sets is driven by that of a given time-dependent set, which has to include the admissible sets and hence is to be understood as an external loading. The process is driven by the competition between perimeter minimization and minimization of volume changes. \par In the mathematical modeling of this process, we distinguish the adhesive case, in which the constraint that the (complement of) the `external load' contains the evolving sets is penalized by a term contributing to the driving energy functional, from the brittle case, enforcing this constraint. The existence of Energetic solutions for the adhesive system is proved by passing to the limit in the associated time-incremental minimization scheme. In the brittle case, this time-discretization procedure gives rise to evolving sets satisfying the stability condition, but it remains an open problem to additionally deduce energy-dissipation balance in the time-continuous limit. This can be obtained under some suitable quantification of data. The properties of the brittle evolution law are illustrated by numerical examples in two space dimensions.

math.AP↗

arXiv2018

Rate-independent damage in thermo-viscoelastic materials with inertia

We present a model for rate-independent, unidirectional, partial damage in visco-elastic materials with inertia and thermal effects. The damage process is modeled by means of an internal variable, governed by a rate-independent flow rule. The heat equation and the momentum balance for the displacements are coupled in a highly nonlinear way. Our assumptions on the corresponding energy functional also comprise the case of the Ambrosio-Tortorelli phase-field model (without passage to the brittle limit). We discuss a suitable weak formulation and prove an existence theorem obtained with the aid of a (partially) decoupled time-discrete scheme and variational convergence methods. We also carry out the asymptotic analysis for vanishing viscosity and inertia and obtain a fully rate-independent limit model for displacements and damage, which is independent of temperature.

math.AP↗

arXiv2016

From Adhesive to Brittle Delamination in Visco-Elastodynamics

In this paper we analyze a system for brittle delamination between two visco-elastic bodies, also subject to inertia, which can be interpreted as a model for dynamic fracture. The rate-independent flow rule for the delamination parameter is coupled with the momentum balance for the displacement, including inertia. This model features a nonsmooth constraint ensuring the continuity of the displacements outside the crack set, which is marked by the support of the delamination parameter. A weak solvability concept, generalizing the notion of energetic solution for rate-independent systems to the present mixed rate-dependent/rate-independent frame, is proposed. Via refined variational convergence techniques, existence of solutions is proved by passing to the limit in approximating systems which regularize the nonsmooth constraint by conditions for adhesive contact. The presence of the inertial term requires the design of suitable recovery spaces small enough to provide compactness but large enough to recover the information on the crack set in the limit.

math.AP↗

arXiv2015

A rate-independent gradient system in damage coupled with plasticity via structured strains

This contribution deals with a class of models combining isotropic damage with plasticity. We are inspired by It has been inspired by a work by Freddi and Royer-Carfagni, including the case where the inelastic part of the strain only evolves in regions where the material is damaged. The evolution both of the damage and of the plastic variable is assumed to be rate-independent. Existence of solutions is established in the abstract energetic framework elaborated by Mielke and coworkers.

math.AP↗