arXiv subjects

Joanna Rencławowicz

Publications and source records attributed to Joanna Rencławowicz.

5 recordsLinked to original sources

arXiv2025

Weak solutions to incompressible heat-conducting motions with large flux

We analyze the problem for viscous incompressible heat-conduc\-ting fluid in a finite cylinder with large inflow and outflow, modelled with Navier-Stokes equations coupled with the heat equation. We prove energy estimate without restrictions on the magnitudes of the external force, initial data, inflow and outflow. In order to estimate nonlinear terms we use weighted spaces. Next, using Galerkin approximation, we show existence of weak solutions.

math.AP↗

arXiv2024

Local solutions for nonhomogeneous Navier-Stokes equations with large flux

The local existence of solutions to nonhomogeneous Navier-Stokes equations in cylindrical domains with arbitrary large flux is demonstrated. The existence is proved by the method of successive approximations. To show the existence with the lowest possible regularity the special Besov spaces called the Sobolev-Slobodetskii spaces are used. The inflow and outflow are prescribed on the parts of the boundary which are perpendicular to the $x_3$-axis. Since the inflow and outflow are positive the crucial point of this paper is to verify that $x_3$-coordinate of velocity is also positive. Finally, we conclude the local existence such that the velocity belongs to $W_σ ^{2+{s},1+{s}/2}(Ω^t)$, the gradient of pressure to $W_σ ^{{s},{s}/2}(Ω^t)$ and the density to $W_{r,\infty}^{1,1}(Ω^t)$, where ${s}\in(0,1)$, $σ >3/{s}$, $r>5/{s}$, $r>σ $.

math.AP↗

arXiv2024

Global motions for nonhomogeneous Navier-Stokes equations with large flux

The nonhomogeneous Navier-Stokes equations are considered in a cylindrical domain in ${\mathbb R}^3$, parallel to the $x_3$-axis with large inflow and outflow on the top and the bottom. Moreover, on the lateral part of the cylinder the slip boundary conditions are assumed. The global existence of regular solutions is proved under assumptions that inflow and outflow are close to homogeneous and norms of derivative with respect to $x_3$ of the external force and initial velocity are sufficiently small. The key point of this paper is to verify that $x_3$-coordinate of velocity remains positive.

math.AP↗

arXiv2021

On the Stokes system in cylindrical domains

The existence of solutions to some initial-boundary value problem for the Stokes system is proved. The result is shown in Sobolev-Slobodetskii spaces such that the velocity belongs to $W_r^{2+σ,1+σ/2}(Ω^T)$ and gradient of pressure to $W_r^{σ,σ/2}(Ω^T)$, where $r\in(1,\infty)$, $σ\in(0,1)$, $Ω^T=Ω\times(0,T)$. These are special Besov spaces: $B_{r,r}^{2+σ,1+σ/2}(Ω^T)$ and $B_{r,r}^{σ,σ/2}(Ω^T)$, respectively. The existence is proved by the technique of regularizer.

math.AP↗

arXiv2018

On some regularity criteria for axisymmetric Navier-Stokes equations

We point out some criteria that imply regularity of axisymmetric solutions to Navier-Stokes equations. We show that boundedness of $\|{v_{r}}/{\sqrt{r^3}}\|_{L_2({\rm R}^3\times (0,T))}$ as well as boundedness of $\|{ω_φ}/{\sqrt{r}}\|_{L_2({\rm R}^3\times (0,T))},$ where $v_r$ is the radial component of velocity and $ω_φ$ is the angular component of vorticity, imply regularity of weak solutions.

math.AP↗