arXiv subjects

Maja Szlenk

Publications and source records attributed to Maja Szlenk.

9 recordsLinked to original sources

arXiv2026

Dissipative Solutions to a Compressible Non-Newtonian Korteweg System with Density-Dependent Viscous Stress Tensor

The main objective of this paper is to prove that if capillarity effect is taken into account then there exist dissipative solutions to a system describing viscoplastic compressible flows with density dependent viscosities in a periodic domain $\T^d$ with $d=2,3$. We calculate the relative entropy inequality and in consequence show existence of dissipative solutions and the weak-strong uniqueness for this system. Our result extends the recent result concerning the link between Euler--Korteweg and Navier--Stokes--Korteweg systems for Newtonian flows (when the viscosity depends on the density) [See D.~Bresch, M. Gisclon, I. Lacroix-Violet, {\it Arch. Rational Mech. Anal.} (2019)] to non-Newtonian flows.

math.AP↗

arXiv2026

Weak solutions to the Navier-Stokes equations for steady compressible non-Newtonian fluids

We prove the existence of weak solutions to steady, compressible non-Newtonian Navier-Stokes system on a bounded, two- or three-dimensional domain. Assuming the viscous stress tensor is monotone satisfying a power-law growth with power $r$ and the pressure is given by $\varrho^γ$, we construct a solution provided that $r>\frac{3d}{d+2}$ and $γ$ is sufficiently large, depending on the values of $r$. Additionally, we also show the existence for time-discretized model for Herschel-Bulkley fluids, where the viscosity has a singular part.

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↗

arXiv2025

Discontinuous shear-thickening asymptotic for power-law systems related to compressible flows

In this paper we study the convergence of a power-law model for dilatant compressible fluids to a class of models exhibiting a maximum admissible shear rate, called thick compressible fluids. These kinds of problems were studied previously for elliptic equations, stating with the work of Bhattacharya, E. DiBenedetto and J. Manfredi [Rend. Sem. Mat. Univ. Politec. Torino 1989], and more recently for incompressible fluids by J.F. Rodrigues [J. Math. Sciences 2015]. Our result may be seen as an extension to the compressible setting of these previous works. Physically, this is motivated by the fact that the pressures generated during a squeezing flow are often large, potentially requiring the consideration of compressibility, see M. Fang and R. Gilbert [Z. Anal. Anwend 2004]. Mathematically, the main difficulty in the compressible setting concerns the strong hyperbolicparabolic coupling between the density and velocity field. We obtain two main results, the first concerning the one-dimensional non-stationary compressible power-law system while the second one concerns the semi-stationary multi-dimensional case. Finally, we present an extension in onedimension for a viscous Cauchy stress with singular dependence on the shear rate.

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

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 $K$.

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↗

arXiv2022

Weak solutions for the Stokes system for compressible non-Newtonian fluids with unbounded divergence

We investigate the existence of weak solutions to a certain system of partial differential equations, modelling the behaviour of a compressible non-Newtonian fluid for small Reynolds number. We construct the weak solutions despite the lack of the $L^\infty$ estimate on the divergence of the velocity field. The result was obtained by combining the regularity theory for singular operators with a certain logarithmic integral inequality for $BMO$ functions, which allowed us to adjust the method from (Feireisl et al., 2015) to more relaxed conditions on the velocity.

math.AP↗

arXiv2021

Weak solutions for the Stokes system for compressible fluids with general pressure

We prove existence and uniqueness of global in time weak solutions for the Stokes system for compressible fluids with a general, non-monotone pressure. We construct the solution at the level of Lagrangian formulation and then define the transformation to the original Eulerian coordinates. For nonnegative and bounded initial density the solution is also nonnegative for all $t$ and belongs to $L^\infty([0,\infty)\times\mathbb{T}^d)$. A key point of our considerations is the uniqueness of such transformation. Since the velocity might not be Lipschitz continuous, we develop a method which relies on the results of Crippa \& De Lellis, concerning regular Lagriangian flows. The uniqueness is obtained thanks to the application of a certain weighted flow and detail analysis based on the properties of the $BMO$ space.

math.AP↗