arXiv subjects

Tomasz Dębiec

Publications and source records attributed to Tomasz Dębiec.

17 recordsLinked to original sources

arXiv2026

Incompressible limit for a two-species Brinkman model with drift

We study a two-species model for tissue growth in which both populations are transported by an external drift and by a velocity potential determined through Brinkman's law. The pressure is generated by a stiff constitutive relation depending on the total density. Our main result establishes the incompressible limit as the stiffness exponent tends to infinity, in arbitrary space dimension and for merely integrable initial data. The limit system consists of the two balance laws coupled to Brinkman's equation, the hard-congestion constraint $0\leq n_\infty\leq 1$, the graph relation $p_\infty(1-n_\infty)=0$, and the corresponding complementarity relation. A key point of the analysis is a new $L^2$-based compactness theory that avoids both uniform $L^\infty$-bounds on the pressure and the kinetic reformulation used in earlier approaches. We first construct weak solutions for bounded data and then remove the boundedness assumption by means of a weighted compactness argument inspired by Bresch--Jabin. We also prove an Aubin--Lions--Simon type lemma based on oscillation control, yielding time continuity of the constructed solutions. Finally, a refined dissipation estimate for the Bresch--Jabin compactness functional gives strong compactness of the pressure in the stiff limit and implies a regularising effect: the limiting pressure is bounded even when the approximating pressures are only integrable.

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arXiv2025

Lipschitz stability for Bayesian inference in porous medium tissue growth models

We consider a macroscopic model for the dynamics of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Given a power-law constitutive relation between the pressure and cell density, the model can be written as a porous medium equation with a growth term. We prove Lipschitz continuity of the mild solutions of the model with respect to the diffusion parameter (the exponent $γ$ in the pressure-density law) in the $L_1$ norm. While of independent analytical interest, our motivation for this result is to provide a vital step towards using Bayesian inverse problem methodology for parameter estimation based on experimental data -- such stability estimates are indispensable for applying sampling algorithms which rely on the gradient of the likelihood function.

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arXiv2024

Nonlocal approximation of an anisotropic cross-diffusion system

Localisation limits and nonlocal approximations of degenerate parabolic systems have experienced a renaissance in recent years. However, only few results cover anisotropic systems. This work addresses this gap by establishing the nonlocal-to-limit for a specific anisotropic cross-diffusion system encountered in population dynamics featuring phase-separation phenomena, i.e., internal layers between different species. A critical element of the proof is an entropy dissipation identity, which we show to hold for any weak solution.

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arXiv2024

From Finite to Continuous Phenotypes in (Visco-)Elastic Tissue Growth Models

In this study, we explore a mathematical model for tissue growth focusing on the interplay between multiple cell subpopulations with distinct phenotypic characteristics. The model addresses the dynamics of tissue growth influenced by phenotype-dependent growth rates and collective population pressure, governed by Brinkman's law. We examine two primary objectives: the joint limit where viscosity tends to zero while the number of species approaches infinity, yielding an inviscid Darcy-type model with a continuous phenotype variable, and the continuous phenotype limit where the number of species becomes infinite with a fixed viscosity, resulting in a novel viscoelastic tissue growth model. In this sense, this paper provides a comprehensive framework that elucidates the relationships between four different modelling paradigms in tissue growth.

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arXiv2024

Stability of solutions of the porous medium equation with growth with respect to the diffusion exponent

We consider a macroscopic model for the growth of living tissues incorporating pressure-driven dispersal and pressure-modulated proliferation. Assuming a power-law relation between the mechanical pressure and the cell density, the model can be expressed as the porous medium equation with a growth term. We prove Hölder continuous dependence of the solutions of the model on the diffusion exponent. The main difficulty lies in the degeneracy of the porous medium equations at vacuum. To deal with this issue, we first regularise the equation by shifting the initial data away from zero and then optimise the stability estimate derived in the regular setting.

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arXiv2023

On a class of generalised solutions to the kinetic Hookean dumbbell model for incompressible dilute polymeric fluids: existence and macroscopic closure

We consider the Hookean dumbbell model, a system of nonlinear PDEs arising in the kinetic theory of homogeneous dilute polymeric fluids. It consists of the unsteady incompressible Navier-Stokes equations in a bounded Lipschitz domain, coupled to a Fokker-Planck-type parabolic equation with a centre-of-mass diffusion term, for the probability density function, modelling the evolution of the configuration of noninteracting polymer molecules in the solvent. The micro-macro interaction is reflected by the presence of a drag term in the Fokker-Planck equation and the divergence of a polymeric extra-stress tensor in the Navier-Stokes balance of momentum equation. We introduce the concept of generalised dissipative solution - a relaxation of the usual notion of weak solution, allowing for the presence of a, possibly nonzero, defect measure in the momentum equation. This defect measure accounts for the lack of compactness in the polymeric extra-stress tensor. We prove global existence of generalised dissipative solutions satisfying additionally an energy inequality for the macroscopic deformation tensor. Using this inequality, we establish a conditional regularity result: any generalised dissipative solution with a sufficiently regular velocity field is a weak solution to the Hookean dumbbell model. Additionally, in two space dimensions we provide a rigorous derivation of the macroscopic closure of the Hookean model and discuss its relationship with the Oldroyd-B model with stress diffusion. Finally, we improve an earlier result by Barrett and Süli by establishing the global existence of weak solutions for a larger class of initial data.

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arXiv2023

A Degenerate Cross-Diffusion System as the Inviscid Limit of a Nonlocal Tissue Growth Model

In recent years, there has been a spike in the interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relationship between them. In this paper, we want to address this dearth in the literature, providing a rigorous argument that bridges the gap between a viscoelastic tumour model (of Brinkman type) and an inviscid tumour model (of Darcy type).

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arXiv2021

Dissipative measure-valued solutions to the Euler-Poisson equation

We consider several pressureless variants of the compressible Euler equation driven by nonlocal repulsionattraction and alignment forces with Poisson interaction. Under an energy admissibility criterion, we prove existence of global measure-valued solutions, i.e., very weak solutions described by a classical Young measure together with appropriate concentration defects. We then investigate the evolution of a relative energy functional to compare a measure-valued solution to a regular solution emanating from the same initial datum. This leads to a (partial) weak-strong uniqueness principle.

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arXiv2021

Convergence rate for the incompressible limit of nonlinear diffusion-advection equations

The incompressible limit of nonlinear diffusion equations of porous medium type has attracted a lot of attention in recent years, due to its ability to link the weak formulation of cell-population models to free boundary problems of Hele-Shaw type. Although vast literature is available on this singular limit, little is known on the convergence rate of the solutions. In this work, we compute the convergence rate in a negative Sobolev norm and, upon interpolating with BV-uniform bounds, we deduce a convergence rate in appropriate Lebesgue spaces.

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arXiv2021

A General Convex Integration Scheme for the Isentropic Compressible Euler Equations

We prove via convex integration a result that allows to pass from a so-called subsolution of the isentropic Euler equations (in space dimension at least $2$) to exact weak solutions. The method is closely related to the incompressible scheme established by De Lellis--Székelyhidi, in particular we only perturb momenta and not densities. Surprisingly, though, this turns out not to be a restriction, as can be seen from our simple characterization of the $Λ$-convex hull of the constitutive set. An important application of our scheme will be exhibited in forthcoming work by Gallenmüller--Wiedemann.

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arXiv2019

Incompressible limit for a two-species tumour model with coupling through Brinkman's law in one dimension

We present a two-species model with applications in tumour modelling. The main novelty is the coupling of both species through the so-called Brinkman law which is typically used in the context of visco-elastic media, where the velocity field is linked to the total population pressure via an elliptic equation. The same model for only one species has been studied by Perthame and Vauchelet in the past. The first part of this paper is dedicated to establishing existence of solutions to the problem, while the second part deals with the incompressible limit as the stiffness of the pressure law tends to infinity. Here we present a novel approach in one spatial dimension that differs from the kinetic reformulation used in the aforementioned study and, instead, relies on uniform BV-estimates.

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arXiv2019

On entropy conservation for general systems of conservation laws

In this work we consider companion conservation laws to general systems of conservation laws. We investigate sufficient regularity for weak solutions to satisfy companion laws, assuming the fluxes to be $C^{1,γ}$, $0<γ<1$, regular. We discuss applications of the general framework to nonlinear elasticity and compressible Euler system.

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arXiv2018

Energy Conservation for the Compressible Euler and Navier-Stokes Equations with Vacuum

We consider the compressible isentropic Euler equations on $\mathbb{T}^d\times [0,T]$ with a pressure law $p\in C^{1,γ-1}$, where $1\le γ<2$. This includes all physically relevant cases, e.g.\ the monoatomic gas. We investigate under what conditions on its regularity a weak solution conserves the energy. Previous results have crucially assumed that $p\in C^2$ in the range of the density, however, for realistic pressure laws this means that we must exclude the vacuum case. Here we improve these results by giving a number of sufficient conditions for the conservation of energy, even for solutions that may exhibit vacuum: Firstly, by assuming the velocity to be a divergence-measure field; secondly, imposing extra integrability on $1/ρ$ near a vacuum; thirdly, assuming $ρ$ to be quasi-nearly subharmonic near a vacuum; and finally, by assuming that $u$ and $ρ$ are Hölder continuous. We then extend these results to show global energy conservation for the domain $Ω\times [0,T]$ where $Ω$ is bounded with a $C^2$ boundary. We show that we can extend these results to the compressible Navier-Stokes equations, even with degenerate viscosity.

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arXiv2018

Relative Entropy Method for Measure Solutions of the Growth-Fragmentation Equation

The aim of this study is to generalise recent results of the two last authors on en-tropy methods for measure solutions of the renewal equation to other classes of structured population problems. Specifically, we develop a generalised relative entropy inequality for the growth-fragmentation equation and prove asymptotic convergence to a steady-state solution, even when the initial datum is only a non-negative measure.

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arXiv2017

Conservation of energy for the Euler-Korteweg equations

In this article we study the principle of energy conservation for the Euler-Korteweg system. We formulate an Onsager-type sufficient regularity condition for weak solutions of the Euler-Korteweg system to conserve the total energy. The result applies to the system of Quantum Hydrodynamics.

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arXiv2017

Relative entropy method for measure-valued solutions in natural sciences

In this article we describe the applications of the relative entropy framework. In particular uniqueness of an entropy solution is proven for a scalar conservation law, using the notion of measure-valued entropy solutions. Further we survey recent results concerning measure-valued-strong uniqueness for a number of physical systems - incompressible and compressible Euler equations, compressible Navier-Stokes, polyconvex elastodynamics and general hyperbolic conservation laws, as well as long-time asymptotics of the McKendrick-Von Foerster equation.

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arXiv2017

A tribute to conservation of energy for weak solutions

In this article we focus our attention on the principle of energy conservation within the context of systems of fluid dynamics. We give an overview of results concerning the resolution of the famous Onsager conjecture - which states regularity requirements for weak solutions of the incompressible Euler system to conserve energy. Further we survey results providing optimal sufficient regularity conditions for energy conservation for other balance laws: compressible Euler, Navier-Stokes, magnetohydrodynamics and general conservation laws.

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