arXiv subjects

Piotr B. Mucha

Publications and source records attributed to Piotr B. Mucha.

At least 19 recordsLinked to original sources

arXiv2025

Free Boundary Problem for inhomogeneous Navier-Stokes equations

We study free boundary problems for incompressible inhomogeneous flows governed by the Navier--Stokes equations, focusing on the regularity and global-in-time well-posedness of solutions in critical functional frameworks for small initial data. We introduce a novel analytical framework for free boundary problems formulated as perturbations of the half-space. Our approach relies on the natural Lagrangian change of coordinates and a detailed analysis of the linearized problem (the Stokes system) in the maximal regularity regime, formulated in the Lebesgue spaces $L_p(0,T; L_q)$, including time-weighted variants. The main difficulty lies in the treatment of boundary terms, for which we apply a new technique based on complex interpolation to control nonlinear terms in fractional Sobolev spaces. This strategy also allows us to handle the case of variable density, which is not easily addressed by approaches based on Besov spaces. Using this framework and real interpolation techniques, we construct also solutions in the Lorentz class $L_{p,1}(0,T; L_q)$ in time. The method further enables a rigorous study of the stability of equilibrium configurations. In particular, we resolve the problem in two spatial dimensions, where the interplay between geometry and regularity is especially subtle. Beyond these specific applications, the proposed approach provides a powerful tool for broader classes of nonlinear PDEs and further developments in maximal regularity theory.

math.AP↗

arXiv2025

A multifluid model with chemically reacting components -- construction of weak solutions

We investigate the existence of weak solutions to a multi-component system, consisting of compressible chemically reacting components, coupled with the compressible Stokes equation for the velocity. Specifically, we consider the case of irreversible chemical reactions and assume a nonlinear relation between the pressure and the particular densities. These assumptions cause the additional difficulties in the mathematical analysis, due to the possible presence of vacuum. It is shown that there exists a global weak solution, satisfying the $L^\infty$ bounds for all the components. We obtain strong compactness of the sequence of densities in $L^p$ spaces, under the assumption that all components are strictly positive. The applied method captures the properties of models of high generality, which admit an arbitrary number of components. Furthermore, the framework that we develop can handle models that contain both diffusing and non-diffusing elements.

math.AP↗

arXiv2024

Inhomogenous Navier--Stokes equations with unbounded density

In the current state of the art regarding the Navier--Stokes equations, the existence of unique solutions for incompressible flows in two spatial dimensions is already well-established. Recently, these results have been extended to models with variable density, maintaining positive outcomes for merely bounded densities, even in cases with large vacuum regions. However, the study of incompressible Navier-Stokes equations with unbounded densities remains incomplete. Addressing this gap is the focus of the present paper. Our main result demonstrates the global existence of a unique solution for flows initiated by unbounded density, whose regularity/integrability is characterized within a specific subset of the Yudovich class of unbounded functions. The core of our proof lies in the application of Desjardins' inequality, combined with a blow-up criterion for ordinary differential equations. Furthermore, we derive time-weighted estimates that guarantee the existence of a $C^1$ velocity field and ensure the equivalence of Eulerian and Lagrangian formulations of the equations. Finally, by leveraging results from \cite{DanMu}, we conclude the uniqueness of the solution.

math.AP↗

arXiv2024

Construction of weak solutions to the equations of a compressible viscous model

The paper aims on the construction of weak solutions to equations of a model of compressible viscous fluids, being a simplification of the classical compressible Navier-Stokes system. We present a novel scheme for approximating systems that preserves structural integrity by avoiding classical regularization with $ - \varepsilon Δ\varrho $, thus maintaining the transport character of the continuity equation. Our approach, which necessitates specific conditions on the constitutive equation, accommodates physically relevant models such as isentropic and van der Waals gases, and globally handles non-monotone pressures. From an analytical perspective, our method synthesizes techniques from Feireisl-Lions and Bresch-Jabin to demonstrate the convergence of approximate densities using compensated compactness techniques. We also apply renormalization of the continuity equations and utilize weight techniques to manage unfavorable terms.

math.AP↗

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.

math.AP↗

arXiv2024

Comparative Analysis of Obstacle Approximation Strategies for the Steady Incompressible Navier-Stokes Equations

This paper aims to compare and evaluate various obstacle approximation techniques employed in the context of the steady incompressible Navier-Stokes equations. Specifically, we investigate the effectiveness of a standard volume penalization approximation and an approximation method utilizing high viscosity inside the obstacle region, as well as their composition. Analytical results concerning the convergence rate of these approaches are provided, and extensive numerical experiments are conducted to validate their performance.

math.AP↗

arXiv2023

A Polarization Opinion Model Inspired by Bounded Confidence Communications

We present an opinion model founded upon the principles of the bounded confidence interaction among agents. Our objective is to explain the polarization effects inherent to vector-valued opinions. The evolutionary process adheres to the rule where each agent aspires to increase polarization through communication with a single friend during each discrete time step. The dynamics ensure that agents' ultimate (temporal) configuration will encompass a finite number of outlier states. We introduce deterministic and stochastic models, accompanied by a comprehensive mathematical analysis of their inherent properties. Additionally, we provide compelling illustrative examples and introduce a stochastic solver tailored for scenarios featuring an extensive set of agents. Furthermore, in the context of smaller agent populations, we scrutinize the suitability of neural networks for the rapid inference of limit configurations.

cs.MA↗

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.

math.AP↗

arXiv2022

A fuzzy $q$-closest alignment model

The paper examines the problems related to the well-posedness of the Cucker-Smale model with communication restricted to the $q$-closest neighbors, known also as the Cucker-Dong model. With agents oscillating on the boundary of different clusters, the system becomes difficult to precisely define, which leads to further problems with kinetic limits as the number of agents tends to infinity. We introduce the fuzzy $q$-closest system, which circumvents the issues with well-posedness. For such a system we prove a stability estimate for measure-valued solutions and perform the kinetic mean-field limit.

math.AP↗

arXiv2022

Reacting multi-component fluids -- regular solutions in Lorentz spaces

The paper deals with analysis of a model of a multi-component fluid admitting chemical reactions. The flow is considered in the incompressible regime. The main result shows global existence of regular solutions under assumption of suitable smallness conditions. In order to control the solutions a special structure condition on the derivatives of chemical production functions determining the reactions is required. The existence is shown in a new critical functional framework of Lorentz spaces of type $L_{p,r}(0,T;L_q)$, which allows to control the integral $\int_0^\infty \|\nabla u(t)\|_{\infty} dt$.

math.AP↗

arXiv2021

Stability of the density patches problem with vacuum for incompressible inhomogeneous viscous flows

We consider the inhomogeneous incompressible Navier-Stokes system in a smooth two or three dimensional bounded domain, in the case where the initial density is only bounded. Existence and uniqueness for such initial data was shown recently in [10], but the stability issue was left open. After observing that the solutions constructed in [10] have exponential decay, a result of independent interest, we prove the stability with respect to initial data, first in Lagrangian coordinates, and then in the Eulerian frame. We actually obtain stability in $L_2({\mathbb R}_+;H^1(Ω))$ for the velocity and in a negative Sobolev space for the density. Let us underline that, as opposed to prior works, in case of vacuum, our stability estimates are not weighted by the initial densities. Hence, our result applies in particular to the classical density patches problem, where the density is a characteristic function.

math.AP↗

arXiv2021

Free Boundary Problems via Da Prato-Grisvard Theory

An $\mathrm{L}_1$-maximal regularity theory for parabolic evolution equations inspired by the pioneering work of Da Prato and Grisvard is developed. Besides of its own interest, the approach yields a framework allowing global-in-time control of the change of Eulerian to Lagrangian coordinates in various problems related to fluid mechanics. This property is of course decisive for free boundary problems. This concept is illustrated by the analysis of the free boundary value problem describing the motion of viscous, incompressible Newtonian fluids without surface tension and, secondly, the motion of compressible pressureless gases. For this purpose, an endpoint maximal $\mathrm{L}_1$-regularity approach to the Stokes and Lamé systems is developed. It is applied then to establish global, strong well-posedness results for the free boundary problems described above in the case where the initial domain coincides with the half-space, and the initial velocity is small with respect to a suitable scaling invariant norm.

math.AP↗

arXiv2020

A new SEIR type model including quarantine effects and its application to analysis of Covid-19 pandemia in Poland in March-April 2020

Contact tracing and quarantine are well established non-pharmaceutical epidemic control tools. The paper aims to clarify the impact of these measures in COVID-19 epidemic. A new deterministic model is introduced (SEIRQ: susceptible, exposed, infectious, removed, quarantined) with Q compartment capturing individuals and releasing them with delay. We obtain a simple rule defining the reproduction number $\mathcal{R}$ in terms of quarantine parameters, ratio of diagnosed cases and transmission parameters. The model is applied to the epidemic in Poland in March - April 2020, when social distancing measures were in place. We investigate 3 scenarios corresponding to different ratios of diagnosed cases. Our results show that depending on the scenario contact tracing could have prevented from 50\% to over 90\% of cases. The effects of quarantine are limited by fraction of undiagnosed cases. Taking into account the transmission intensity in Poland prior to introduction of social restrictions it is unlikely that the control of the epidemic could be achieved without any social distancing measures.

q-bio.PE↗

arXiv2020

Density-induced Consensus Protocol

The paper introduces a model of collective behavior where agents receive information only from sufficiently dense crowds in their immediate vicinity. The system is an asymmetric, density-induced version of the Cucker-Smale model with short-range interactions. We prove the basic mathematical properties of the system and concentrate on the presentation of interesting behaviors of the solutions. The results are illustrated by numerical simulations.

math.AP↗

arXiv2020

Boussinesq system with measure forcing

We address a question concerning the issue of existence to a Boussinesq type system with a heat source. The problem is studied in the whole two dimensional plane and the heat source is a measure transported by the flow. For arbitrary initial data, we prove global in time existence of unique regular solutions. Measure being a heat source limits regularity of constructing solutions and make us work in a non-standard framework of inhomogeneous Besov spaces of the $L^\infty(0,T;B^s_{p,\infty})$-type. Application of the Lagrangian coordinates yields uniqueness omitting difficulties with comparison of measures.

math.AP↗

arXiv2019

Stationary compressible Navier Stokes Equations with inflow condition in domains with piecewise analytical boundaries

We show the existence of strong solutions in Sobolev-Slobodetskii spaces to the stationary compressible Navier-Stokes equations with inflow boundary condition. Our result holds provided certain condition on the shape of the boundary around the points where characteristics of the continuity equation are tangent to the boundary, which holds in particular for piecewise analytical boundaries. The mentioned situation creates a singularity which limits regularity at such points. We show the existence and uniqueness of regular solutions in a vicinity of given laminar solutions under the assumption that the pressure is a linear function of the density. The proofs require the language of suitable fractional Sobolev spaces. In other words our result is an example where application of fractional spaces is irreplaceable, although the subject is a classical system.

math.AP↗

arXiv2019

Between homogeneous and inhomogeneous Navier-Stokes systems: the issue of stability

We construct large velocity vector solutions to the three dimensional inhomogeneous Navier-Stokes system. The result is proved via the stability of two dimensional solutions with constant density, under the assumption that initial density is point-wisely close to a constant. Key elements of our approach are estimates in the maximal regularity regime and the Lagrangian coordinates. Considerations are done in the whole $\R^3$.

math.AP↗

arXiv2018

Finite-energy solutions for compressible two-fluid Stokes system

We prove existence of global in time weak solutions to a compressible two-fluid Stokes system with a single velocity field and algebraic closure for the pressure law. The constitutive relation involves densities of both fluids through an implicit function. The system appears to be outside the class of problems that can be treated using the classical Lions-Feireisl approach. Adapting the novel compactness tool developed by the first author and P.-E. Jabin in the mono-fluid compressible Navier-Stokes setting, we first prove the weak sequential stability of solutions. Next, we construct weak solutions via unconventional approximation using the Lagrangian formulation, truncations and stability result of trajectories for rough velocity fields.

math.AP↗