arXiv subjects

Wojciech Ożański

Publications and source records attributed to Wojciech Ożański.

7 recordsLinked to original sources

arXiv2026

An analysis of aerodynamic properties of delta wings

We consider a sharp-edge delta wing of small aspect ratio $A>0$, which is an example of a 3D airfoil whose aerodynamic properties cannot be modeled using the potential lift}only. An important role is played by a pair of attached vortices, which generate the vortex lift. We review in detail the leading-edge suction analogy, developed by Edward Polhamus (NASA Technical Note D-3767, 1966), which shows that the most accurate description of the vortex lift can be obtained by avoiding any direct analysis of vorticity of the flow around the wing. We also describe a 2D slender wing theory, which was introduced by Jones (NACA report no. 835, 1946) to approximate the potential lift. We show that, neglecting certain two borderline nonintegrability issues and an elliptic regularity issue, the slender wing theory is sufficient to yield accurate prediction of the potential lift as $A\to 0$. We also show that this 2D approximation is necessary to justify the modelling assumptions of the leading-edge suction analogy, so that the slender wing model can also be used to approximate the vortex lift as $A\to 0$. For completeness, we provide additional explanations of the necessary airfoil theory.

math.AP↗

arXiv2026

Gevrey instability in the inviscid inflow-outflow problem

We consider the 2D incompressible Euler equations on a periodic channel $\mathbb{T}\times (0,1)$ with inflow-outflow boundary condition $u=(0,1)$ on $\mathbb{T} \times \{0,1 \}$. We also impose the incoming vorticity boundary condition $ω=η$ on $\mathbb{T}\times \{ 0 \}$, where $η$ is prescribed. We show that the problem is globally well-posed in Gevrey spaces (for any value of the Gevrey exponent $s>1$) as long as $η$ remains Gevrey. This proves that the inflow-outflow velocity boundary condition determines the solution locally in time if and only if the solution is considered in an analytic class. In particular, leaving the analytic class, nonuniquness of solutions occurs already in any Gevrey class, by prescribing $η$. Hence, prescribing an analytic inflow-outflow velocity leads to precisely one analytic and continuum $s$-Gevrey solutions for every $s>1$. Furthermore, the result implies that if $η$ is analytic, then the unique global solution can lose analyticity in space for all $t>0$, but remain $s$-Gevrey regular for all $s$.

math.AP↗

arXiv2025

Linear and nonlinear instability of vortex columns

We consider vortex column solutions $v = V(r) e_θ+ W(r) e_z$ to the $3$D Euler equations. We give a mathematically rigorous construction of the countable family of unstable modes discovered by Liebovich and Stewartson (J. Fluid Mech. 126, 1983) via formal asymptotic analysis. The unstable modes exhibit $O(1)$ growth rates and concentrate on a ring $r= r_0$ asymptotically as the azimuthal and axial wavenumbers $n, α\to \infty$ with a fixed ratio. We construct these so-called ring modes with an inner-outer gluing procedure. Finally, we prove that each linear instability implies nonlinear instability for vortex columns. In particular, our analysis yields nonlinear instability for the Batchelor trailing line vortex $V(r) :=\frac{q}{r} (1-\mathrm{e}^{-r^2})$ and $W(r) :=\mathrm{e}^{-r^2}$ when $0<q <\log 2 / \sqrt{1-\log 2} \approx 1.251$.

math.AP↗

arXiv2025

A simple proof of linear instability of shear flows with application to vortex sheets

We consider the construction of linear instability of parallel shear flows, which was developed by Zhiwu Lin (SIAM J. Math. Anal. 35(2), 2003). We give an alternative simple proof in Sobolev setting of the problem, which exposes the mathematical role of the Plemelj-Sochocki formula in the emergence of the instability, as well as does not require the cone condition. Moreover, we localize this approach to obtain an approximation of the Kelvin-Helmholtz instability of a flat vortex sheet.

math.AP↗

arXiv2025

The inviscid inflow-outflow problem via analyticity

We consider the incompressible Euler equation on an analytic domain $Ω$ with nonhomogeneous boundary condition $u\cdot \mathsf{n} = \overline{u} \cdot \mathsf{n}$ on $\partial Ω$, where $\overline{u}$ is a given divergence-free analytic vector field. We establish local well-posedness for $u$ in analytic spaces without any compatibility conditions in all space dimensions. We also prove global well-posedness in the 2D case if $\overline{u}$ decays in time sufficiently fast.

math.AP↗

arXiv2024

An improvement to Prandtl's 1933 model for minimizing induced drag

We consider Prandtl's 1933 model for calculating circulation distribution function $Γ$ of a finite wing which minimizes induced drag, under the constraints of prescribed total lift and moment of inertia. We prove existence of a global minimizer of the problem without the restriction of nonnegativity $Γ\geq 0$ in an appropriate function space. We also consider an improved model, where the prescribed moment of inertia takes into account the bending moment due to the weight of the wing itself, which leads to a more efficient solution than Prandtl's 1933 result.

math.AP↗

arXiv2023

Instantaneous gap loss of Sobolev regularity for the 2D incompressible Euler equations

We construct solutions of the 2D incompressible Euler equations in $\mathds{R}^2\times [0,\infty)$ such that initially the velocity is in the super-critical Sobolev space $H^β$ for $1<β<2$, but are not in $H^{β'}$ for $β'>1+\frac{(3-β)(β-1)}{2 - (β-1)^2}$ for $0<t<\infty$. These solutions are not in the Yudovich class, but they exists globally in time and they are unique in a determined family of classical solutions.

math.AP↗