arXiv subjects

Ewelina Zatorska

Publications and source records attributed to Ewelina Zatorska.

At least 19 recordsLinked to original sources

arXiv2026

Dissipative measure-valued solutions and weak--strong uniqueness for a viscous Baer--Nunziato system with pressure relaxation

We study a viscous one-velocity Baer--Nunziato system for two barotropic compressible fluids in a bounded three-dimensional domain. In contrast with models in which the volume fraction is merely transported or determined by an algebraic equilibrium constraint, it satisfies a differential closure driven by the pressure gap between the phases. For arbitrary finite-energy initial data and adiabatic exponents $γ^\pm>1$, we construct global-in-time dissipative measure-valued solutions and prove dissipative measure-valued--strong uniqueness relative to any sufficiently regular solution with the same initial data. The principal difficulties are the singular and non-continuous behavior of the pressure-relaxation source at vanishing volume fractions, the associated concentration defects, and the lack of direct coercivity of the standard thermodynamic relative energy with respect to the volume fraction. These are resolved by an endpoint cutoff argument and an augmented relative energy coupled to the renormalized volume-fraction equation.

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arXiv2026

Low Mach Number Limit and Convergence Rates for a Compressible Two-Fluid Model with Algebraic Pressure Closure

We study the low Mach number limit for a viscous compressible two-fluid model with algebraic pressure closure in the three-dimensional torus $\mathbb{T}^3$. The pressure is determined implicitly through the densities of the two phases, which makes the singular limit substantially more delicate than for models with explicit pressure laws. Working in the framework of local-in-time strong solutions, we prove that, for well-prepared initial data, solutions to the rescaled compressible two-fluid system exist on a time interval independent of the Mach number and converge to the solution of the incompressible Navier--Stokes equations as the Mach number tends to zero. In addition, we establish explicit convergence rates for the densities and the velocity field. The proof relies on uniform high-order energy estimates and a relative energy argument adapted to the implicit structure of the pressure law. These results provide a rigorous justification of the low Mach number limit for the compressible two-fluid model with algebraic pressure closure.

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arXiv2026

On a PDE model for Learning in Stochastic Market Entry Games

We study a continuum model for stochastic reinforcement learning in repeated market entry games. Starting from a discrete-time microscopic learning rule, we derive a Fokker--Planck-type equation for the distribution of agents' propensities and, using a kinetic closure, obtain a nonlinear one-particle equation of a mean-field type. For the resulting Cauchy problem, we prove existence and uniqueness of solutions and analyze their long-time behavior. The PDE captures two key phenomena observed in market entry dynamics: aggregate learning (the average number of entrants approaches market capacity) and sorting (propensities concentrate near extreme behaviors). The model also yields explicit characteristic time scales, showing that aggregate learning occurs faster than sorting, in agreement with experimental and computational evidence.

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arXiv2026

Weak-strong uniqueness for bi-fluid compressible system with algebraic closure

We consider a real two-fluid system of compressible viscous fluids with a common velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The existence of global-in-time finite energy weak solutions to this system is known since the work of Novotný and Pokorný [Arch. Rational Mech. Anal., 2020]. On the other hand, existence of local-in-time strong solutions is due to Piasecki and Zatorska [J. Math Fluid Mech., 2022]. In this paper, we establish the weak-strong uniqueness principle using the relative entropy method. In sharp contrast to the two-phase model of Baer-Nunziato type, the volume fraction of phase $+$ obeys a transport equation with an additional nonlinear term. This gives rise to troublesome terms in the relative entropy inequality. We are able to close the estimate by making an elaborate use of the structure of the system.

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arXiv2026

Microscopic derivation of a one-dimensional lubrication model with roughness

We derive a hydrodynamic model for the motion of inertial particles with a spherical hard core, interacting through lubrication forces and pairwise repulsive forces. The repulsion arises from the assumption that each particle is surrounded by a thin rough layer of reduced permeability. We prove that, as the number of particles tends to infinity (and their size tends to 0), the microscopic dynamics converges to a macroscopic hydrodynamic model in which congestion effects are encoded directly into the macroscopic interaction forces, depending on a local critical density transported by the flow. In particular, we extend the work of Lefebvre-Lepot and Maury where non-inertial particles, submitted to only a lubrication force were considered, and present the convergence proof when inertial effects and roughness are taken into account.

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arXiv2025

Anelastic approximation for the degenerate compressible Navier--Stokes equations revisited

In this paper, we revisit the joint low-Mach and low-Frode number limit for the compressible Navier-Stokes equations with degenerate, density-dependent viscosity. Employing the relative entropy framework based on the concept of $κ$-entropy, we rigorously justify the convergence of weak solutions toward the generalized anelastic system in a three-dimensional periodic domain for well-prepared initial data. For general ill-prepared initial data, we establish a similar convergence result in the whole space, relying essentially on dispersive estimates for acoustic waves. Compared with the work of Fanelli and Zatorska [Commun. Math. Phys., 400 (2023), pp. 1463-1506], our analysis is conducted for the standard isentropic pressure law, thereby eliminating the need for the cold pressure term that played a crucial role in the previous approach. To the best of our knowledge, this is the first rigorous singular limit result for the compressible Navier-Stokes equations with degenerate viscosity that requires no additional regularization of the system.

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arXiv2025

Existence and weak-strong uniqueness of measure solutions to Euler-alignment/Aw--Rascle--Zhang model of collective behaviour

We study the multi-dimensional Euler-alignment system with a matrix-valued communication kernel, motivated by models of anticipation dynamics in collective behaviour. A key feature of this system is its formal equivalence to a nonlocal variant of the Aw--Rascle--Zhang (ARZ) traffic model, in which the desired velocity is modified by a nonlocal gradient interaction. We prove the global-in-time existence of measure solutions to both formulations, obtained via a single degenerate pressureless Navier--Stokes approximation. Furthermore, we establish a weak-strong uniqueness principle adapted to the pressureless setting and to nonlocal alignment forces. As a consequence, we rigorously justify the formal correspondence between the nonlocal ARZ and Euler-alignment models: they arise from the same inviscid limit, and the weak-strong uniqueness property ensures that, whenever a classical solution exists, both formulations coincide with it.

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arXiv2025

Recent advances in the analysis of the dissipative Aw-Rascle system

The one-dimensional Aw-Rascle (AR) system has become a cornerstone of macroscopic models for single-lane vehicular traffic. A possible generalization of this model to a multi-dimensional setting is the so-called dissipative AR model, which is more suited to capturing crowd dynamics. This review summarizes recent studies that analyze the dissipative AR model, its hard congestion limit, the non-uniqueness of weak solutions, the existence and asymptotics of solutions within the duality framework, non-local interactions, and the existence of regular solutions.

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arXiv2024

Regular solutions to the dissipative Aw-Rascle system

In this paper we prove the local-in-time existence of regular solutions to dissipative Aw-Rascle system with the offset equal to gradient of some increasing and regular function of density. It is a mixed degenerate parabolic-hyperbolic hydrodynamic model, and we extend the techniques previously developed for compressible Navier-Stokes equations to show the well-posedness of the system in the $L_2-L_2$ setting. We also discuss relevant existence results for offset involving singular or nonlocal functions of density.

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arXiv2024

Non-local dissipative Aw-Rascle model and its relation with Matrix-valued communication in Euler alignment

We compare the multi-dimensional generalisation of the Aw-Rascle model with the pressureless Euler-alignment system, in which the communication weight is matrix-valued. Our generalisation includes the velocity offset in the form of a gradient of a non-local density function, given by the convolution with a kernel $K$. We investigate connections between these models at the macroscopic, mesoscopic and macroscopic (hydrodynamic) level, and overview the results on the mean-field limit for various assumptions on $K$.

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arXiv2024

On thermally driven fluid flows arising in astrophysics

We consider the Navier-Stokes-Fourier-Poisson system driven by an inhomogeneous temperature distribution on the boundary of an exterior fluid domain. We impose the finite mass constraint, positive far field condition for the temperature as well as the no--slip boundary conditions for the velocity. The existence of global--in--time weak solutions and the weak-strong principle are proved.

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arXiv2024

Global solutions of the one-dimensional compressible Euler equations with nonlocal interactions via the inviscid limit

We are concerned with the global existence of finite-energy entropy solutions of the one-dimensional compressible Euler equations with (possibly) damping, alignment forces, and nonlocal interactions: Newtonian repulsion and quadratic confinement. Both the polytropic gas law and the general gas law are analyzed. This is achieved by constructing a sequence of solutions of the one-dimensional compressible Navier-Stokes-type equations with density-dependent viscosity under the stress-free boundary condition and then taking the vanishing viscosity limit. The main difficulties in this paper arise from the appearance of the nonlocal terms. In particular, some uniform higher moment estimates for the compressible Navier-Stokes equations on expanding intervals with stress-free boundary conditions are obtained by careful design of the approximate initial data.

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arXiv2024

Construction of weak solutions to a pressureless viscous model driven by nonlocal attraction-repulsion

We analyze the pressureless Navier-Stokes system with nonlocal attraction-repulsion forces. Such systems appear in the context of models of collective behavior. We prove the existence of weak solutions on the whole space $\mathbb{R}^3$ in the case of density-dependent degenerate viscosity. For the nonlocal term it is assumed that the interaction kernel has the quadratic growth at infinity and almost quadratic singularity at zero. Under these assumptions, we derive the analog of the Bresch-Desjardins and Mellet-Vasseur estimates for the nonlocal system. In particular, we are able to adapt the approach of Vasseur and Yu to construct a weak solution.

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arXiv2024

Duality solutions to the hard-congestion model for the dissipative Aw-Rascle system

We introduce the notion of duality solution for the hard-congestion model on the real line, and additionally prove an existence result for this class of solutions. Our study revolves around the analysis of a generalised Aw-Rascle system, where the offset function is replaced by the gradient of a singular function, such as $ρ$ $γ$ n , where $γ$ $\rightarrow$ $\infty$. We prove that under suitable assumptions on the initial data, solutions to the Aw-Rascle system converge towards the so-called duality solutions, which have previously found applications in other systems which exhibit compressive dynamics. We also prove that one can obtain weak solutions to the limiting system under stricter assumptions on the initial data. Finally, we discuss (non-)uniqueness issues.

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arXiv2024

Existence of weak solutions and long-time asymptotics for hydrodynamic model of swarming

We consider a one-dimensional hydrodynamic model featuring nonlocal attraction-repulsion interactions and singular velocity alignment. We introduce a two-velocity reformulation and the corresponding energy-type inequality, in the spirit of the Bresch-Desjardins estimate. We identify a dependence between the communication weight and interaction kernel and between the pressure and viscosity term allowing for this inequality to be uniform in time. It is then used to study long-time asymptotics of solutions.

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arXiv2023

Analysis of the generalised Aw-Rascle model

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.

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arXiv2023

A new construction of weak solutions to compressible Navier-Stokes equations

We prove the existence of the weak solutions to the compressible Navier--Stokes system with barotropic pressure $p(\varrho)=\varrho^γ$ for $γ\geq 9/5$ in three space dimension. The novelty of the paper is the approximation scheme that instead of the classical regularization of the continuity equation (based on the viscosity approximation $\ep Δ\varrho$) uses more direct truncation and regularisation of nonlinear terms an the pressure. This scheme is compatible with the Bresch-Jabin compactness criterion for the density. We revisit this criterion and prove, in full rigour, that it can be applied in our approximation at any level.

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arXiv2022

Nonuniqueness of weak solutions to the dissipative Aw-Rascle model

We prove nonuniqueness of weak solutions to multi-dimensional generalisation of the Aw-Rascle model of vehicular traffic. Our generalisation includes the velocity offset in a form of gradient of density function, which results in a dissipation effect, similar to viscous dissipation in the compressible viscous fluid models. We show that despite this dissipation, the extension of the method of convex integration can be applied to generate infinitely many weak solutions connecting arbitrary initial and final states. We also show that for certain choice of data, ill posedness holds in the class of admissible weak solutions.

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