arXiv subjects

Esther S. Daus

Publications and source records attributed to Esther S. Daus.

16 recordsLinked to original sources

arXiv2025

Nonisothermal Richards flow in porous media with cross diffusion

The existence of large-data weak entropy solutions to a nonisothermal immiscible compressible two-phase unsaturated flow model in porous media is proved. The model is thermodynamically consistent and includes temperature gradients and cross-diffusion effects. Due to the fact that some terms from the total energy balance are non-integrable in the classical weak sense, we consider so-called variational entropy solutions. A priori estimates are derived from the entropy balance and the total energy balance. The compactness is achieved by using the Div-Curl lemma.

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arXiv2021

Derivation of a Fractional Cross-Diffusion System as the Limit of a Stochastic Many-Particle System Driven by Lévy Noise

In this article a fractional cross-diffusion system is derived as the rigorous many-particle limit of a multi-species system of moderately interacting particles that is driven by Lévy noise. The form of the mutual interaction is motivated by the porous medium equation with fractional potential pressure. Our approach is based on the techniques developed by Oelschläger (1989) and Stevens (2000), in the latter of which the convergence of a regularization of the empirical measure to the solution of a correspondingly regularized macroscopic system is shown. A well-posedness result and the non-negativity of solutions are proved for the regularized macroscopic system, which then yields the same results for the non-regularized fractional cross-diffusion system in the limit.

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arXiv2021

Rigorous derivation of population cross-diffusion systems from moderately interacting particle systems

Population cross-diffusion systems of Shigesada-Kawasaki-Teramoto type are derived in a mean-field-type limit from stochastic, moderately interacting many-particle systems for multiple population species in the whole space. The diffusion term in the stochastic model depends nonlinearly on the interactions between the individuals, and the drift term is the gradient of the environmental potential. In the first step, the mean-field limit leads to an intermediate nonlocal model. The local cross-diffusion system is derived in the second step in a moderate scaling regime, when the interaction potentials approach the Dirac delta distribution. The global existence of strong solutions to the intermediate and the local diffusion systems is proved for sufficiently small initial data. Furthermore, numerical simulations on the particle level are presented.

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arXiv2021

Random-batch method for multi-species stochastic interacting particle systems

A random-batch method for multi-species interacting particle systems is proposed, extending the method of S. Jin, L. Li, and J.-G. Liu [J. Comput. Phys. 400 (2020), 108877]. The idea of the algorithmus is to randomly divide, at each time step, the ensemble of particles into small batches and then to evolve the interaction of each particle within the batches until the next time step. This reduces the computational cost by one order of magnitude, while keeping a certain accuracy. It is proved that the $L^2$ error of the error process behaves like the square root of the time step size, uniformly in time, thus providing the convergence of the scheme. The numerical efficiency is tested for some examples, and numerical simulations of the opinion dynamics in a hierarchical company, consisting of workers, managers, and CEOs, are presented.

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arXiv2021

Non-Local Porous Media Equations with Fractional Time Derivative

In this paper we investigate existence of solutions for the system: \begin{equation*} \left\{ \begin{array}{l} D^α_tu=\textrm{div}(u \nabla p),\\ D^α_tp=-(-Δ)^{s}p+u^{2}, \end{array} \right. \end{equation*} in $\mathbb{T}^3$ for $0< s \leq 1$, and $0< α\le 1$. The term $D^α_t u$ denotes the Caputo derivative, which models memory effects in time. The fractional Laplacian $(-Δ)^{s}$ represents the Lévy diffusion. We prove global existence of nonnegative weak solutions that satisfy a variational inequality. The proof uses several approximations steps, including an implicit Euler time discretization. We show that the proposed discrete Caputo derivative satisfies several important properties, including positivity preserving, convexity and rigorous convergence towards the continuous Caputo derivative. Most importantly, we give a strong compactness criteria for piecewise constant functions, in the spirit of Aubin-Lions theorem, based on bounds of the discrete Caputo derivative.

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arXiv2021

On the multi-species Boltzmann equation with uncertainty and its stochastic Galerkin approximation

In this paper the nonlinear multi-species Boltzmann equation with random uncertainty coming from the initial data and collision kernel is studied. Well-posedness and long-time behavior - exponential decay to the global equilibrium - of the analytical solution, and spectral gap estimate for the corresponding linearized gPC-based stochastic Galerkin system are obtained, by using and extending the analytical tools provided in [M. Briant and E. S. Daus, {\it Arch. Ration. Mech. Anal.}, 3, 1367-1443, 2016] for the deterministic problem in the perturbative regime, and in [E. S. Daus, S. Jin and L. Liu, {\it Kinet. Relat. Models}, 12, 909-922, 2019] for the single-species problem with uncertainty. The well-posedness result of the sensitivity system presented here has not been obtained so far even for the single-species case.

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arXiv2020

Convergence of a finite-volume scheme for a degenerate-singular cross-diffusion system for biofilms

An implicit Euler finite-volume scheme for a cross-diffusion system modeling biofilm growth is analyzed by exploiting its formal gradient-flow structure. The numerical scheme is based on a two-point flux approximation that preserves the entropy structure of the continuous model. Assuming equal diffusivities, the existence of nonnegative and bounded solutions to the scheme and its convergence are proved. Finally, we supplement the study by numerical experiments in one and two space dimensions.

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arXiv2019

Global existence for a two-phase flow model with cross diffusion

In this work we study a degenerate pseudo-parabolic system with cross diffusion describing the evolution of the densities of an unsaturated two-phase flow mixture with dynamic capillary pressure in porous medium with saturation-dependent relaxation parameter and hypocoercive diffusion operator modeling cross diffusion. The equations are derived in a thermodynamically correct way from mass conservation laws. Global-in-time existence of weak solutions to the system in a bounded domain with equilibrium boundary conditions is shown. The main tools of the analysis are an entropy inequality and a crucial apriori bound which allows for controlling the degeneracy.

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arXiv2019

Spectral Convergence of the Stochastic Galerkin Approximation to the Boltzmann Equation with Multiple Scales and Large Random Perturbation in the Collision Kernel

In [L. Liu and S. Jin, Multiscale Model. Simult., 16, 1085-1114, 2018], spectral convergence and long-time decay of the numerical solution towards the global equilibrium of the stochastic Galerkin approximation for the Boltzmann equation with random inputs in the initial data and collision kernel for hard potentials and Maxwellian molecules under Grad's angular cutoff were established using the hypocoercive properties of the collisional kinetic model. One assumption for the random perturbation of the collision kernel is that the perturbation is in the order of the Knudsen number, which can be very small in the fluid dynamical regime. In this article, we remove this smallness assumption, and establish the same results but now for random perturbations of the collision kernel that can be of order one. The new analysis relies on the establishment of a spectral gap for the numerical collision operator.

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arXiv2018

Longtime behavior and weak-strong uniqueness for a nonlocal porous media equation

In this manuscript we consider a non-local porous medium equation with non-local diffusion effects given by a fractional heat operator \begin{equation*} \partial_t u = \mbox{div}(u\nabla p),\qquad \partial_t p = -(-Δ)^s p + u^2, \end{equation*} in three space dimensions for $3/4\le s < 1$ and analyze the long time asymptotics. The proof is based on energy methods and leads to algebraic decay towards the stationary solution $u=0$ and $\nabla p=0$ in the $L^2(\mathbb{R}^3)$-norm. The decay rate depends on the exponent $s$. We also show weak-strong uniqueness of solutions and continuous dependence from the initial data. As a side product of our analysis we also show that existence of weak solutions, previously shown in [Caffarelli, Gualdani, Zamponi 2018] for $3/4\le s \le 1$, holds for $1/2 < s\le 1$ if we consider our problem in the torus.

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arXiv2018

Rigorous mean-field limit and cross diffusion

The mean-field limit in a weakly interacting stochastic many-particle system for multiple population species in the whole space is proved. The limiting system consists of cross-diffusion equations, modeling the segregation of populations. The mean-field limit is performed in two steps: First, the many-particle system leads in the large population limit to an intermediate nonlocal diffusion system. The local cross-diffusion system is then obtained from the nonlocal system when the interaction potentials approach the Dirac delta distribution. The global existence of the limiting and the intermediate diffusion systems is shown for small initial data, and an error estimate is given.

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arXiv2018

About the entropic structure of detailed balanced multi-species cross-diffusion equations

This paper links at the formal level the entropy structure of a multi-species cross-diffusion system of Shigesada-Kawasaki-Teramoto (SKT) type satisfying the detailed balance condition with the entropy structure of a reversible microscopic many-particle Markov process on a discretised space. The link is established by first performing a mean-field limit to a master equation over discretised space. Then the spatial discretisation limit is performed in a completely rigorous way. This by itself provides a novel strategy for proving global existence of weak solutions to a class of cross-diffusion systems.

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arXiv2018

Trend to equilibrium of renormalized solutions to reaction-cross-diffusion systems

The convergence to equilibrium of renormalized solutions to reaction-cross-diffusion systems in a bounded domain under no-flux boundary conditions is studied. The reactions model complex balanced chemical reaction networks coming from mass-action kinetics and thus do not obey any growth condition, while the diffusion matrix is of cross-diffusion type and hence nondiagonal and neither symmetric nor positive semi-definite, but the system admits a formal gradient-flow or entropy structure. The diffusion term generalizes the population model of Shigesada, Kawasaki and Teramoto to an arbitrary number of species. By showing that any renormalized solution satisfies the conservation of masses and a weak entropy-entropy production inequality, it can be proved under the assumption of no boundary equilibria that {\it all} renormalized solutions converge exponentially to the complex balanced equilibrium with a rate which is explicit up to a finite dimensional inequality.

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arXiv2018

Analysis of a degenerate and singular volume-filling cross-diffusion system modeling biofilm growth

We analyze the mathematical properties of a multi-species biofilm cross-diffusion model together with very general reaction terms and mixed Dirichlet-Neumann boundary conditions on a bounded domain. This model belongs to the class of volume-filling type cross-diffusion systems which exhibit a porous medium-type degeneracy when the total biomass vanishes as well as a superdiffusion-type singularity when the biomass reaches its maximum cell capacity, which make the analysis extremely challenging. The equations also admit a very interesting non-standard entropy structure. We prove the existence of global-in-time weak solutions, study the asymptotic behavior and the uniqueness of the solutions, and complement the analysis by numerical simulations that illustrate the theoretically obtained results.

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arXiv2018

Exponential time decay of solutions to reaction-cross-diffusion systems of Maxwell-Stefan type

The large-time asymptotics of weak solutions to Maxwell--Stefan diffusion systems for chemically reacting fluids with different molar masses and reversible reactions are investigated. The diffusion matrix of the system is generally neither symmetric nor positive definite, but the equations admit a formal gradient-flow structure which provides entropy (free energy) estimates. The main result is the exponential decay to the unique equilibrium with a rate that is constructive up to a finite-dimensional inequality. The key elements of the proof are the existence of a unique detailed-balanced equilibrium and the derivation of an inequality relating the entropy and the entropy production. The main difficulty comes from the fact that the reactions are represented by molar fractions while the conservation laws hold for the concentrations. The idea is to enlarge the space of $n$ partial concentrations by adding the total concentration, viewed as an independent variable, thus working with $n+1$ variables. Further results concern the existence of global bounded weak solutions to the parabolic system and an extension of the results to complex-balanced systems.

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arXiv2016

Global existence analysis of cross-diffusion population systems for multiple species

The existence of global-in-time weak solutions to reaction-cross-diffusion systems for an arbitrary number of competing population species is proved. The equations can be derived from an on-lattice random-walk model with general transition rates. In the case of linear transition rates, it extends the two-species population model of Shigesada, Kawasaki, and Teramoto. The equations are considered in a bounded domain with homogeneous Neumann boundary conditions. The existence proof is based on a refined entropy method and a new approximation scheme. Global existence follows under a detailed balance or weak cross-diffusion condition. The detailed balance condition is related to the symmetry of the mobility matrix, which mirrors Onsager's principle in thermodynamics. Under detailed balance (and without reaction), the entropy is nonincreasing in time, but counter-examples show that the entropy may increase initially if detailed balance does not hold.

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