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Tomasz Piasecki

Publications and source records attributed to Tomasz Piasecki.

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

Steady compressible Navier-Stokes-Fourier system with general temperature dependent viscosities I: density estimates based on Bogovskii operator

The aim of this paper is to reconsider the existence theory for steady compressible Navier--Stokes--Fourier system assuming more general condition of the dependence of the viscosities on the temperature in the form $\mu(\vartheta)$, $\xi(\vartheta) \sim (1+\vartheta)^\alpha$ for $0\leq \alpha \leq 1$. This extends the known theory for $\alpha=1$ from and improves significantly the results for $\alpha =0$. This paper is the first of a series of two papers dealing with this problem and is connected with the Bogovskii-type estimates of the sequence of densities. This leads, among others, to the limitation $\gamma >\frac 32$ for the pressure law $p(\varrho,\vartheta) \sim \varrho^\gamma + \varrho\vartheta$. The paper considers both the heat-flux (Robin) and Dirichlet boundary conditions for the temperature as well as both the homogeneous Dirichlet and zero inflow/outflow Navier boundary conditions for the velocity. Further extension for $\gamma >1$ only is based on different type of pressure estimates and will be the content of the subsequent paper.

math.AP↗
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.

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↗
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 $\int_0^\infty \|\nabla u(t)\|_{\infty} dt$.

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arXiv2022

Maximal Regularity for Compressible Two-Fluid System

We investigate a compressible two-fluid Navier-Stokes type 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. We are interested in regular solutions in a $L_p-L_q$ maximal regularity setting. We show that such solutions exists locally in time and, under additional smallness assumptions on the initial data, also globally. Our proof rely on appropriate transformation of the original problem, application of Lagrangian coordinates and maximal regularity estimates for associated linear problem.

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↗
arXiv2020

Compressible Navier-Stokes system on a moving domain in the $L_p-L_q$ framework

We prove the local well-posedness for the barotropic compressible Navier-Stokes system on a moving domain, a motion of which is determined by a given vector field ${\bf V}$, in a maximal $L_p-L_q$ regularity framework. Under additional smallness assumptions on the data we show that our solution exists globally in time and satisfies a decay estimate. In particular, for the global well-posedness we don't require exponential decay or smallness of ${\bf V}$ in $L_p(L_q)$. However, we require exponential decay and smallness of its derivatives.

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↗
arXiv2019

On the isothermal compressible multi-component mixture flow: the local existence and maximal $L_p-L_q$ regularity of solutions

We consider the initial-boundary value problem for the system of equations describing the flow of compressible isothermal mixture of arbitrary large number of components. The system consists of the compressible Navier-Stokes equations and a subsystem of diffusion equations for the species. The subsystems are coupled by the form of the pressure and the strong cross-diffusion effects in the diffusion fluxes of the species. Assuming the existence of solutions to the symmetrized and linearized equations, proven in [35], we derive the estimates for the nonlinear equations and prove the local-in-time existence and maximal $L_p-L_q$ regularity of solutions.

math.AP↗
arXiv2019

On the maximal $L_p$-$L_q$ regularity of solutions to a general linear parabolic system

We show the existence of solution in the maximal $L_p-L_q$ regularity framework to a class of symmetric parabolic problems on a uniformly $C^2$ domain in ${\mathcal R}$. Our approach consist in showing ${\mathcal R}$ - boundedness of families of solution operators to corresponding resolvent problems first in the whole space, then in half-space, perturbed half-space and finally, using localization arguments, on the domain. Assuming additionally boudedness of the domain we also show exponential decay of the solution. In particular, our approach does not require assuming a priori the uniform Lopatinskii - Shapiro condition.

math.AP↗
arXiv2019

Weak-strong uniqueness for the compressible fluid-rigid body interaction

In this work we study the coupled system of partial and ordinary differential equations describing the interaction between a compressible isentropic viscous fluid and a rigid body moving freely inside the fluid. In particular the position and velocity of the rigid body in the fluid are unknown and the motion of the rigid body is driven by the normal stress forces of the fluid acting on the boundary of the body. We prove that the strong solution, which is known to exist under certain smallness assumptions, is unique in the class of weak solutions to the problem. The proof relies on a correct definition of the relative energy, to use this tool we then have to introduce a change of coordinates to transform the strong solution to the domain of the weak solution in order to use it as a test function in the relative energy inequality. Estimating all arising terms we prove that the weak solution has to coincide with the transformed strong solution and finally that the transformation has to be in fact an identity.

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.

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arXiv2018

On strong dynamics of compressible two-component mixture flow

We investigate a system describing the flow of a compressible two-component mixture. The system is composed of the compressible Navier-Stokes equations coupled with non-symmetric reaction-diffusion equations describing the evolution of fractional masses. We show the local existence and, under certain smallness assumptions, also the global existence of unique strong solutions in $L_p-L_q$ framework. Our approach is based on so called entropic variables which enable to rewrite the system in a symmetric form. Then, applying Lagrangian coordinates, we show the local existence of solutions applying the $L_p$-$L_q$ maximal regularity estimate. Next, applying exponential decay estimate we show that the solution exists globally in time provided the initial data is sufficiently close to some constants. The nonlinear estimates impose restrictions $2<p<\infty, \ 3<q<\infty$. However, for the purpose of generality we show the linear estimates for wider range of $p$ and $q$.

math.AP↗
arXiv2018

Local existence of strong solutions and weak-strong uniqueness for the compressible Navier-Stokes system on moving domains

We consider the compressible Navier-Stokes system on time-dependent domains with prescribed motion of the boundary. For both the no-slip boundary conditions as well as slip boundary conditions we prove local-in-time existence of strong solutions. These results are obtained using a transformation of the problem to a fixed domain and an existence theorem for Navier-Stokes like systems with lower order terms and perturbed boundary conditions. We also show the weak-strong uniqueness principle for slip boundary conditions which remained so far open question.

math.AP↗
arXiv2017

On steady solutions to a model of chemically reacting heat conducting compressible mixture with slip boundary conditions

We consider a model of chemically reacting heat conducting compressible mixture. We investigate the corresponding system of partial differential equations in the steady regime with slip boundary conditions for the velocity and, in dependence on the model parameters, we establish existence of either weak or variational entropy solutions. The results extend the range of parameters for which the existence of weak solutions is known in the case of homogeneous Dirichlet boundary conditions for the velocity.

math.AP↗
arXiv2016

Weak and variational entropy solutions to the system describing steady flow of a compressible reactive mixture

We consider a system of partial differential equations which describes steady flow of a compressible heat conducting chemically reacting gaseous mixture. We extend the result from Giovangigli, Pokorný, Zatorska (2015) in the sense that we introduce the variational entropy solution for this model and prove existence of a weak solution for $γ>\frac 43$ and existence of a variational entropy solution for any $γ>1$. The proof is based on improved density estimates.

math.AP↗