arXiv subjects
Publications and source records attributed to Piotr Gwiazda.
In this paper we apply the newly developed theory of extended graphons to a certain class of non-exchangeable, non-conservative problems. We establish the mean-field limit by proving the convergence of the associated generalized weighted empirical measures.
This paper studies the asymptotic behaviour of a structured population model on the space of nonnegative Radon measures. Such formulations naturally arise when solutions develop concentration phenomena or when the population is represented by discrete cohorts. Asynchronous exponential convergence of measure solutions towards a one-dimensional global attractor is established. While such results are classical in the $L^1$ setting, their extension to measure spaces requires different compactness and spectral arguments. We identify conditions under which the classical asymptotic behaviour persists in the space of Radon measures endowed with the flat metric, thereby extending the theory of asynchronous exponential growth beyond the classical $L^1$ framework.
Given two continuity equations with density-dependent velocities, we provide a new formula for the Wasserstein distance between the solutions in terms of the difference of velocities evaluated at the same density. The formula is particularly attractive to deduce quantitative estimates and rates of convergence for singular limits. We illustrate it using several examples. For the porous medium equation with exponent $m$, we prove that solutions are Lipschitz continuous with respect to $m$, providing a quantitative version of the result of Bénilan and Crandall. This result can be extended to a general aggregation-diffusion equation. We also study the limit $m \to \infty$ (the so-called mesa problem or the incompressible limit) and we recover the rate of convergence $1/{\sqrt{m}}$. Last but not least, we improve the rate of nonlocal-to-local convergence for the quadratic porous medium equation from recently obtained $\sqrt{\varepsilon}$ to numerically conjectured $\varepsilon$.
In this paper, we propose a numerical scheme for structured population models defined on a separable and complete metric space. In particular, we consider a generalized version of a transport equation with additional growth and non-local interaction terms in the space of nonnegative Radon measures equipped with the flat metric. The solutions, given by families of Radon measures, are approximated by linear combinations of Dirac measures. For this purpose, we introduce a finite-range approximation of the measure-valued model functions, provided that they are linear. By applying an operator splitting technique, we are able to separate the effects of the transport from those of growth and the non-local interaction. We derive the order of convergence of the numerical scheme, which is linear in the spatial discretization parameters and polynomial of order $α$ in the time step size, assuming that the model functions are $α$ Hölder regular in time. In a second step, we show that our proposed algorithm can approximate the posterior measure of Bayesian inverse models, which will allow us to link model parameters to measured data in the future.
Data assimilation plays a crucial role in modern weather prediction, providing a systematic way to incorporate observational data into complex dynamical models. The paper addresses continuous data assimilation for a model arising as a singular limit of the three-dimensional compressible Navier-Stokes-Fourier system with rotation driven by temperature gradient. The limit system preserves the essential physical mechanisms of the original model, while exhibiting a reduced, effectively two-and-a-half-dimensional structure. This simplified framework allows for a rigorous analytical study of the data assimilation process while maintaining a direct physical connection to the full compressible model. We establish well posedness of global-in-time solutions and a compact trajectory attractor, followed by the stability and convergence results for the nudging scheme applied to the limiting system. Finally, we demonstrate how these results can be combined with a relative entropy argument to extend the assimilation framework to the full three-dimensional compressible setting, thereby establishing a rigorous connection between the reduced and physically complete models.
Calibrating mathematical models of biological processes is essential for achieving predictive accuracy and gaining mechanistic insight. However, this task remains challenging due to limited and noisy data, significant biological variability, and the computational complexity of the models themselves. In this method's article, we explore a range of approaches for parameter inference in partial differential equation (PDE) models of biological systems. We introduce a unified mathematical framework, the Correlation Integral Likelihood (CIL) method, for parameter estimation in systems exhibiting heterogeneous or chaotic dynamics, encompassing both pattern formation models and individual-based models. Departing from classical Bayesian inverse problem methodologies, we motivate the development of the CIL method, demonstrate its versatility, and highlight illustrative applications within mathematical biology. Furthermore, we compare stochastic sampling strategies, such as Markov Chain Monte Carlo (MCMC), with deterministic gradient flow approaches, highlighting how these methods can be integrated within the proposed framework to enhance inference performance. Our work provides a practical and theoretically grounded toolbox for researchers seeking to calibrate complex biological models using incomplete, noisy, or heterogeneous data, thereby advancing both the predictive capability and mechanistic understanding of such systems.
We consider a stochastic parabolic partial differential equation with Dirichlet boundary conditions, multiplicative stochastic noise, and a monotone parabolic operator A. The growth and coercivity of A is controlled by a general N-function M, which depends on time, and spatial variable, but we do not assume any regularity with respect to the former. We show the existence of weak solutions to such system. As auxiliary result, we also provide the proof for the Itô's formula in Orlicz spaces. This general result applies to the ones studied in the literature, such as p(t, x)-Laplacian and double phase problems.
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.
We consider a general compressible, viscous, heat and magnetically conducting fluid described by the compressible Navier-Stokes-Fourier system coupled with induction equation. In particular, we do not assume conservative boundary conditions for the temperature and allow heating or cooling on the surface of the domain. We are interested in the mathematical analysis when the Mach, Froude, and Alfvén numbers are small, converging to zero at a specific rate. We give a rigorous mathematical justification that in the limit, in case of low stratification, one obtains a modified Oberbeck-Boussinesq-MHD system with a non-local term or a non-local boundary condition for the temperature deviation. Choosing a domain confined between parallel plates, one finds also that the flow is horizontal, and the magnetic field is perpendicular to it. The proof is based on the analysis of weak solutions to a primitive system and the relative entropy method.
We show that certain radially symmetric steady states of compressible viscous fluids in domains with inflow/outflow boundary conditions are unconditionally stable. This means that any not necessarily radially symmetric solution of the associated evolutionary problem converges to a single radially symmetric steady state.
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.
It is well-known that one can construct solutions to the nonlocal Cahn-Hilliard equation with singular potentials via Yosida approximation with parameter $λ\to 0$. The usual method is based on compactness arguments and does not provide any rate of convergence. Here, we fill the gap and we obtain an explicit convergence rate $\sqrtλ$. The proof is based on the theory of maximal monotone operators and an observation that the nonlocal operator is of Hilbert-Schmidt type. Our estimate can provide convergence result for the Galerkin methods where the parameter $λ$ could be linked to the discretization parameters, yielding appropriate error estimates.
This paper presents a mathematical framework for modeling the dynamics of heterogeneous populations. Models describing local and non-local growth and transport processes, dependent on dynamically changing population structures, appear in a variety of applications such as crowd dynamics, tissue regeneration, cancer development, and coagulation-fragmentation processes. The current body of literature regarding mathematical modeling presents common challenges to mathematicians due to the multiscale nature of the structures that underlie self-organisation and control within complex, heterogeneous systems. In various applications, similar, abstract mathematical concepts arise through problem formulation and the assimilation of mathematical depictions into the language of measure evolution on a multi-faceted state space. In view of the above observations, we propose an overarching mathematical framework for nonlinear structured population models on abstract metric spaces, which are only assumed to be separable and complete. To achieve this, we exploit the structure of the space of non-negative Radon measures under the dual bounded Lipschitz distance (flat metric), a generalization of the Wasserstein distance that is capable of addressing non-conservative problems. The formulation of models on generic metric spaces facilitates the study on infinite-dimensional state spaces or graphs with a combination of discrete and continuous structures. This opens up exciting possibilities for modeling single cell data, crowd dynamics or coagulation-fragmentation processes.
Several recent papers considered the high-friction limit for systems arising in fluid mechanics. Following this approach, we rigorously derive the nonlocal Cahn-Hilliard equation as a limit of the nonlocal Euler-Korteweg equation using the relative entropy method. Applying the recent result by the first and third author, we also derive rigorously the local degenerate Cahn-Hilliard equation. The proof is formulated for dissipative measure-valued solutions of the nonlocal Euler-Korteweg equation which are known to exist on arbitrary intervals of time. Our work provides a new method to derive equations not enjoying classical solutions via relative entropy method by introducing the nonlocal effect in the fluid equation.
We consider a combined system of Euler--Korteweg and Euler--Poisson equations with friction and exponential pressure with exponent $γ> 1$. We show the existence of dissipative measure-valued solutions in the cases of repulsive and attractive potential in Euler--Poisson system. The latter case requires additional restriction on $γ$. Furthermore in case of $γ\geq 2$ we show that the strong solutions to the Cahn--Hillard--Keller--Segel system are a high-friction limit of the dissipative measure-valued solutions to Euler--Korteweg--Poisson equations.
We propose a new concept of weak solution to the equations of compressible magnetohydrodynamics driven by large boundary data. The system of the underlying field equations is solvable globally in time in the out of equilibrium regime characteristic for turbulence. The weak solutions comply with the weak-strong uniqueness principle; they coincide with the classical solution of the problem as long as the latter exists. The choice of constitutive relations is motivated by applications in stellar magnetoconvection.
We consider the multi-dimensional generalization of the Aw-Rascle system for vehicular traffic. For an arbitrary large class of initial data and the periodic domain, we prove the existence of global-in-time measure-valued solutions. Moreover, using the relative energy technique, we show that the measure-valued solutions coincide with the classical solutions as long as the latter exist.
We consider the system of equations describing the flow of incompressible fluids in bounded domain. In the considered setting, the Cauchy stress tensor is a monotone mapping and has asymptotically $(s-1)$-growth with the parameter $s$ depending on the spatial and time variable. We do not assume any smoothness of $s$ with respect to time variable and assume the log-Hölder continuity with respect to spatial variable. Such a setting is a natural choice if the material properties are instantaneously, e.g. by the switched electric field. We establish the long time and the large data existence of weak solution provided that $s\ge(3d+2)(d+2)$.