arXiv subjects

W. S. Ożański

Publications and source records attributed to W. S. Ożański.

9 recordsLinked to original sources

arXiv2026

The analytic method of constructing local-in-time solutions of the incompressible Euler equations in Sobolev spaces

We introduce a new method for constructing local-in-time solutions of the incompressible Euler equations in Sobolev spaces on an arbitrary Sobolev bounded domain. The method is based on a construction of an analytic solution in an analytically approximated domain, after which we apply analytic persistence to extend the analytic solution using given a priori bounds in Sobolev spaces. The method does not introduce any modification or regularization of the equations themselves and appears applicable to many other PDEs.

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arXiv2026

Stabilization of an incompressible fluid-elastic structure system using a vacuum bubble

We prove a priori estimates for the system of partial differential equations modeling the interaction between an elastic body and an incompressible fluid in a 3D curved domain. The fluid is governed by the incompressible Navier-Stokes equations and contains a bubble whose interior is a vacuum. The elastic body is described by a damped wave equation, and interaction with the fluid takes place along a free interface whose initial domain is curved. We show that the presence of the vacuum bubble stabilizes the system in the sense that it provides control of the average of the pressure function, and hence allows global existence and exponential decay of smooth solutions for small data.

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arXiv2025

A unified theory of existence of suitable weak solutions to the 3D incompressible Navier-Stokes equations for non-decaying initial data

We consider any cover $\mathscr{C}$ of $\mathbb{R}^3$ by balls of radius bigger or equal $1$ satisfying two conditions: (i) any ball intersects at most $σ>0$ other balls, and (ii) intersecting balls have comparable sizes. We consider a natural Morrey-type space such that the $L^2_{\mathrm{uloc}}$ setting of Lemarié-Rieusset (Recent Developments in the Navier-Stokes Problem, 2002) and the dyadic-type space considered by Bradshaw and Kukavica (J. Math. Fluid Mech., 22(1), 2020) are particular cases. We provide a priori estimates and prove local existence of weak solutions in two cases; first, when there exists $ε>0$ such that $|B|^{1/3} \lesssim |x_B|^{1-ε}$ for all $B\in \mathscr{C}$, where $x_B$ denotes the center of~$B$, or when $|B|^{1/3} \gtrsim 1+ |x_B|$ for all $B\in\mathscr{C}$. In particular, we introduce a new non-divergence-free approach to the construction of weak solutions, which simplifies the existence proof in the $L^2_{\mathrm{uloc}}$ setting. In addition, for the dyadic setting, we do not require vanishing at the spatial infinity. The constructed solutions are suitable in the sense of Caffarelli, Kohn, and Nirenberg, thus allowing an application of the partial regularity theory.

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arXiv2025

On the regularity of axially-symmetric solutions to the incompressible Navier-Stokes equations in a cylinder

We consider the axisymmetric Navier-Stokes equations in a finite cylinder $Ω\subset\mathbb{R}^3$. We assume that $v_r$, $v_φ$, $ω_φ$ vanish on the lateral boundary $\partial Ω$ of the cylinder, and that $v_z$, $ω_φ$, $\partial_z v_φ$ vanish on the top and bottom parts of the boundary $\partial Ω$, where we used standard cylindrical coordinates, and we denoted by $ω=\mathrm{curl}\, v$ the vorticity field. We use weighted estimates and $H^3$ Sobolev estimate on the modified stream function to derive three order-reduction estimates. These enable one to reduce the order of the nonlinear estimates of the equations, and help observe that the solutions to the equations are ``almost regular''. We use the order-reduction estimates to show that the solution to the equations remains regular as long as, for any $p\in (6,\infty)$, $\| v_φ\|_{L^\infty_t L^p_x}/\| v_φ\|_{L^\infty_t L^\infty_x}$ remains bounded below by a positive number.

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arXiv2025

Existence of nonsymmetric logarithmic spiral vortex sheet solutions to the 2D Euler equations

We consider solutions of the 2D incompressible Euler equation in the form of $M\geq 1$ cocentric logarithmic spirals. We prove the existence of a generic family of spirals that are nonsymmetric in the sense that the angles of the individual spirals are not uniformly distributed over the unit circle. Namely, we show that if $M=2$ or $M\geq 3 $ is an odd integer such that certain non-degeneracy conditions hold, then, for each $n \in \{ 1,2 \}$, there exists a logarithmic spiral with $M$ branches of relative angles arbitrarily close to $\barθ_{k} = knπ/M$ for $k=0,1,\ldots , M-1$, which include halves of the angles of the Alexander spirals. We show that the non-degeneracy conditions are satisfied if $M\in \{ 2, 3,5,7,9 \}$, and that the conditions hold for all odd $M>9$ given a certain gradient matrix is invertible, which appears to be true by numerical computations.

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arXiv2023

Quantitative control of solutions to the axisymmetric Navier-Stokes equations in terms of the weak $L^3$ norm

We are concerned with strong axisymmetric solutions to the $3$D incompressible Navier-Stokes equations. We show that if the weak $L^3$ norm of a strong solution $u$ on the time interval $[0,T]$ is bounded by $A \gg 1$ then for each $k\geq 0 $ there exists $C_k>1$ such that $\| D^k u (t) \|_{L^\infty (\mathbb{R}^3) } \leq t^{-(1+k)/2}\exp \exp A^{C_k}$ for all $t\in (0,T]$.

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arXiv2023

Well-posedness of logarithmic spiral vortex sheets

We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vorträge aus dem Gebiete der Hydro- und Aero-dynamik, 1922) and by Alexander (Phys. Fluids, 1971). We prove that for each such spiral the normal component of the velocity field remains continuous across the spiral. We give sufficient conditions for spiral vortex sheets to be weak solutions of the 2D incompressible Euler equations. Namely, we show that a spiral gives rise to such a solution if and only if two conditions hold across every spiral: a velocity matching condition and a pressure matching condition. Furthermore we show that these two conditions are equivalent to the imaginary part and the real part, respectively, of a single complex constraint on the coefficients of the spirals. This in particular provides a rigorous mathematical framework for logarithmic spirals, an issue that has remained open since their introduction by Prandtl in 1922. Another consequence of the main result is well-posedness of the symmetric Alexander spiral with two branches, despite recent evidence for the contrary. Moreover, our result implies a sharpness result of Delort's theorem on global existence of solutions to the Euler equations with initial vorticity measure. Our main tools are new explicit formulas for the velocity field and for the pressure function, as well as a notion of a \emph{winding number} of a spiral, which not only gives a robust way of localizing the spirals' arms with respect to a given point in the plane, but also ensures correct asymptotic behaviour near $0$.

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arXiv2022

Global-in-time stability of ground states of a pressureless hydrodynamic model of collective behaviour

We consider a pressureless hydrodynamic model of collective behaviour, which is concerned with a density function $ρ$ and a velocity field $v$ on the torus, and is described by the continuity equation for $ρ$, $\partial_t ρ+ \mathrm{div} (vρ)=0$, and a compressible hydrodynamic equation for $v$, $ρv_t + ρv\cdot \nabla v - Δv = -ρ\nabla K ρ$ with a forcing modelling collective behaviour related to the density $ρ$, where $K$ stands for the interaction potential, defined as the solution to the Poisson equation on $\mathbb{T}^d$. We show global-in-time stability of the ground state $(ρ, v)=(1,0)$ if the perturbation $(ρ_0-1 ,v_0)$ satisfies $\| v_0 \|_{B^{d/p-1}_{p,1}(\mathbb{T}^d )} + \| ρ_0-1 \|_{B^{d/p}_{p,1}(\mathbb{T}^d )} \leq ε$ for sufficiently small $ε>0$.

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arXiv2018

Partial regularity for a surface growth model

We prove two partial regularity results for the scalar equation $u_t+u_{xxxx}+\partial_{xx}u_x^2=0$, a model of surface growth arising from the physical process of molecular epitaxy. We show that the set of space-time singularities has (upper) box-counting dimension no larger than $7/6$ and $1$-dimensional (parabolic) Hausdorff measure zero. These parallel the results available for the three-dimensional Navier--Stokes equations. In fact the mathematical theory of the surface growth model is known to share a number of striking similarities with the Navier--Stokes equations, and the partial regularity results are the next step towards understanding this remarkable similarity. As far as we know the surface growth model is the only lower-dimensional "mini-model" of the Navier--Stokes equations for which such an analogue of the partial regularity theory has been proved. In the course of our proof, which is inspired by the rescaling analysis of Lin (1998) and Ladyzhenskaya & Seregin (1999), we develop certain nonlinear parabolic Poincaré inequality, which is a concept of independent interest. We believe that similar inequalities could be applicable in other parabolic equations.

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