arXiv subjects

Julian Fischer

Publications and source records attributed to Julian Fischer.

36 records · Page 2Linked to original sources

arXiv2021

Global renormalised solutions and equilibration of reaction-diffusion systems with non-linear diffusion

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.

math.AP↗

arXiv2020

Sharp criteria for the waiting time phenomenon in solutions to the thin-film equation

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.

math.AP↗

arXiv2020

Optimal homogenization rates in stochastic homogenization of nonlinear uniformly elliptic equations and systems

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 (i.e. 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 $d\geq 3$, respectively 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 $\varepsilon^δ$. 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) $C^{1,α}$ regularity theory is available.

math.AP↗

arXiv2020

A priori error analysis of a numerical stochastic homogenization method

This paper provides an a~priori error analysis of a localized orthogonal decomposition method (LOD) 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 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.

math.NA↗

arXiv2020

Convergence rates of the Allen-Cahn equation to mean curvature flow: A short proof based on relative entropies

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.

math.AP↗

arXiv2019

The waiting time phenomenon in spatially discretized porous medium and thin film equations

Various degenerate diffusion equations exhibit a waiting time phenomenon: Dependening 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 suffcient criteria for the occurrence of waiting times that are consistent with the known ones for the original PDEs. 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.

math.AP↗

arXiv2019

External forces in the continuum limit of discrete systems with non-convex interaction potentials: Compactness for a $Γ$-development

This paper is concerned with equilibrium configurations of one-dimensional particle system 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. Our result is based on a new technique for deriving compactness results which are required for calculating the first-order $Γ$-limit: 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. This new technique is required due to the additional presence of forces with non-convex potentials. The paper is complemented with the study of the minimizers of the $Γ$-limit.

math.AP↗

arXiv2019

Variance reduction for effective energies of random lattices in the Thomas-Fermi-von Weizsäcker model

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.

math.AP↗

arXiv2019

Quantitative normal approximation for sums of random variables with multilevel local dependence structure

We establish a quantitative normal approximation result for sums of random variables with multilevel local dependencies. As a corollary, we obtain a quantitative normal approximation result for linear functionals of random fields which may be approximated well by random fields with finite dependency range. Such random fields occur for example in the homogenization of linear elliptic equations with random coefficient fields. In particular, our result allows for the derivation of a quantitative normal approximation result for the approximation of effective coefficients in stochastic homogenization in the setting of coefficient fields with finite range of dependence. The proof of our normal approximation theorem is based on a suitable adaption of Stein's method and requires a different estimation strategy for terms originating from long-range dependencies as opposed to terms stemming from short-range dependencies.

math.PR↗

arXiv2019

Weak-strong uniqueness for the Navier-Stokes equation for two fluids with surface tension

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 [2] 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.

math.AP↗

arXiv2018

The choice of representative volumes in the approximation of effective properties of random materials

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 material 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.

math.AP↗

arXiv2017

Weak-strong uniqueness of solutions to entropy-dissipating reaction-diffusion equations

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.

math.AP↗

arXiv2016

Stochastic homogenization of linear elliptic equations: Higher-order error estimates in weak norms via second-order correctors

We are concerned with the homogenization of second-order linear elliptic equations with random coefficient fields. For symmetric coefficient fields with only short-range correlations, quantified through a logarithmic Sobolev inequality for the ensemble, we prove that when measured in weak spatial norms, the solution to the homogenized equation provides a higher-order approximation of the solution to the equation with oscillating coefficients. In the case of nonsymmetric coefficient fields, we provide a higher-order approximation (in weak spatial norms) of the solution to the equation with oscillating coefficients in terms of solutions to constant-coefficient equations. In both settings, we also provide optimal error estimates for the two-scale expansion truncated at second order. Our results rely on novel estimates on the second-order homogenization corrector, which we establish via sensitivity estimates for the second-order corrector and a large-scale $L^p$ theory for elliptic equations with random coefficients. Our results also cover the case of elliptic systems.

math.AP↗

arXiv2016

Liouville principles and a large-scale regularity theory for random elliptic operators on the half-space

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 $C^{1,α}$-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.

math.AP↗

arXiv2016

Bi-Sobolev Solutions to the Prescribed Jacobian Inequality in the Plane with $L^p$ Data

We construct planar bi-Sobolev mappings whose local volume distortion is bounded from below by a given function $f\in L^p$ with $p>1$, i.e. bi-Sobolev solutions for the prescribed Jacobian inequality in the plane for right-hand sides $f\in L^p$. More precisely, for any $1<q<(p+1)/2$ we construct a solution which belongs to $W^{1,q}$ and which preserves the boundary pointwise. For bounded right-hand sides $f\in L^{\infty}$, we provide bi-Lipschitz solutions. The basic building block of our construction are 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 covering result of Alberti, Csörnyei, and Preiss for measurable planar sets in the case of compact sets.

math.AP↗

arXiv2016

A Coherent Polariton Laser

The semiconductor polariton laser promises a new source of coherent light, which, compared to conventional semiconductor photon lasers, has input-energy threshold orders of magnitude lower. However, intensity stability, a defining feature of a coherent state, has remained poor. Intensity noise at many times of the shot-noise of a coherent state has persisted, which has been attributed to multiple mechanisms that are difficult to separate in conventional polariton systems. The large intensity noise in turn limited the phase coherence. These limit the capability of the polariton laser as a source of coherence light. Here, we demonstrate a polariton laser with shot-noise limited intensity stability, as expected of a fully coherent state. This is achieved by using an optical cavity with high mode selectivity to enforce single-mode lasing, suppress condensate depletion, and establish gain saturation. The absence of spurious intensity fluctuations moreover enabled measurement of a transition from exponential to Gaussian decay of the phase coherence of the polariton laser. It suggests large self-interaction energies in the polariton condensate, exceeding the laser bandwidth. Such strong interactions are unique to matter-wave laser and important for nonlinear polariton devices. The results will guide future development of polariton lasers and nonlinear polariton devices.

cond-mat.quant-gas↗

arXiv2015

Sublinear growth of the corrector in stochastic homogenization: Optimal stochastic estimates for slowly decaying correlations

We establish sublinear growth of correctors in the context of stochastic homogenization of linear elliptic PDEs. In case of weak decorrelation and "essentially Gaussian" coefficient fields, we obtain optimal (stretched exponential) stochastic moments for the minimal radius above which the corrector is sublinear. Our estimates also capture the quantitative sublinearity of the corrector (caused by the quantitative decorrelation on larger scales) correctly. The result is based on estimates on the Malliavin derivative for certain functionals which are basically averages of the gradient of the corrector, on concentration of measure, and on a mean value property for $a$-harmonic functions.

math.AP↗

arXiv2015

A higher-order large-scale regularity theory for random elliptic operators

We develop a large-scale regularity theory of higher order for divergence-form elliptic equations with heterogeneous coefficient fields $a$ in the context of stochastic homogenization. The large-scale regularity of $a$-harmonic functions is encoded by Liouville principles: The space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the same dimension as in the constant-coefficient case. This result can be seen as the qualitative side of a large-scale $C^{k,α}$-regularity theory, which in the present work is developed in the form of a corresponding $C^{k,α}$-"excess decay" estimate: For a given $a$-harmonic function $u$ on a ball $B_R$, its energy distance on some ball $B_r$ to the above space of $a$-harmonic functions that grow at most like a polynomial of degree $k$ has the natural decay in the radius $r$ above some minimal radius $r_0$. Though motivated by stochastic homogenization, the contribution of this paper is of purely deterministic nature: We work under the assumption that for the given realization $a$ of the coefficient field, the couple $(ϕ,σ)$ of scalar and vector potentials of the harmonic coordinates, where $ϕ$ is the usual corrector, grows sublinearly in a mildly quantified way. We then construct "$k$th-order correctors" and thereby the space of $a$-harmonic functions that grow at most like a polynomial of degree $k$, establish the above excess decay and then the corresponding Liouville principle.

math.AP↗