@article{8792,
abstract = {This paper is concerned with a non-isothermal Cahn-Hilliard model based on a microforce balance. The model was derived by A. Miranville and G. Schimperna starting from the two fundamental laws of Thermodynamics, following M. Gurtin's two-scale approach. The main working assumptions are made on the behaviour of the heat flux as the absolute temperature tends to zero and to infinity. A suitable Ginzburg-Landau free energy is considered. Global-in-time existence for the initial-boundary value problem associated to the entropy formulation and, in a subcase, also to the weak formulation of the model is proved by deriving suitable a priori estimates and by showing weak sequential stability of families of approximating solutions. At last, some highlights are given regarding a possible approximation scheme compatible with the a-priori estimates available for the system.},
author = {Marveggio, Alice and Schimperna, Giulio},
issn = {10902732},
journal = {Journal of Differential Equations},
number = {2},
pages = {924--970},
publisher = {Elsevier},
title = {{On a non-isothermal Cahn-Hilliard model based on a microforce balance}},
doi = {10.1016/j.jde.2020.10.030},
volume = {274},
year = {2021},
}
@article{9307,
abstract = {We establish finite time extinction with probability one for weak solutions of the Cauchy–Dirichlet problem for the 1D stochastic porous medium equation with Stratonovich transport noise and compactly supported smooth initial datum. Heuristically, this is expected to hold because Brownian motion has average spread rate O(t12) whereas the support of solutions to the deterministic PME grows only with rate O(t1m+1). The rigorous proof relies on a contraction principle up to time-dependent shift for Wong–Zakai type approximations, the transformation to a deterministic PME with two copies of a Brownian path as the lateral boundary, and techniques from the theory of viscosity solutions.},
author = {Hensel, Sebastian},
issn = {2194041X},
journal = {Stochastics and Partial Differential Equations: Analysis and Computations},
publisher = {Springer Nature},
title = {{Finite time extinction for the 1D stochastic porous medium equation with transport noise}},
doi = {10.1007/s40072-021-00188-9},
year = {2021},
}
@article{9335,
abstract = {Various degenerate diffusion equations exhibit a waiting time phenomenon: depending on the “flatness” of the compactly supported initial datum at the boundary of the support, the support of the solution may not expand for a certain amount of time. We show that this phenomenon is captured by particular Lagrangian discretizations of the porous medium and the thin film equations, and we obtain sufficient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. For the spatially discrete solution, the waiting time phenomenon refers to a deviation of the edge of support from its original position by a quantity comparable to the mesh width, over a mesh-independent time interval. Our proof is based on estimates on the fluid velocity in Lagrangian coordinates. Combining weighted entropy estimates with an iteration technique à la Stampacchia leads to upper bounds on free boundary propagation. Numerical simulations show that the phenomenon is already clearly visible for relatively coarse discretizations.},
author = {Fischer, Julian L and Matthes, Daniel},
issn = {0036-1429},
journal = {SIAM Journal on Numerical Analysis},
number = {1},
pages = {60--87},
publisher = {Society for Industrial and Applied Mathematics},
title = {{The waiting time phenomenon in spatially discretized porous medium and thin film equations}},
doi = {10.1137/19M1300017},
volume = {59},
year = {2021},
}
@article{9352,
abstract = {This paper provides an a priori error analysis of a localized orthogonal decomposition method for the numerical stochastic homogenization of a model random diffusion problem. If the uniformly elliptic and bounded random coefficient field of the model problem is stationary and satisfies a quantitative decorrelation assumption in the form of the spectral gap inequality, then the expected $L^2$ error of the method can be estimated, up to logarithmic factors, by $H+(\varepsilon/H)^{d/2}$, $\varepsilon$ being the small correlation length of the random coefficient and $H$ the width of the coarse finite element mesh that determines the spatial resolution. The proof bridges recent results of numerical homogenization and quantitative stochastic homogenization.},
author = {Fischer, Julian L and Gallistl, Dietmar and Peterseim, Dietmar},
issn = {0036-1429},
journal = {SIAM Journal on Numerical Analysis},
number = {2},
pages = {660--674},
publisher = {Society for Industrial and Applied Mathematics},
title = {{A priori error analysis of a numerical stochastic homogenization method}},
doi = {10.1137/19M1308992},
volume = {59},
year = {2021},
}
@article{9240,
abstract = {A stochastic PDE, describing mesoscopic fluctuations in systems of weakly interacting inertial particles of finite volume, is proposed and analysed in any finite dimension . It is a regularised and inertial version of the Dean–Kawasaki model. A high-probability well-posedness theory for this model is developed. This theory improves significantly on the spatial scaling restrictions imposed in an earlier work of the same authors, which applied only to significantly larger particles in one dimension. The well-posedness theory now applies in d-dimensions when the particle-width ϵ is proportional to for and N is the number of particles. This scaling is optimal in a certain Sobolev norm. Key tools of the analysis are fractional Sobolev spaces, sharp bounds on Bessel functions, separability of the regularisation in the d-spatial dimensions, and use of the Faà di Bruno's formula.},
author = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes},
issn = {1090-2732},
journal = {Journal of Differential Equations},
number = {5},
pages = {253--283},
publisher = {Elsevier},
title = {{Well-posedness for a regularised inertial Dean–Kawasaki model for slender particles in several space dimensions}},
doi = {10.1016/j.jde.2021.02.048},
volume = {284},
year = {2021},
}
@unpublished{10011,
abstract = {We propose a new weak solution concept for (two-phase) mean curvature flow which enjoys both (unconditional) existence and (weak-strong) uniqueness properties. These solutions are evolving varifolds, just as in Brakke's formulation, but are coupled to the phase volumes by a simple transport equation. First, we show that, in the exact same setup as in Ilmanen's proof [J. Differential Geom. 38, 417-461, (1993)], any limit point of solutions to the Allen-Cahn equation is a varifold solution in our sense. Second, we prove that any calibrated flow in the sense of Fischer et al. [arXiv:2003.05478] - and hence any classical solution to mean curvature flow - is unique in the class of our new varifold solutions. This is in sharp contrast to the case of Brakke flows, which a priori may disappear at any given time and are therefore fatally non-unique. Finally, we propose an extension of the solution concept to the multi-phase case which is at least guaranteed to satisfy a weak-strong uniqueness principle.},
author = {Hensel, Sebastian and Laux, Tim},
booktitle = {arXiv},
keywords = {Mean curvature flow, gradient flows, varifolds, weak solutions, weak-strong uniqueness, calibrated geometry, gradient-flow calibrations},
title = {{A new varifold solution concept for mean curvature flow: Convergence of the Allen-Cahn equation and weak-strong uniqueness}},
year = {2021},
}
@article{10005,
abstract = {We study systems of nonlinear partial differential equations of parabolic type, in which the elliptic operator is replaced by the first-order divergence operator acting on a flux function, which is related to the spatial gradient of the unknown through an additional implicit equation. This setting, broad enough in terms of applications, significantly expands the paradigm of nonlinear parabolic problems. Formulating four conditions concerning the form of the implicit equation, we first show that these conditions describe a maximal monotone p-coercive graph. We then establish the global-in-time and large-data existence of a (weak) solution and its uniqueness. To this end, we adopt and significantly generalize Minty’s method of monotone mappings. A unified theory, containing several novel tools, is developed in a way to be tractable from the point of view of numerical approximations.},
author = {Bulíček, Miroslav and Maringová, Erika and Málek, Josef},
issn = {1793-6314},
journal = {Mathematical Models and Methods in Applied Sciences},
keywords = {Nonlinear parabolic systems, implicit constitutive theory, weak solutions, existence, uniqueness},
number = {09},
publisher = {World Scientific},
title = {{On nonlinear problems of parabolic type with implicit constitutive equations involving flux}},
doi = {10.1142/S0218202521500457},
volume = {31},
year = {2021},
}
@phdthesis{10007,
abstract = {The present thesis is concerned with the derivation of weak-strong uniqueness principles for curvature driven interface evolution problems not satisfying a comparison principle. The specific examples being treated are two-phase Navier-Stokes flow with surface tension, modeling the evolution of two incompressible, viscous and immiscible fluids separated by a sharp interface, and multiphase mean curvature flow, which serves as an idealized model for the motion of grain boundaries in an annealing polycrystalline material. Our main results - obtained in joint works with Julian Fischer, Tim Laux and Theresa M. Simon - state that prior to the formation of geometric singularities due to topology changes, the weak solution concept of Abels (Interfaces Free Bound. 9, 2007) to two-phase Navier-Stokes flow with surface tension and the weak solution concept of Laux and Otto (Calc. Var. Partial Differential Equations 55, 2016) to multiphase mean curvature flow (for networks in R^2 or double bubbles in R^3) represents the unique solution to these interface evolution problems within the class of classical solutions, respectively. To the best of the author's knowledge, for interface evolution problems not admitting a geometric comparison principle the derivation of a weak-strong uniqueness principle represented an open problem, so that the works contained in the present thesis constitute the first positive results in this direction. The key ingredient of our approach consists of the introduction of a novel concept of relative entropies for a class of curvature driven interface evolution problems, for which the associated energy contains an interfacial contribution being proportional to the surface area of the evolving (network of) interface(s). The interfacial part of the relative entropy gives sufficient control on the interface error between a weak and a classical solution, and its time evolution can be computed, at least in principle, for any energy dissipating weak solution concept. A resulting stability estimate for the relative entropy essentially entails the above mentioned weak-strong uniqueness principles. The present thesis contains a detailed introduction to our relative entropy approach, which in particular highlights potential applications to other problems in curvature driven interface evolution not treated in this thesis.},
author = {Hensel, Sebastian},
issn = {2663-337X},
pages = {300},
publisher = {IST Austria},
title = {{Curvature driven interface evolution: Uniqueness properties of weak solution concepts}},
doi = {10.15479/at:ista:10007},
year = {2021},
}
@unpublished{10013,
abstract = {We derive a weak-strong uniqueness principle for BV solutions to multiphase mean curvature flow of triple line clusters in three dimensions. Our proof is based on the explicit construction of a gradient-flow calibration in the sense of the recent work of Fischer et al. [arXiv:2003.05478] for any such cluster. This extends the two-dimensional construction to the three-dimensional case of surfaces meeting along triple junctions.},
author = {Hensel, Sebastian and Laux, Tim},
booktitle = {arXiv},
title = {{Weak-strong uniqueness for the mean curvature flow of double bubbles}},
year = {2021},
}
@article{8697,
abstract = {In the computation of the material properties of random alloys, the method of 'special quasirandom structures' attempts to approximate the properties of the alloy on a finite volume with higher accuracy by replicating certain statistics of the random atomic lattice in the finite volume as accurately as possible. In the present work, we provide a rigorous justification for a variant of this method in the framework of the Thomas–Fermi–von Weizsäcker (TFW) model. Our approach is based on a recent analysis of a related variance reduction method in stochastic homogenization of linear elliptic PDEs and the locality properties of the TFW model. Concerning the latter, we extend an exponential locality result by Nazar and Ortner to include point charges, a result that may be of independent interest.},
author = {Fischer, Julian L and Kniely, Michael},
issn = {13616544},
journal = {Nonlinearity},
number = {11},
pages = {5733--5772},
publisher = {IOP Publishing},
title = {{Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model}},
doi = {10.1088/1361-6544/ab9728},
volume = {33},
year = {2020},
}
@article{7637,
abstract = {The evolution of finitely many particles obeying Langevin dynamics is described by Dean–Kawasaki equations, a class of stochastic equations featuring a non-Lipschitz multiplicative noise in divergence form. We derive a regularised Dean–Kawasaki model based on second order Langevin dynamics by analysing a system of particles interacting via a pairwise potential. Key tools of our analysis are the propagation of chaos and Simon's compactness criterion. The model we obtain is a small-noise stochastic perturbation of the undamped McKean–Vlasov equation. We also provide a high-probability result for existence and uniqueness for our model.},
author = {Cornalba, Federico and Shardlow, Tony and Zimmer, Johannes},
issn = {13616544},
journal = {Nonlinearity},
number = {2},
pages = {864--891},
publisher = {IOP Publishing},
title = {{From weakly interacting particles to a regularised Dean-Kawasaki model}},
doi = {10.1088/1361-6544/ab5174},
volume = {33},
year = {2020},
}
@article{7866,
abstract = {In this paper, we establish convergence to equilibrium for a drift–diffusion–recombination system modelling the charge transport within certain semiconductor devices. More precisely, we consider a two-level system for electrons and holes which is augmented by an intermediate energy level for electrons in so-called trapped states. The recombination dynamics use the mass action principle by taking into account this additional trap level. The main part of the paper is concerned with the derivation of an entropy–entropy production inequality, which entails exponential convergence to the equilibrium via the so-called entropy method. The novelty of our approach lies in the fact that the entropy method is applied uniformly in a fast-reaction parameter which governs the lifetime of electrons on the trap level. Thus, the resulting decay estimate for the densities of electrons and holes extends to the corresponding quasi-steady-state approximation.},
author = {Fellner, Klemens and Kniely, Michael},
issn = {22969039},
journal = {Journal of Elliptic and Parabolic Equations},
pages = {529--598},
publisher = {Springer Nature},
title = {{Uniform convergence to equilibrium for a family of drift–diffusion models with trap-assisted recombination and the limiting Shockley–Read–Hall model}},
doi = {10.1007/s41808-020-00068-8},
volume = {6},
year = {2020},
}
@article{9039,
abstract = {We give a short and self-contained proof for rates of convergence of the Allen--Cahn equation towards mean curvature flow, assuming that a classical (smooth) solution to the latter exists and starting from well-prepared initial data. Our approach is based on a relative entropy technique. In particular, it does not require a stability analysis for the linearized Allen--Cahn operator. As our analysis also does not rely on the comparison principle, we expect it to be applicable to more complex equations and systems.},
author = {Fischer, Julian L and Laux, Tim and Simon, Theresa M.},
issn = {10957154},
journal = {SIAM Journal on Mathematical Analysis},
number = {6},
pages = {6222--6233},
publisher = {Society for Industrial and Applied Mathematics},
title = {{Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies}},
doi = {10.1137/20M1322182},
volume = {52},
year = {2020},
}
@article{9196,
abstract = {In order to provide a local description of a regular function in a small neighbourhood of a point x, it is sufficient by Taylor’s theorem to know the value of the function as well as all of its derivatives up to the required order at the point x itself. In other words, one could say that a regular function is locally modelled by the set of polynomials. The theory of regularity structures due to Hairer generalizes this observation and provides an abstract setup, which in the application to singular SPDE extends the set of polynomials by functionals constructed from, e.g., white noise. In this context, the notion of Taylor polynomials is lifted to the notion of so-called modelled distributions. The celebrated reconstruction theorem, which in turn was inspired by Gubinelli’s \textit {sewing lemma}, is of paramount importance for the theory. It enables one to reconstruct a modelled distribution as a true distribution on Rd which is locally approximated by this extended set of models or “monomials”. In the original work of Hairer, the error is measured by means of Hölder norms. This was then generalized to the whole scale of Besov spaces by Hairer and Labbé. It is the aim of this work to adapt the analytic part of the theory of regularity structures to the scale of Triebel–Lizorkin spaces.},
author = {Hensel, Sebastian and Rosati, Tommaso},
issn = {0039-3223},
journal = {Studia Mathematica},
keywords = {General Mathematics},
number = {3},
pages = {251--297},
publisher = {Institute of Mathematics, Polish Academy of Sciences},
title = {{Modelled distributions of Triebel–Lizorkin type}},
doi = {10.4064/sm180411-11-2},
volume = {252},
year = {2020},
}
@article{7489,
abstract = {In the present work, we consider the evolution of two fluids separated by a sharp interface in the presence of surface tension—like, for example, the evolution of oil bubbles in water. Our main result is a weak–strong uniqueness principle for the corresponding free boundary problem for the incompressible Navier–Stokes equation: as long as a strong solution exists, any varifold solution must coincide with it. In particular, in the absence of physical singularities, the concept of varifold solutions—whose global in time existence has been shown by Abels (Interfaces Free Bound 9(1):31–65, 2007) for general initial data—does not introduce a mechanism for non-uniqueness. The key ingredient of our approach is the construction of a relative entropy functional capable of controlling the interface error. If the viscosities of the two fluids do not coincide, even for classical (strong) solutions the gradient of the velocity field becomes discontinuous at the interface, introducing the need for a careful additional adaption of the relative entropy.},
author = {Fischer, Julian L and Hensel, Sebastian},
issn = {14320673},
journal = {Archive for Rational Mechanics and Analysis},
pages = {967--1087},
publisher = {Springer Nature},
title = {{Weak–strong uniqueness for the Navier–Stokes equation for two fluids with surface tension}},
doi = {10.1007/s00205-019-01486-2},
volume = {236},
year = {2020},
}
@unpublished{10012,
abstract = {We prove that in the absence of topological changes, the notion of BV solutions to planar multiphase mean curvature flow does not allow for a mechanism for (unphysical) non-uniqueness. Our approach is based on the local structure of the energy landscape near a classical evolution by mean curvature. Mean curvature flow being the gradient flow of the surface energy functional, we develop a gradient-flow analogue of the notion of calibrations. Just like the existence of a calibration guarantees that one has reached a global minimum in the energy landscape, the existence of a "gradient flow calibration" ensures that the route of steepest descent in the energy landscape is unique and stable.},
author = {Fischer, Julian L and Hensel, Sebastian and Laux, Tim and Simon, Thilo},
booktitle = {arXiv},
title = {{The local structure of the energy landscape in multiphase mean curvature flow: weak-strong uniqueness and stability of evolutions}},
year = {2020},
}
@article{6617,
abstract = {The effective large-scale properties of materials with random heterogeneities on a small scale are typically determined by the method of representative volumes: a sample of the random material is chosen—the representative volume—and its effective properties are computed by the cell formula. Intuitively, for a fixed sample size it should be possible to increase the accuracy of the method by choosing a material sample which captures the statistical properties of the material particularly well; for example, for a composite material consisting of two constituents, one would select a representative volume in which the volume fraction of the constituents matches closely with their volume fraction in the overall material. Inspired by similar attempts in materials science, Le Bris, Legoll and Minvielle have designed a selection approach for representative volumes which performs remarkably well in numerical examples of linear materials with moderate contrast. In the present work, we provide a rigorous analysis of this selection approach for representative volumes in the context of stochastic homogenization of linear elliptic equations. In particular, we prove that the method essentially never performs worse than a random selection of the material sample and may perform much better if the selection criterion for the material samples is chosen suitably.},
author = {Fischer, Julian L},
issn = {1432-0673},
journal = {Archive for Rational Mechanics and Analysis},
number = {2},
pages = {635–726},
publisher = {Springer},
title = {{The choice of representative volumes in the approximation of effective properties of random materials}},
doi = {10.1007/s00205-019-01400-w},
volume = {234},
year = {2019},
}
@article{6762,
abstract = {We present and study novel optimal control problems motivated by the search for photovoltaic materials with high power-conversion efficiency. The material must perform the first step: convert light (photons) into electronic excitations. We formulate various desirable properties of the excitations as mathematical control goals at the Kohn-Sham-DFT level
of theory, with the control being given by the nuclear charge distribution. We prove that nuclear distributions exist which give rise to optimal HOMO-LUMO excitations, and present illustrative numerical simulations for 1D finite nanocrystals. We observe pronounced goal-dependent features such as large electron-hole separation, and a hierarchy of length scales: internal HOMO and LUMO wavelengths < atomic spacings < (irregular) fluctuations of the doping profiles < system size.},
author = {Friesecke, Gero and Kniely, Michael},
issn = {15403467},
journal = {Multiscale Modeling and Simulation},
number = {3},
pages = {926--947},
publisher = {SIAM},
title = {{New optimal control problems in density functional theory motivated by photovoltaics}},
doi = {10.1137/18M1207272},
volume = {17},
year = {2019},
}
@article{151,
abstract = {We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function f∈Lp with p>1. More precisely, for any 1<q<(p+1)/2 we construct W1,q-bi-Sobolev maps with identity boundary conditions; for f∈L∞, we provide bi-Lipschitz maps. The basic building block of our construction are bi-Lipschitz maps which stretch a given compact subset of the unit square by a given factor while preserving the boundary. The construction of these stretching maps relies on a slight strengthening of the celebrated covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets. We apply our result to a model functional in nonlinear elasticity, the integrand of which features fast blowup as the Jacobian determinant of the deformation becomes small. For such functionals, the derivation of the equilibrium equations for minimizers requires an additional regularization of test functions, which our maps provide.},
author = {Fischer, Julian L and Kneuss, Olivier},
journal = {Journal of Differential Equations},
number = {1},
pages = {257 -- 311},
publisher = {Elsevier},
title = {{Bi-Sobolev solutions to the prescribed Jacobian inequality in the plane with L p data and applications to nonlinear elasticity}},
doi = {10.1016/j.jde.2018.07.045},
volume = {266},
year = {2019},
}
@article{606,
abstract = {We establish the existence of a global solution for a new family of fluid-like equations, which are obtained in certain regimes in as the mean-field evolution of the supercurrent density in a (2D section of a) type-II superconductor with pinning and with imposed electric current. We also consider general vortex-sheet initial data, and investigate the uniqueness and regularity properties of the solution. For some choice of parameters, the equation under investigation coincides with the so-called lake equation from 2D shallow water fluid dynamics, and our analysis then leads to a new existence result for rough initial data.},
author = {Duerinckx, Mitia and Fischer, Julian L},
journal = {Annales de l'Institut Henri Poincare (C) Non Linear Analysis},
number = {5},
pages = {1267--1319},
publisher = {Elsevier},
title = {{Well-posedness for mean-field evolutions arising in superconductivity}},
doi = {10.1016/j.anihpc.2017.11.004},
volume = {35},
year = {2018},
}