@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{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{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}, } @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{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}, } @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}, } @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{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{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{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}, } @article{1014, abstract = {We consider the large-scale regularity of solutions to second-order linear elliptic equations with random coefficient fields. In contrast to previous works on regularity theory for random elliptic operators, our interest is in the regularity at the boundary: We consider problems posed on the half-space with homogeneous Dirichlet boundary conditions and derive an associated C1,α-type large-scale regularity theory in the form of a corresponding decay estimate for the homogenization-adapted tilt-excess. This regularity theory entails an associated Liouville-type theorem. The results are based on the existence of homogenization correctors adapted to the half-space setting, which we construct-by an entirely deterministic argument-as a modification of the homogenization corrector on the whole space. This adaption procedure is carried out inductively on larger scales, crucially relying on the regularity theory already established on smaller scales.}, author = {Fischer, Julian L and Raithel, Claudia}, issn = {00361410}, journal = {SIAM Journal on Mathematical Analysis}, number = {1}, pages = {82 -- 114}, publisher = {Society for Industrial and Applied Mathematics }, title = {{Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space}}, doi = {10.1137/16M1070384}, volume = {49}, year = {2017}, }