@article{14797, abstract = {We study a random matching problem on closed compact 2 -dimensional Riemannian manifolds (with respect to the squared Riemannian distance), with samples of random points whose common law is absolutely continuous with respect to the volume measure with strictly positive and bounded density. We show that given two sequences of numbers n and m=m(n) of points, asymptotically equivalent as n goes to infinity, the optimal transport plan between the two empirical measures μn and νm is quantitatively well-approximated by (Id,exp(∇hn))#μn where hn solves a linear elliptic PDE obtained by a regularized first-order linearization of the Monge-Ampère equation. This is obtained in the case of samples of correlated random points for which a stretched exponential decay of the α -mixing coefficient holds and for a class of discrete-time Markov chains having a unique absolutely continuous invariant measure with respect to the volume measure.}, author = {Clozeau, Nicolas and Mattesini, Francesco}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{Annealed quantitative estimates for the quadratic 2D-discrete random matching problem}}, doi = {10.1007/s00440-023-01254-0}, year = {2024}, } @article{14884, abstract = {We perform a stochastic homogenization analysis for composite materials exhibiting a random microstructure. Under the assumptions of stationarity and ergodicity, we characterize the Gamma-limit of a micromagnetic energy functional defined on magnetizations taking value in the unit sphere and including both symmetric and antisymmetric exchange contributions. This Gamma-limit corresponds to a micromagnetic energy functional with homogeneous coefficients. We provide explicit formulas for the effective magnetic properties of the composite material in terms of homogenization correctors. Additionally, the variational analysis of the two exchange energy terms is performed in the more general setting of functionals defined on manifold-valued maps with Sobolev regularity, in the case in which the target manifold is a bounded, orientable smooth surface with tubular neighborhood of uniform thickness. Eventually, we present an explicit characterization of minimizers of the effective exchange in the case of magnetic multilayers, providing quantitative evidence of Dzyaloshinskii’s predictions on the emergence of helical structures in composite ferromagnetic materials with stochastic microstructure.}, author = {Davoli, Elisa and D’Elia, Lorenza and Ingmanns, Jonas}, issn = {1432-1467}, journal = {Journal of Nonlinear Science}, number = {2}, publisher = {Springer Nature}, title = {{Stochastic homogenization of micromagnetic energies and emergence of magnetic skyrmions}}, doi = {10.1007/s00332-023-10005-3}, volume = {34}, year = {2024}, } @article{12485, abstract = {In this paper we introduce the critical variational setting for parabolic stochastic evolution equations of quasi- or semi-linear type. Our results improve many of the abstract results in the classical variational setting. In particular, we are able to replace the usual weak or local monotonicity condition by a more flexible local Lipschitz condition. Moreover, the usual growth conditions on the multiplicative noise are weakened considerably. Our new setting provides general conditions under which local and global existence and uniqueness hold. Moreover, we prove continuous dependence on the initial data. We show that many classical SPDEs, which could not be covered by the classical variational setting, do fit in the critical variational setting. In particular, this is the case for the Cahn-Hilliard equations, tamed Navier-Stokes equations, and Allen-Cahn equation.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1432-2064}, journal = {Probability Theory and Related Fields}, publisher = {Springer Nature}, title = {{The critical variational setting for stochastic evolution equations}}, doi = {10.1007/s00440-023-01249-x}, year = {2024}, } @article{15098, abstract = {The paper is devoted to the analysis of the global well-posedness and the interior regularity of the 2D Navier–Stokes equations with inhomogeneous stochastic boundary conditions. The noise, white in time and coloured in space, can be interpreted as the physical law describing the driving mechanism on the atmosphere–ocean interface, i.e. as a balance of the shear stress of the ocean and the horizontal wind force.}, author = {Agresti, Antonio and Luongo, Eliseo}, issn = {1432-1807}, journal = {Mathematische Annalen}, publisher = {Springer Nature}, title = {{Global well-posedness and interior regularity of 2D Navier-Stokes equations with stochastic boundary conditions}}, doi = {10.1007/s00208-024-02812-0}, year = {2024}, } @article{15119, abstract = {In this paper we consider an SPDE where the leading term is a second order operator with periodic boundary conditions, coefficients which are measurable in (t,ω) , and Hölder continuous in space. Assuming stochastic parabolicity conditions, we prove Lp((0,T)×Ω,tκdt;Hσ,q(Td)) -estimates. The main novelty is that we do not require p=q . Moreover, we allow arbitrary σ∈R and weights in time. Such mixed regularity estimates play a crucial role in applications to nonlinear SPDEs which is clear from our previous work. To prove our main results we develop a general perturbation theory for SPDEs. Moreover, we prove a new result on pointwise multiplication in spaces with fractional smoothness.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {0246-0203}, journal = {Annales de l'institut Henri Poincare Probability and Statistics}, number = {1}, pages = {413--430}, publisher = {Institute of Mathematical Statistics}, title = {{Stochastic maximal Lp(Lq)-regularity for second order systems with periodic boundary conditions}}, doi = {10.1214/22-AIHP1333}, volume = {60}, year = {2024}, } @article{10550, abstract = {The global existence of renormalised solutions and convergence to equilibrium for reaction-diffusion systems with non-linear diffusion are investigated. The system is assumed to have quasi-positive non-linearities and to satisfy an entropy inequality. The difficulties in establishing global renormalised solutions caused by possibly degenerate diffusion are overcome by introducing a new class of weighted truncation functions. By means of the obtained global renormalised solutions, we study the large-time behaviour of complex balanced systems arising from chemical reaction network theory with non-linear diffusion. When the reaction network does not admit boundary equilibria, the complex balanced equilibrium is shown, by using the entropy method, to exponentially attract all renormalised solutions in the same compatibility class. This convergence extends even to a range of non-linear diffusion, where global existence is an open problem, yet we are able to show that solutions to approximate systems converge exponentially to equilibrium uniformly in the regularisation parameter.}, author = {Fellner, Klemens and Fischer, Julian L and Kniely, Michael and Tang, Bao Quoc}, issn = {1432-1467}, journal = {Journal of Nonlinear Science}, publisher = {Springer Nature}, title = {{Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion}}, doi = {10.1007/s00332-023-09926-w}, volume = {33}, year = {2023}, } @article{13043, 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. (2020) 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}, issn = {1463-9971}, journal = {Interfaces and Free Boundaries}, number = {1}, pages = {37--107}, publisher = {EMS Press}, title = {{Weak-strong uniqueness for the mean curvature flow of double bubbles}}, doi = {10.4171/IFB/484}, volume = {25}, year = {2023}, } @article{13129, abstract = {We study the representative volume element (RVE) method, which is a method to approximately infer the effective behavior ahom of a stationary random medium. The latter is described by a coefficient field a(x) generated from a given ensemble ⟨⋅⟩ and the corresponding linear elliptic operator −∇⋅a∇. In line with the theory of homogenization, the method proceeds by computing d=3 correctors (d denoting the space dimension). To be numerically tractable, this computation has to be done on a finite domain: the so-called representative volume element, i.e., a large box with, say, periodic boundary conditions. The main message of this article is: Periodize the ensemble instead of its realizations. By this, we mean that it is better to sample from a suitably periodized ensemble than to periodically extend the restriction of a realization a(x) from the whole-space ensemble ⟨⋅⟩. We make this point by investigating the bias (or systematic error), i.e., the difference between ahom and the expected value of the RVE method, in terms of its scaling w.r.t. the lateral size L of the box. In case of periodizing a(x), we heuristically argue that this error is generically O(L−1). In case of a suitable periodization of ⟨⋅⟩ , we rigorously show that it is O(L−d). In fact, we give a characterization of the leading-order error term for both strategies and argue that even in the isotropic case it is generically non-degenerate. We carry out the rigorous analysis in the convenient setting of ensembles ⟨⋅⟩ of Gaussian type, which allow for a straightforward periodization, passing via the (integrable) covariance function. This setting has also the advantage of making the Price theorem and the Malliavin calculus available for optimal stochastic estimates of correctors. We actually need control of second-order correctors to capture the leading-order error term. This is due to inversion symmetry when applying the two-scale expansion to the Green function. As a bonus, we present a stream-lined strategy to estimate the error in a higher-order two-scale expansion of the Green function.}, author = {Clozeau, Nicolas and Josien, Marc and Otto, Felix and Xu, Qiang}, issn = {1615-3383}, journal = {Foundations of Computational Mathematics}, publisher = {Springer Nature}, title = {{Bias in the representative volume element method: Periodize the ensemble instead of its realizations}}, doi = {10.1007/s10208-023-09613-y}, year = {2023}, } @article{10173, abstract = {We study the large scale behavior of elliptic systems with stationary random coefficient that have only slowly decaying correlations. To this aim we analyze the so-called corrector equation, a degenerate elliptic equation posed in the probability space. In this contribution, we use a parabolic approach and optimally quantify the time decay of the semigroup. For the theoretical point of view, we prove an optimal decay estimate of the gradient and flux of the corrector when spatially averaged over a scale R larger than 1. For the numerical point of view, our results provide convenient tools for the analysis of various numerical methods.}, author = {Clozeau, Nicolas}, issn = {2194-0401}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, pages = {1254–1378}, publisher = {Springer Nature}, title = {{Optimal decay of the parabolic semigroup in stochastic homogenization for correlated coefficient fields}}, doi = {10.1007/s40072-022-00254-w}, volume = {11}, year = {2023}, } @article{12429, abstract = {In this paper, we consider traces at initial times for functions with mixed time-space smoothness. Such results are often needed in the theory of evolution equations. Our result extends and unifies many previous results. Our main improvement is that we can allow general interpolation couples. The abstract results are applied to regularity problems for fractional evolution equations and stochastic evolution equations, where uniform trace estimates on the half-line are shown.}, author = {Agresti, Antonio and Lindemulder, Nick and Veraar, Mark}, issn = {1522-2616}, journal = {Mathematische Nachrichten}, number = {4}, pages = {1319--1350}, publisher = {Wiley}, title = {{On the trace embedding and its applications to evolution equations}}, doi = {10.1002/mana.202100192}, volume = {296}, year = {2023}, } @article{14451, abstract = {We investigate the potential of Multi-Objective, Deep Reinforcement Learning for stock and cryptocurrency single-asset trading: in particular, we consider a Multi-Objective algorithm which generalizes the reward functions and discount factor (i.e., these components are not specified a priori, but incorporated in the learning process). Firstly, using several important assets (BTCUSD, ETHUSDT, XRPUSDT, AAPL, SPY, NIFTY50), we verify the reward generalization property of the proposed Multi-Objective algorithm, and provide preliminary statistical evidence showing increased predictive stability over the corresponding Single-Objective strategy. Secondly, we show that the Multi-Objective algorithm has a clear edge over the corresponding Single-Objective strategy when the reward mechanism is sparse (i.e., when non-null feedback is infrequent over time). Finally, we discuss the generalization properties with respect to the discount factor. The entirety of our code is provided in open-source format.}, author = {Cornalba, Federico and Disselkamp, Constantin and Scassola, Davide and Helf, Christopher}, issn = {1433-3058}, journal = {Neural Computing and Applications}, publisher = {Springer Nature}, title = {{Multi-objective reward generalization: improving performance of Deep Reinforcement Learning for applications in single-asset trading}}, doi = {10.1007/s00521-023-09033-7}, year = {2023}, } @article{14554, abstract = {The Regularised Inertial Dean–Kawasaki model (RIDK) – introduced by the authors and J. Zimmer in earlier works – is a nonlinear stochastic PDE capturing fluctuations around the meanfield limit for large-scale particle systems in both particle density and momentum density. We focus on the following two aspects. Firstly, we set up a Discontinuous Galerkin (DG) discretisation scheme for the RIDK model: we provide suitable definitions of numerical fluxes at the interface of the mesh elements which are consistent with the wave-type nature of the RIDK model and grant stability of the simulations, and we quantify the rate of convergence in mean square to the continuous RIDK model. Secondly, we introduce modifications of the RIDK model in order to preserve positivity of the density (such a feature only holds in a “high-probability sense” for the original RIDK model). By means of numerical simulations, we show that the modifications lead to physically realistic and positive density profiles. In one case, subject to additional regularity constraints, we also prove positivity. Finally, we present an application of our methodology to a system of diffusing and reacting particles. Our Python code is available in open-source format.}, author = {Cornalba, Federico and Shardlow, Tony}, issn = {2804-7214}, journal = {ESAIM: Mathematical Modelling and Numerical Analysis}, number = {5}, pages = {3061--3090}, publisher = {EDP Sciences}, title = {{The regularised inertial Dean' Kawasaki equation: Discontinuous Galerkin approximation and modelling for low-density regime}}, doi = {10.1051/m2an/2023077}, volume = {57}, year = {2023}, } @article{14042, abstract = {Long-time and large-data existence of weak solutions for initial- and boundary-value problems concerning three-dimensional flows of incompressible fluids is nowadays available not only for Navier–Stokes fluids but also for various fluid models where the relation between the Cauchy stress tensor and the symmetric part of the velocity gradient is nonlinear. The majority of such studies however concerns models where such a dependence is explicit (the stress is a function of the velocity gradient), which makes the class of studied models unduly restrictive. The same concerns boundary conditions, or more precisely the slipping mechanisms on the boundary, where the no-slip is still the most preferred condition considered in the literature. Our main objective is to develop a robust mathematical theory for unsteady internal flows of implicitly constituted incompressible fluids with implicit relations between the tangential projections of the velocity and the normal traction on the boundary. The theory covers numerous rheological models used in chemistry, biorheology, polymer and food industry as well as in geomechanics. It also includes, as special cases, nonlinear slip as well as stick–slip boundary conditions. Unlike earlier studies, the conditions characterizing admissible classes of constitutive equations are expressed by means of tools of elementary calculus. In addition, a fully constructive proof (approximation scheme) is incorporated. Finally, we focus on the question of uniqueness of such weak solutions.}, author = {Bulíček, Miroslav and Málek, Josef and Maringová, Erika}, issn = {1422-6952}, journal = {Journal of Mathematical Fluid Mechanics}, number = {3}, publisher = {Springer Nature}, title = {{On unsteady internal flows of incompressible fluids characterized by implicit constitutive equations in the bulk and on the boundary}}, doi = {10.1007/s00021-023-00803-w}, volume = {25}, year = {2023}, } @article{12486, abstract = {This paper is concerned with the problem of regularization by noise of systems of reaction–diffusion equations with mass control. It is known that strong solutions to such systems of PDEs may blow-up in finite time. Moreover, for many systems of practical interest, establishing whether the blow-up occurs or not is an open question. Here we prove that a suitable multiplicative noise of transport type has a regularizing effect. More precisely, for both a sufficiently noise intensity and a high spectrum, the blow-up of strong solutions is delayed up to an arbitrary large time. Global existence is shown for the case of exponentially decreasing mass. The proofs combine and extend recent developments in regularization by noise and in the Lp(Lq)-approach to stochastic PDEs, highlighting new connections between the two areas.}, author = {Agresti, Antonio}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, publisher = {Springer Nature}, title = {{Delayed blow-up and enhanced diffusion by transport noise for systems of reaction-diffusion equations}}, doi = {10.1007/s40072-023-00319-4}, year = {2023}, } @article{14755, abstract = {We consider the sharp interface limit for the scalar-valued and vector-valued Allen–Cahn equation with homogeneous Neumann boundary condition in a bounded smooth domain Ω of arbitrary dimension N ⩾ 2 in the situation when a two-phase diffuse interface has developed and intersects the boundary ∂ Ω. The limit problem is mean curvature flow with 90°-contact angle and we show convergence in strong norms for well-prepared initial data as long as a smooth solution to the limit problem exists. To this end we assume that the limit problem has a smooth solution on [ 0 , T ] for some time T > 0. Based on the latter we construct suitable curvilinear coordinates and set up an asymptotic expansion for the scalar-valued and the vector-valued Allen–Cahn equation. In order to estimate the difference of the exact and approximate solutions with a Gronwall-type argument, a spectral estimate for the linearized Allen–Cahn operator in both cases is required. The latter will be shown in a separate paper, cf. (Moser (2021)).}, author = {Moser, Maximilian}, issn = {1875-8576}, journal = {Asymptotic Analysis}, keywords = {General Mathematics}, number = {3-4}, pages = {297--383}, publisher = {IOS Press}, title = {{Convergence of the scalar- and vector-valued Allen–Cahn equation to mean curvature flow with 90°-contact angle in higher dimensions, part I: Convergence result}}, doi = {10.3233/asy-221775}, volume = {131}, year = {2023}, } @article{14661, abstract = {This paper is concerned with equilibrium configurations of one-dimensional particle systems with non-convex nearest-neighbour and next-to-nearest-neighbour interactions and its passage to the continuum. The goal is to derive compactness results for a Γ-development of the energy with the novelty that external forces are allowed. In particular, the forces may depend on Lagrangian or Eulerian coordinates and thus may model dead as well as live loads. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order Γ-limit in the presence of external forces: instead of comparing a configuration of n atoms to a global minimizer of the Γ-limit, we compare the configuration to a minimizer in some subclass of functions which in some sense are "close to" the configuration. The paper is complemented with the study of the minimizers of the Γ-limit.}, author = {Carioni, Marcello and Fischer, Julian L and Schlömerkemper, Anja}, issn = {2363-6394}, journal = {Journal of Convex Analysis}, number = {1}, pages = {217--247}, publisher = {Heldermann Verlag}, title = {{External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a Γ-development}}, volume = {30}, year = {2023}, } @article{13135, abstract = {In this paper we consider a class of stochastic reaction-diffusion equations. We provide local well-posedness, regularity, blow-up criteria and positivity of solutions. The key novelties of this work are related to the use transport noise, critical spaces and the proof of higher order regularity of solutions – even in case of non-smooth initial data. Crucial tools are Lp(Lp)-theory, maximal regularity estimates and sharp blow-up criteria. We view the results of this paper as a general toolbox for establishing global well-posedness for a large class of reaction-diffusion systems of practical interest, of which many are completely open. In our follow-up work [8], the results of this paper are applied in the specific cases of the Lotka-Volterra equations and the Brusselator model.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1090-2732}, journal = {Journal of Differential Equations}, number = {9}, pages = {247--300}, publisher = {Elsevier}, title = {{Reaction-diffusion equations with transport noise and critical superlinear diffusion: Local well-posedness and positivity}}, doi = {10.1016/j.jde.2023.05.038}, volume = {368}, year = {2023}, } @article{10551, abstract = {The Dean–Kawasaki equation—a strongly singular SPDE—is a basic equation of fluctuating hydrodynamics; it has been proposed in the physics literature to describe the fluctuations of the density of N independent diffusing particles in the regime of large particle numbers N≫1. The singular nature of the Dean–Kawasaki equation presents a substantial challenge for both its analysis and its rigorous mathematical justification. Besides being non-renormalisable by the theory of regularity structures by Hairer et al., it has recently been shown to not even admit nontrivial martingale solutions. In the present work, we give a rigorous and fully quantitative justification of the Dean–Kawasaki equation by considering the natural regularisation provided by standard numerical discretisations: We show that structure-preserving discretisations of the Dean–Kawasaki equation may approximate the density fluctuations of N non-interacting diffusing particles to arbitrary order in N−1 (in suitable weak metrics). In other words, the Dean–Kawasaki equation may be interpreted as a “recipe” for accurate and efficient numerical simulations of the density fluctuations of independent diffusing particles.}, author = {Cornalba, Federico and Fischer, Julian L}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, number = {5}, publisher = {Springer Nature}, title = {{The Dean-Kawasaki equation and the structure of density fluctuations in systems of diffusing particles}}, doi = {10.1007/s00205-023-01903-7}, volume = {247}, year = {2023}, } @phdthesis{14587, abstract = {This thesis concerns the application of variational methods to the study of evolution problems arising in fluid mechanics and in material sciences. The main focus is on weak-strong stability properties of some curvature driven interface evolution problems, such as the two-phase Navier–Stokes flow with surface tension and multiphase mean curvature flow, and on the phase-field approximation of the latter. Furthermore, we discuss a variational approach to the study of a class of doubly nonlinear wave equations. First, we consider the two-phase Navier–Stokes flow with surface tension within a bounded domain. The two fluids are immiscible and separated by a sharp interface, which intersects the boundary of the domain at a constant contact angle of ninety degree. We devise a suitable concept of varifolds solutions for the associated interface evolution problem and we establish a weak-strong uniqueness principle in case of a two dimensional ambient space. In order to focus on the boundary effects and on the singular geometry of the evolving domains, we work for simplicity in the regime of same viscosities for the two fluids. The core of the thesis consists in the rigorous proof of the convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow for a suitable class of multi- well potentials and for well-prepared initial data. We even establish a rate of convergence. Our relative energy approach relies on the concept of gradient-flow calibration for branching singularities in multiphase mean curvature flow and thus enables us to overcome the limitations of other approaches. To the best of the author’s knowledge, our result is the first quantitative and unconditional one available in the literature for the vectorial/multiphase setting. This thesis also contains a first study of weak-strong stability for planar multiphase mean curvature flow beyond the singularity resulting from a topology change. Previous weak-strong results are indeed limited to time horizons before the first topology change of the strong solution. We consider circular topology changes and we prove weak-strong stability for BV solutions to planar multiphase mean curvature flow beyond the associated singular times by dynamically adapting the strong solutions to the weak one by means of a space-time shift. In the context of interface evolution problems, our proofs for the main results of this thesis are based on the relative energy technique, relying on novel suitable notions of relative energy functionals, which in particular measure the interface error. Our statements follow from the resulting stability estimates for the relative energy associated to the problem. At last, we introduce a variational approach to the study of nonlinear evolution problems. This approach hinges on the minimization of a parameter dependent family of convex functionals over entire trajectories, known as Weighted Inertia-Dissipation-Energy (WIDE) functionals. We consider a class of doubly nonlinear wave equations and establish the convergence, up to subsequences, of the associated WIDE minimizers to a solution of the target problem as the parameter goes to zero.}, author = {Marveggio, Alice}, issn = {2663 - 337X}, pages = {228}, publisher = {Institute of Science and Technology Austria}, title = {{Weak-strong stability and phase-field approximation of interface evolution problems in fluid mechanics and in material sciences}}, doi = {10.15479/at:ista:14587}, year = {2023}, } @article{14772, abstract = {Many coupled evolution equations can be described via 2×2-block operator matrices of the form A=[ A B C D ] in a product space X=X1×X2 with possibly unbounded entries. Here, the case of diagonally dominant block operator matrices is considered, that is, the case where the full operator A can be seen as a relatively bounded perturbation of its diagonal part with D(A)=D(A)×D(D) though with possibly large relative bound. For such operators the properties of sectoriality, R-sectoriality and the boundedness of the H∞-calculus are studied, and for these properties perturbation results for possibly large but structured perturbations are derived. Thereby, the time dependent parabolic problem associated with A can be analyzed in maximal Lpt -regularity spaces, and this is applied to a wide range of problems such as different theories for liquid crystals, an artificial Stokes system, strongly damped wave and plate equations, and a Keller-Segel model.}, author = {Agresti, Antonio and Hussein, Amru}, issn = {0022-1236}, journal = {Journal of Functional Analysis}, keywords = {Analysis}, number = {11}, publisher = {Elsevier}, title = {{Maximal Lp-regularity and H∞-calculus for block operator matrices and applications}}, doi = {10.1016/j.jfa.2023.110146}, volume = {285}, year = {2023}, } @article{10548, abstract = {Consider a linear elliptic partial differential equation in divergence form with a random coefficient field. The solution operator displays fluctuations around its expectation. The recently developed pathwise theory of fluctuations in stochastic homogenization reduces the characterization of these fluctuations to those of the so-called standard homogenization commutator. In this contribution, we investigate the scaling limit of this key quantity: starting from a Gaussian-like coefficient field with possibly strong correlations, we establish the convergence of the rescaled commutator to a fractional Gaussian field, depending on the decay of correlations of the coefficient field, and we investigate the (non)degeneracy of the limit. This extends to general dimension $d\ge1$ previous results so far limited to dimension $d=1$, and to the continuum setting with strong correlations recent results in the discrete iid case.}, author = {Duerinckx, Mitia and Fischer, Julian L and Gloria, Antoine}, issn = {1050-5164}, journal = {Annals of applied probability}, number = {2}, pages = {1179--1209}, publisher = {Institute of Mathematical Statistics}, title = {{Scaling limit of the homogenization commutator for Gaussian coefficient fields}}, doi = {10.1214/21-AAP1705}, volume = {32}, year = {2022}, } @article{10547, abstract = {We establish global-in-time existence results for thermodynamically consistent reaction-(cross-)diffusion systems coupled to an equation describing heat transfer. Our main interest is to model species-dependent diffusivities, while at the same time ensuring thermodynamic consistency. A key difficulty of the non-isothermal case lies in the intrinsic presence of cross-diffusion type phenomena like the Soret and the Dufour effect: due to the temperature/energy dependence of the thermodynamic equilibria, a nonvanishing temperature gradient may drive a concentration flux even in a situation with constant concentrations; likewise, a nonvanishing concentration gradient may drive a heat flux even in a case of spatially constant temperature. We use time discretisation and regularisation techniques and derive a priori estimates based on a suitable entropy and the associated entropy production. Renormalised solutions are used in cases where non-integrable diffusion fluxes or reaction terms appear.}, author = {Fischer, Julian L and Hopf, Katharina and Kniely, Michael and Mielke, Alexander}, issn = {0036-1410}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Energy-Reaction-Diffusion Systems, Cross Diffusion, Global-In-Time Existence of Weak/Renormalised Solutions, Entropy Method, Onsager System, Soret/Dufour Effect}, number = {1}, pages = {220--267}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Global existence analysis of energy-reaction-diffusion systems}}, doi = {10.1137/20M1387237}, volume = {54}, year = {2022}, } @article{11701, abstract = {In this paper we develop a new approach to nonlinear stochastic partial differential equations with Gaussian noise. Our aim is to provide an abstract framework which is applicable to a large class of SPDEs and includes many important cases of nonlinear parabolic problems which are of quasi- or semilinear type. This first part is on local existence and well-posedness. A second part in preparation is on blow-up criteria and regularization. Our theory is formulated in an Lp-setting, and because of this we can deal with nonlinearities in a very efficient way. Applications to several concrete problems and their quasilinear variants are given. This includes Burgers' equation, the Allen–Cahn equation, the Cahn–Hilliard equation, reaction–diffusion equations, and the porous media equation. The interplay of the nonlinearities and the critical spaces of initial data leads to new results and insights for these SPDEs. The proofs are based on recent developments in maximal regularity theory for the linearized problem for deterministic and stochastic evolution equations. In particular, our theory can be seen as a stochastic version of the theory of critical spaces due to Prüss–Simonett–Wilke (2018). Sharp weighted time-regularity allow us to deal with rough initial values and obtain instantaneous regularization results. The abstract well-posedness results are obtained by a combination of several sophisticated splitting and truncation arguments.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1361-6544}, journal = {Nonlinearity}, number = {8}, pages = {4100--4210}, publisher = {IOP Publishing}, title = {{Nonlinear parabolic stochastic evolution equations in critical spaces Part I. Stochastic maximal regularity and local existence}}, doi = {10.1088/1361-6544/abd613}, volume = {35}, year = {2022}, } @article{11858, abstract = {This paper is a continuation of Part I of this project, where we developed a new local well-posedness theory for nonlinear stochastic PDEs with Gaussian noise. In the current Part II we consider blow-up criteria and regularization phenomena. As in Part I we can allow nonlinearities with polynomial growth and rough initial values from critical spaces. In the first main result we obtain several new blow-up criteria for quasi- and semilinear stochastic evolution equations. In particular, for semilinear equations we obtain a Serrin type blow-up criterium, which extends a recent result of Prüss–Simonett–Wilke (J Differ Equ 264(3):2028–2074, 2018) to the stochastic setting. Blow-up criteria can be used to prove global well-posedness for SPDEs. As in Part I, maximal regularity techniques and weights in time play a central role in the proofs. Our second contribution is a new method to bootstrap Sobolev and Hölder regularity in time and space, which does not require smoothness of the initial data. The blow-up criteria are at the basis of these new methods. Moreover, in applications the bootstrap results can be combined with our blow-up criteria, to obtain efficient ways to prove global existence. This gives new results even in classical 𝐿2-settings, which we illustrate for a concrete SPDE. In future works in preparation we apply the results of the current paper to obtain global well-posedness results and regularity for several concrete SPDEs. These include stochastic Navier–Stokes equations, reaction– diffusion equations and the Allen–Cahn equation. Our setting allows to put these SPDEs into a more flexible framework, where less restrictions on the nonlinearities are needed, and we are able to treat rough initial values from critical spaces. Moreover, we will obtain higher-order regularity results.}, author = {Agresti, Antonio and Veraar, Mark}, issn = {1424-3202}, journal = {Journal of Evolution Equations}, keywords = {Mathematics (miscellaneous)}, number = {2}, publisher = {Springer Nature}, title = {{Nonlinear parabolic stochastic evolution equations in critical spaces part II}}, doi = {10.1007/s00028-022-00786-7}, volume = {22}, year = {2022}, } @article{12079, abstract = {We extend the recent rigorous convergence result of Abels and Moser (SIAM J Math Anal 54(1):114–172, 2022. https://doi.org/10.1137/21M1424925) concerning convergence rates for solutions of the Allen–Cahn equation with a nonlinear Robin boundary condition towards evolution by mean curvature flow with constant contact angle. More precisely, in the present work we manage to remove the perturbative assumption on the contact angle being close to 90∘. We establish under usual double-well type assumptions on the potential and for a certain class of boundary energy densities the sub-optimal convergence rate of order ε12 for general contact angles α∈(0,π). For a very specific form of the boundary energy density, we even obtain from our methods a sharp convergence rate of order ε; again for general contact angles α∈(0,π). Our proof deviates from the popular strategy based on rigorous asymptotic expansions and stability estimates for the linearized Allen–Cahn operator. Instead, we follow the recent approach by Fischer et al. (SIAM J Math Anal 52(6):6222–6233, 2020. https://doi.org/10.1137/20M1322182), thus relying on a relative entropy technique. We develop a careful adaptation of their approach in order to encode the constant contact angle condition. In fact, we perform this task at the level of the notion of gradient flow calibrations. This concept was recently introduced in the context of weak-strong uniqueness for multiphase mean curvature flow by Fischer et al. (arXiv:2003.05478v2).}, author = {Hensel, Sebastian and Moser, Maximilian}, issn = {1432-0835}, journal = {Calculus of Variations and Partial Differential Equations}, number = {6}, publisher = {Springer Nature}, title = {{Convergence rates for the Allen–Cahn equation with boundary contact energy: The non-perturbative regime}}, doi = {10.1007/s00526-022-02307-3}, volume = {61}, year = {2022}, } @article{12305, abstract = {This paper is concerned with the sharp interface limit for the Allen--Cahn equation with a nonlinear Robin boundary condition in a bounded smooth domain Ω⊂\R2. We assume that a diffuse interface already has developed and that it is in contact with the boundary ∂Ω. The boundary condition is designed in such a way that the limit problem is given by the mean curvature flow with constant α-contact angle. For α close to 90° we prove a local in time convergence result for well-prepared initial data for times when a smooth solution to the limit problem exists. Based on the latter we construct a suitable curvilinear coordinate system and carry out a rigorous asymptotic expansion for the Allen--Cahn equation with the nonlinear Robin boundary condition. Moreover, we show a spectral estimate for the corresponding linearized Allen--Cahn operator and with its aid we derive strong norm estimates for the difference of the exact and approximate solutions using a Gronwall-type argument.}, author = {Abels, Helmut and Moser, Maximilian}, issn = {1095-7154}, journal = {SIAM Journal on Mathematical Analysis}, keywords = {Applied Mathematics, Computational Mathematics, Analysis}, number = {1}, pages = {114--172}, publisher = {Society for Industrial and Applied Mathematics}, title = {{Convergence of the Allen--Cahn equation with a nonlinear Robin boundary condition to mean curvature flow with contact angle close to 90°}}, doi = {10.1137/21m1424925}, volume = {54}, year = {2022}, } @article{12304, abstract = {We establish sharp criteria for the instantaneous propagation of free boundaries in solutions to the thin-film equation. The criteria are formulated in terms of the initial distribution of mass (as opposed to previous almost-optimal results), reflecting the fact that mass is a locally conserved quantity for the thin-film equation. In the regime of weak slippage, our criteria are at the same time necessary and sufficient. The proof of our upper bounds on free boundary propagation is based on a strategy of “propagation of degeneracy” down to arbitrarily small spatial scales: We combine estimates on the local mass and estimates on energies to show that “degeneracy” on a certain space-time cylinder entails “degeneracy” on a spatially smaller space-time cylinder with the same time horizon. The derivation of our lower bounds on free boundary propagation is based on a combination of a monotone quantity and almost optimal estimates established previously by the second author with a new estimate connecting motion of mass to entropy production.}, author = {De Nitti, Nicola and Fischer, Julian L}, issn = {1532-4133}, journal = {Communications in Partial Differential Equations}, keywords = {Applied Mathematics, Analysis}, number = {7}, pages = {1394--1434}, publisher = {Taylor & Francis}, title = {{Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation}}, doi = {10.1080/03605302.2022.2056702}, volume = {47}, year = {2022}, } @article{12178, abstract = {In this paper we consider the stochastic primitive equation for geophysical flows subject to transport noise and turbulent pressure. Admitting very rough noise terms, the global existence and uniqueness of solutions to this stochastic partial differential equation are proven using stochastic maximal L² regularity, the theory of critical spaces for stochastic evolution equations, and global a priori bounds. Compared to other results in this direction, we do not need any smallness assumption on the transport noise which acts directly on the velocity field and we also allow rougher noise terms. The adaptation to Stratonovich type noise and, more generally, to variable viscosity and/or conductivity are discussed as well.}, author = {Agresti, Antonio and Hieber, Matthias and Hussein, Amru and Saal, Martin}, issn = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, keywords = {Applied Mathematics, Modeling and Simulation, Statistics and Probability}, publisher = {Springer Nature}, title = {{The stochastic primitive equations with transport noise and turbulent pressure}}, doi = {10.1007/s40072-022-00277-3}, year = {2022}, } @article{11842, abstract = {We consider the flow of two viscous and incompressible fluids within a bounded domain modeled by means of a two-phase Navier–Stokes system. The two fluids are assumed to be immiscible, meaning that they are separated by an interface. With respect to the motion of the interface, we consider pure transport by the fluid flow. Along the boundary of the domain, a complete slip boundary condition for the fluid velocities and a constant ninety degree contact angle condition for the interface are assumed. In the present work, we devise for the resulting evolution problem a suitable weak solution concept based on the framework of varifolds and establish as the main result a weak-strong uniqueness principle in 2D. The proof is based on a relative entropy argument and requires a non-trivial further development of ideas from the recent work of Fischer and the first author (Arch. Ration. Mech. Anal. 236, 2020) to incorporate the contact angle condition. To focus on the effects of the necessarily singular geometry of the evolving fluid domains, we work for simplicity in the regime of same viscosities for the two fluids.}, author = {Hensel, Sebastian and Marveggio, Alice}, issn = {1422-6952}, journal = {Journal of Mathematical Fluid Mechanics}, number = {3}, publisher = {Springer Nature}, title = {{Weak-strong uniqueness for the Navier–Stokes equation for two fluids with ninety degree contact angle and same viscosities}}, doi = {10.1007/s00021-022-00722-2}, volume = {24}, year = {2022}, } @unpublished{14597, abstract = {Phase-field models such as the Allen-Cahn equation may give rise to the formation and evolution of geometric shapes, a phenomenon that may be analyzed rigorously in suitable scaling regimes. In its sharp-interface limit, the vectorial Allen-Cahn equation with a potential with N≥3 distinct minima has been conjectured to describe the evolution of branched interfaces by multiphase mean curvature flow. In the present work, we give a rigorous proof for this statement in two and three ambient dimensions and for a suitable class of potentials: As long as a strong solution to multiphase mean curvature flow exists, solutions to the vectorial Allen-Cahn equation with well-prepared initial data converge towards multiphase mean curvature flow in the limit of vanishing interface width parameter ε↘0. We even establish the rate of convergence O(ε1/2). Our approach is based on the gradient flow structure of the Allen-Cahn equation and its limiting motion: Building on the recent concept of "gradient flow calibrations" for multiphase mean curvature flow, we introduce a notion of relative entropy for the vectorial Allen-Cahn equation with multi-well potential. This enables us to overcome the limitations of other approaches, e.g. avoiding the need for a stability analysis of the Allen-Cahn operator or additional convergence hypotheses for the energy at positive times.}, author = {Fischer, Julian L and Marveggio, Alice}, booktitle = {arXiv}, title = {{Quantitative convergence of the vectorial Allen-Cahn equation towards multiphase mean curvature flow}}, doi = {10.48550/ARXIV.2203.17143}, year = {2022}, } @unpublished{10174, abstract = {Quantitative stochastic homogenization of linear elliptic operators is by now well-understood. In this contribution we move forward to the nonlinear setting of monotone operators with p-growth. This first work is dedicated to a quantitative two-scale expansion result. Fluctuations will be addressed in companion articles. By treating the range of exponents 2≤p<∞ in dimensions d≤3, we are able to consider genuinely nonlinear elliptic equations and systems such as −∇⋅A(x)(1+|∇u|p−2)∇u=f (with A random, non-necessarily symmetric) for the first time. When going from p=2 to p>2, the main difficulty is to analyze the associated linearized operator, whose coefficients are degenerate, unbounded, and depend on the random input A via the solution of a nonlinear equation. One of our main achievements is the control of this intricate nonlinear dependence, leading to annealed Meyers' estimates for the linearized operator, which are key to the quantitative two-scale expansion result.}, author = {Clozeau, Nicolas and Gloria, Antoine}, booktitle = {arXiv}, title = {{Quantitative nonlinear homogenization: control of oscillations}}, 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}}, doi = {10.48550/arXiv.2109.04233}, year = {2021}, } @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{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}, } @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 = {2194-041X}, journal = {Stochastics and Partial Differential Equations: Analysis and Computations}, pages = {892–939}, publisher = {Springer Nature}, title = {{Finite time extinction for the 1D stochastic porous medium equation with transport noise}}, doi = {10.1007/s40072-021-00188-9}, volume = {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{10549, abstract = {We derive optimal-order homogenization rates for random nonlinear elliptic PDEs with monotone nonlinearity in the uniformly elliptic case. More precisely, for a random monotone operator on \mathbb {R}^d with stationary law (that is spatially homogeneous statistics) and fast decay of correlations on scales larger than the microscale \varepsilon >0, we establish homogenization error estimates of the order \varepsilon in case d\geqq 3, and of the order \varepsilon |\log \varepsilon |^{1/2} in case d=2. Previous results in nonlinear stochastic homogenization have been limited to a small algebraic rate of convergence \varepsilon ^\delta . We also establish error estimates for the approximation of the homogenized operator by the method of representative volumes of the order (L/\varepsilon )^{-d/2} for a representative volume of size L. Our results also hold in the case of systems for which a (small-scale) C^{1,\alpha } regularity theory is available.}, author = {Fischer, Julian L and Neukamm, Stefan}, issn = {1432-0673}, journal = {Archive for Rational Mechanics and Analysis}, keywords = {Mechanical Engineering, Mathematics (miscellaneous), Analysis}, number = {1}, pages = {343--452}, publisher = {Springer Nature}, title = {{Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems}}, doi = {10.1007/s00205-021-01686-9}, volume = {242}, year = {2021}, } @article{10575, abstract = {The choice of the boundary conditions in mechanical problems has to reflect the interaction of the considered material with the surface. Still the assumption of the no-slip condition is preferred in order to avoid boundary terms in the analysis and slipping effects are usually overlooked. Besides the “static slip models”, there are phenomena that are not accurately described by them, e.g. at the moment when the slip changes rapidly, the wall shear stress and the slip can exhibit a sudden overshoot and subsequent relaxation. When these effects become significant, the so-called dynamic slip phenomenon occurs. We develop a mathematical analysis of Navier–Stokes-like problems with a dynamic slip boundary condition, which requires a proper generalization of the Gelfand triplet and the corresponding function space setting.}, author = {Abbatiello, Anna and Bulíček, Miroslav and Maringová, Erika}, issn = {1793-6314}, journal = {Mathematical Models and Methods in Applied Sciences}, number = {11}, pages = {2165--2212}, publisher = {World Scientific Publishing}, title = {{On the dynamic slip boundary condition for Navier-Stokes-like problems}}, doi = {10.1142/S0218202521500470}, volume = {31}, 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 = {Institute of Science and Technology 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}}, doi = {10.48550/arXiv.2108.01733}, year = {2021}, } @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{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{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{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{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{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 = {1730-6337}, journal = {Studia Mathematica}, keywords = {General Mathematics}, number = {3}, pages = {251--297}, publisher = {Instytut Matematyczny}, title = {{Modelled distributions of Triebel–Lizorkin type}}, doi = {10.4064/sm180411-11-2}, volume = {252}, 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{404, abstract = {We construct martingale solutions to stochastic thin-film equations by introducing a (spatial) semidiscretization and establishing convergence. The discrete scheme allows for variants of the energy and entropy estimates in the continuous setting as long as the discrete energy does not exceed certain threshold values depending on the spatial grid size $h$. Using a stopping time argument to prolongate high-energy paths constant in time, arbitrary moments of coupled energy/entropy functionals can be controlled. Having established Hölder regularity of approximate solutions, the convergence proof is then based on compactness arguments---in particular on Jakubowski's generalization of Skorokhod's theorem---weak convergence methods, and recent tools on martingale convergence. }, author = {Fischer, Julian L and Grün, Günther}, journal = {SIAM Journal on Mathematical Analysis}, number = {1}, pages = {411 -- 455}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Existence of positive solutions to stochastic thin-film equations}}, doi = {10.1137/16M1098796}, volume = {50}, year = {2018}, } @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}, } @article{712, abstract = {We establish a weak–strong uniqueness principle for solutions to entropy-dissipating reaction–diffusion equations: As long as a strong solution to the reaction–diffusion equation exists, any weak solution and even any renormalized solution must coincide with this strong solution. Our assumptions on the reaction rates are just the entropy condition and local Lipschitz continuity; in particular, we do not impose any growth restrictions on the reaction rates. Therefore, our result applies to any single reversible reaction with mass-action kinetics as well as to systems of reversible reactions with mass-action kinetics satisfying the detailed balance condition. Renormalized solutions are known to exist globally in time for reaction–diffusion equations with entropy-dissipating reaction rates; in contrast, the global-in-time existence of weak solutions is in general still an open problem–even for smooth data–, thereby motivating the study of renormalized solutions. The key ingredient of our result is a careful adjustment of the usual relative entropy functional, whose evolution cannot be controlled properly for weak solutions or renormalized solutions.}, author = {Fischer, Julian L}, issn = {0362546X}, journal = {Nonlinear Analysis: Theory, Methods and Applications}, pages = {181 -- 207}, publisher = {Elsevier}, title = {{Weak–strong uniqueness of solutions to entropy dissipating reaction–diffusion equations}}, doi = {10.1016/j.na.2017.03.001}, volume = {159}, year = {2017}, }