arXiv subjects

Jakub Woźnicki

Publications and source records attributed to Jakub Woźnicki.

10 recordsLinked to original sources

arXiv2026

On three-dimensional flows of thermo-viscoelastic fluids of Giesekus type

Viscoelastic rate-type fluid models constitute a fundamental framework for the mathematical description of complex materials exhibiting coupled elastic and viscous effects, with a wide range of applications in engineering, biomaterials, and medicine. In realistic regimes, thermal effects are essential and lead to strongly coupled systems in which heat conduction and temperature-dependent constitutive laws play a decisive role. In this paper, we develop a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids. We establish the existence of a global weak solution in the full three-dimensional setting. In contrast to the existing literature, no smallness, regularity, or structural restrictions on the initial data are imposed beyond natural energy and entropy bounds, and no additional regularising mechanisms such as artificial stress diffusion are required. The analysis is based on a weak-strong framework combining energy and entropy balances with compactness tools, allowing us to treat the full nonlinear coupling between the fluid velocity, the temperature, and the elastic stress.

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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

Vlasov equation coupled with non-Newtonian fluids with discontinuous-in-time stress tensor

We analyze the system of equations describing the flow of a dilute particle system coupled with an incompressible non-Newtonian fluid in a bounded domain. In this setting, both PDEs are connected via a drag force, or the friction force. We are interested in a thin spray regime, which means that we neglect the inter-particle interactions. When it comes to the fluid system, the Cauchy stress tensor is supposed to be 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. An example of materials, which satisfy those assumptions, are those whose properties are instantaneous, e.g. changed by the switched electric field. We show the long time and the large data existence of weak solution provided that $s\ge\frac{3d+2}{d+2}$.

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arXiv2025

Strong Convergence of Vorticities in the 2D Viscosity Limit on a Bounded Domain

In the vanishing viscosity limit from the Navier-Stokes to Euler equations on domains with boundaries, a main difficulty comes from the mismatch of boundary conditions and, consequently, the possible formation of a boundary layer. Within a purely interior framework, Constantin and Vicol showed that the two-dimensional viscosity limit is justified for any arbitrary but finite time under the assumption that on each compactly contained subset of the domain, the enstrophies are bounded uniformly along the viscosity sequence. Within this framework, we upgrade to local strong convergence of the vorticities under a similar assumption on the $p$-enstrophies, $p>2$. The key novel idea is the analysis of the evolution of the weak convergence defect.

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arXiv2025

Stochastic parabolic equations in Musielak-Orlicz spaces with discontinuous in time N-function

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.

math.AP↗

arXiv2025

On various models describing behaviour of thermoviscoelastic rate-type fluids

Viscoelastic rate-type fluid models are essential for describing the behavior of a wide range of complex materials, with applications in fields such as engineering, biomaterials, and medicine. These models are particularly useful for understanding the rheological properties of materials that exhibit both elastic and viscous behavior under deformation. However, many real-world applications involve significant thermal effects, where heat conduction and the temperature dependence of material properties must also be considered. In this paper, we introduce a thermodynamically consistent model for heat-conducting viscoelastic rate-type fluids and establish the existence of a global weak solution in a two-dimensional setting. The result holds under the condition that the initial energy and entropy are controlled in appropriate natural norms.

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arXiv2023

Parabolic equations with non-standard growth and measure or integrable data

We consider a parabolic partial differential equation with Dirichlet boundary conditions and measure or $L^1$ data. The key difficulty consists in a presence of a monotone operator~$A$ subjected to a non-standard growth condition, controlled by the exponent $p$ depending on the time and the spatial variable. We show the existence of a weak and an entropy solution to our system, as well as the uniqueness of an entropy solution, under the assumption of boundedness and log-Hölder continuity of the variable exponent~$p$ with respect to the spatial variable. On the other hand, we do not assume any smoothness of~$p$ with respect to the time variable.

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arXiv2023

Cahn-Hillard and Keller-Segel systems as high-friction limits of Euler-Korteweg and Euler-Poisson equations

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.

math.AP↗

arXiv2022

Non-Newtonian fluids with discontinuous-in-time stress tensor

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 $s\ge(3d+2)(d+2)$.

math.AP↗

arXiv2021

Mv-strong uniqueness for density dependent, non-Newtonian, incompressible fluids

We consider density dependent, non-Newtonian, incompressible system with the space being flat torus. The viscious stress in the momentum equation is understood through the rheological law and its connection to the proper convex potential. We define the dissipative measure-valued solutions for the aforementioned equations as well as provide a proof of its existence. The main result of this work is the mv-strong uniqueness of the defined solutions.

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