arXiv subjects

Wojciech S. Ożański

Publications and source records attributed to Wojciech S. Ożański.

At least 19 recordsLinked to original sources

arXiv2025

Instantaneous continuous loss of regularity for the SQG equation

Given $s\in (3/2,2)$ and $\varepsilon >0$, we construct a compactly supported initial data $θ_0$ such that $\| θ_0 \|_{H^s}\leq \varepsilon$ and there exist $T>0$, $c>0$ and a local-in-time solution $θ$ of the SQG equation that is compactly supported in space, continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$, and, for each $t\in [0,T]$, $ θ(\cdot ,t ) \in {H^{s/(1+ct)}}$ and $ θ(\cdot ,t ) \not \in {H^β}$ for any $β> s/(1+ct)$. Moreover, $θ$ is unique among all solutions with initial condition $θ_0$ which belong to $C([0,T];H^{1+α})$ for any $α>0$ and is continuous and differentiable in $t$ and in $x$ on $\mathbb{R}^2\times [0,T]$.

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arXiv2025

Instantaneous continuous loss of Sobolev regularity for the 3D incompressible Euler equation

We prove instantaneous and continuous-in-time loss of supercritical Sobolev regularity for the 3D incompressible Euler equations in $\mathbb{R}^{3}$. Namely, for any $s\in (0,3/2)$ and $\varepsilon >0$, we construct a divergence-free initial vorticity $ω_0$ defined in $\mathbb{R}^{3}$ satisfying $\| ω_0 \|_{H^s}\leq \varepsilon$, as well as $T>0$, $c>0$ and a corresponding local-in-time solution $ω$ such that, for each $t\in [0,T]$, $ω(\cdot ,t ) \in {H^{\frac{s-ct}{1+ct}}}$ and $ ω(\cdot ,t ) \not \in {H^β}$ for any $β> \frac{s-ct}{1+ct} $. Moreover, $ω$ is unique among all solutions with initial condition $ω_0$ which are locally $C^2$ and belong to $C([0,T];L^p )$ for any $p>3 $.

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arXiv2024

Exact boundary controllability of the 3D incompressible ideal MHD system

We consider the three-dimensional ideal MHD system on a domain $Ω' \subset \mathbb{R}^3$ with a part $Γ$ of the boundary~$\partial Ω$, where we prescribe both $u\cdot n$ and $b\cdot n$, while $u\cdot n = b\cdot n =0$ on $\partial Ω' \setminus Γ$. We prove the boundary controllability of the system, namely that we can prescribe the boundary data such that the unique solution of the system with initial state $(u_0,b_0)$ achieves another state $(u_1,b_1)$ in finite time, where $u_0,b_0,u_1,b_1$ are arbitrary divergence-free vector fields satisfying impermeability boundary condition which are extendable to vector fields with the same properties on any bounded domain obtained by extension of $Ω'$ via $Γ$. As a byproduct, we give the first local well-posedness proof of incompressible, ideal MHD system, which does not use Elsasser variables and is thus applicable to any bounded domain with sufficient Sobolev regularity. We also provide a new and simple proof of the $2$D controllability.

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arXiv2023

On a model of an elastic body fully immersed in a viscous incompressible fluid with small data

We consider a model of an elastic body immersed between two layers of incompressible viscous fluid. The elastic displacement $w$ is governed by the damped wave equation $w_{tt} + αw_t + Δw =0$ without any stabilization terms, where $α>0$, and the fluid is modeled by the Navier-Stokes equations. We assume continuity of the displacement and the stresses across the moving interfaces and homogeneous Dirichlet boundary conditions on the outer fluid boundaries. We establish a~priori estimates that provide the global-in-time well-posedness and exponential decay to a final state of the system for small initial data. We prove that the final state must be trivial, except for a possible small displacement of the elastic structure in the horizontal direction.

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arXiv2023

Construction of the free-boundary 3D incompressible Euler flow under limited regularity

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We construct unique local-in-time solutions in the Lagrangian setting for $u_0 \in H^{2.5+δ}$ such that the Rayleigh-Taylor condition holds and $\mathrm{curl}\,u_0 \in H^{2+δ}$ in an arbitrarily small neighborhood of the free boundary. We show that the result is optimal in the sense that $H^{3+δ}$ regularity of the Lagrangian deformation near the free boundary can be ensured if and only if initial vorticity has $H^{2+δ}$ regularity of vorticity near the free boundary.

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arXiv2023

Linear instability of symmetric logarithmic spiral vortex sheets

We consider Alexander spirals with $M\geq 3$ branches, that is symmetric logarithmic spiral vortex sheets. We show that such vortex sheets are linearly unstable in the $L^\infty$ (Kelvin-Helmholtz) sense, as solutions to the Birkhoff-Rott equation. To this end we consider Fourier modes in a logarithmic variable to identify unstable solutions with polynomial growth in time.

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arXiv2022

Global well-posedness and exponential decay for a fluid-structure model with small data

We address the system of partial differential equations modeling motion of an elastic body interacting with an incompressible fluid. The fluid is modeled by the incompressible Navier-Stokes equations while the structure is represented by a damped wave equation $w_{tt}-Δw + αw_t =0$, where $α>0$. We prove the global existence and exponential decay of strong solutions for small initial data in a suitable Sobolev space. We show that the elastic velocity $w_t$ and the acceleration $w_{tt}$ can be controlled by the $H^2$ elliptic estimates and by the dissipation of the fluid via the free interface. We also find that, even though the vanishing of the final displacement $w$ appears invisible in the energy method, it can be deduced from the preservation of total volume. Our approach allows for any superlinear perturbation of the wave equation.

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arXiv2022

A partial uniqueness result and an asymptotically sharp nonuniqueness result for the Zhikov problem on the torus

We consider the stationary diffusion equation $-\mathrm{div} (\nabla u + bu )=f$ in $n$-dimensional torus $\mathbb{T}^n$, where $f\in H^{-1}$ is a given forcing and $b\in L^p$ is a divergence-free drift. Zhikov (Funkts. Anal. Prilozhen., 2004) considered this equation in the case of a bounded, Lipschitz domain $Ω\subset \mathbb{R}^n$, and proved existence of solutions for $b\in L^{2n/(n+2)}$, uniqueness for $b\in L^2$, and has provided a point-singularity counterexample that shows nonuniqueness for $b\in L^{3/2-}$ and $n=3,4,5$. We apply a duality method and a DiPerna-Lions-type estimate to show uniqueness of the solutions constructed by Zhikov for $b\in W^{1,1}$. We use a Nash iteration to demonstrate sharpness of this result, and also show that solutions in $H^1\cap L^{p/(p-1)}$ are flexible for $b\in L^p$, $p\in [1,2(n-1)/(n+1))$; namely we show that the set of $b\in L^p$ for which nonuniqueness in the class $H^1\cap L^{p/(p-1)}$ occurs is dense in the divergence-free subspace of $L^p$.

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arXiv2021

Local regularity of weak solutions of the hypodissipative Navier-Stokes equations

We consider the 3D incompressible hypodissipative Navier-Stokes equations, when the dissipation is given as a fractional Laplacian $(-Δ)^s$ for $s\in (\frac34,1)$, and we provide a new bootstrapping scheme that makes it possible to analyse weak solutions locally in space-time. This includes several homogeneous Kato-Ponce type commutator estimates which we localize in space, and which seems applicable to other parabolic systems with fractional dissipation. We also provide a new estimate on the pressure, $\|(-Δ)^s p \|_{\mathcal{H}^1}\lesssim \| (-Δ)^{\frac s2} u \|^2_{L^2}$. We apply our main result to prove that any suitable weak solution $u$ satisfies $\nabla^n u \in L^{p,\infty }_{\mathrm{loc}}(\mathbb{R}^3\times(0,\infty))$ for $p=\frac{2(3s-1)}{n+2s-1}$, $n=1,2$. As a corollary of our local regularity theorem, we improve the partial regularity result of Tang-Yu [Comm. Math. Phys., 334(30), 2015, pp. 1455--1482], and obtain an estimate on the box-counting dimension of the singular set $S$, $d_B(S\cap \{t\geq t_0 \} )\leq \frac13 (15-2s-8s^2) $ for every $t_0>0$.

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arXiv2021

Partial regularity of Leray-Hopf weak solutions to the incompressible Navier-Stokes equations with hyperdissipation

We show that if $u$ is a Leray-Hopf weak solution to the incompressible Navier--Stokes equations with hyperdissipation $α\in (1,5/4)$ then there exists a set $S\subset \mathbb{R}^3$ such that $u$ remains bounded outside of $S$ at each blow-up time, the Hausdorff dimension of $S$ is bounded above by $ 5-4α$ and its box-counting dimension is bounded by $(-16α^2 + 16α+5)/3$. Our approach is inspired by the ideas of Katz & Pavlović (Geom. Funct. Anal., 2002).

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arXiv2021

Quantitative transfer of regularity of the incompressible Navier-Stokes equations from $\mathbb{R}^3$ to the case of a bounded domain

Let $u_0\in C_0^5 ( B_{R_0})$ be divergence-free and suppose that $u$ is a strong solution of the three-dimensional incompressible Navier-Stokes equations on $[0,T]$ in the whole space $\mathbb{R}^3$ such that $\| u \|_{L^\infty ((0,T);H^5 (\mathbb{R}^3 ))} + \| u \|_{L^\infty ((0,T);W^{5,\infty }(\mathbb{R}^3 ))} \leq M <\infty$. We show that then there exists a unique strong solution $w$ to the problem posed on $B_R$ with the homogeneous Dirichlet boundary conditions, with the same initial data and on the same time interval for $R\geq \max(1+R_0, C(a) C(M)^{1/a} \exp ({CM^4T/a})) )$ for any $a\in [0,3/2)$, and we give quantitative estimates on $u-w$ and the corresponding pressure functions.

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arXiv2021

Local-in-time existence of free-surface 3D Euler flow with $H^{2+δ}$ initial vorticity in a neighborhood of the free boundary

We consider the three-dimensional Euler equations in a domain with a free boundary with no surface tension. We assume that $u_0 \in H^{2.5+δ}$ is such that $\mathrm{curl}\,u_0 \in H^{2+δ}$ in an arbitrarily small neighborhood of the free boundary, and we use Lagrangian approach to derive an a~priori estimate that can be used to prove local-in-time existence and uniqueness of solutions under the Rayleigh-Taylor stability condition.

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arXiv2021

An anisotropic regularity condition for the 3D incompressible Navier-Stokes equations for the entire exponent range

We show that a suitable weak solution to the incompressible Navier-Stokes equations on ${\mathbb{R}^3\times(-1,1)}$ is regular on $\mathbb{R}^3\times(0,1]$ if $\partial_3 u $ belongs to $M^{2p/(2p-3),α} ((-1,0);L^p (\mathbb{R}^3 ))$ for any $α>1$ and $p\in (3/2,\infty)$, which is a logarithmic-type variation of a Morrey space in time. For each $α>1$ this space is, up to a logarithm, critical with respect to the scaling of the equations, and contains all spaces $L^q ((-1,0);L^p (\mathbb{R}^3 ))$ that are subcritical, that is for which $2/q+3/p<2$.

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arXiv2021

On regularity properties of a surface growth model

We show local higher integrability of derivative of a suitable weak solution to the surface growth model, provided a scale-invariant quantity is locally bounded. If additionally our scale-invariant quantity is small, we prove local smoothness of solutions.

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arXiv2020

Global weak solutions of the Navier-Stokes equations for intermittent initial data in half-space

We prove existence of global-in-time weak solutions of the incompressible Navier-Stokes equations in the half-space $\mathbb{R}^3_+$ with initial data in a weighted space that allow non-uniformly locally square integrable functions that grow at spatial infinity in an intermittent sense. The space for initial data is built on cubes whose sides $R$ are proportional to the distance to the origin and the square integral of the data is allowed to grow as a power of $R$. The existence is obtained via a new a priori estimate and stability result in the weighted space, as well as new pressure estimates. Also, we prove eventual regularity of such weak solutions, up to the boundary, for $(x,t)$ satisfying $t>c_1|x|^2 + c_2$, where $c_1,c_2>0$, for a large class of initial data $u_0$, with $c_1$ arbitrarily small. As an application of the existence theorem, we construct global discretely self-similar solutions, thus extending the theory on the half-space to the same generality as the whole space.

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arXiv2019

Weak solutions to the Navier-Stokes inequality with arbitrary energy profiles

In a recent paper, Buckmaster & Vicol (arXiv:1709.10033) used the method of convex integration to construct weak solutions $u$ to the 3D incompressible Navier-Stokes equations such that $\| u(t) \|_{L^2} =e(t)$ for a given non-negative and smooth energy profile $e: [0,T]\to \mathbb{R}$. However, it is not known whether it is possible to extend this method to construct nonunique suitable weak solutions (that is weak solutions satisfying the strong energy inequality (SEI) and the local energy inequality (LEI)), Leray-Hopf weak solutions (that is weak solutions satisfying the SEI), or at least to exclude energy profiles that are not nonincreasing. In this paper we are concerned with weak solutions to the Navier-Stokes inequality on $\mathbb{R}^3$, that is vector fields that satisfy both the SEI and the LEI (but not necessarily solve the Navier-Stokes equations). Given $T>0$ and a nonincreasing energy profile $e\colon [0,T] \to [0,\infty )$ we construct weak solution to the Navier-Stokes inequality that are localised in space and whose energy profile $\| u(t)\|_{L^2 (\mathbb{R}^3 )}$ stays arbitrarily close to $e(t)$ for all $t\in [0,T]$. Our method applies only to nonincreasing energy profiles. The relevance of such solutions is that, despite not satisfying the Navier-Stokes equations, they satisfy the partial regularity theory of Caffarelli, Kohn & Nirenberg (Comm. Pure Appl. Math., 1982). In fact, Scheffer's constructions of weak solutions to the Navier-Stokes inequality with blow-ups (Comm. Math. Phys., 1985 & 1987) show that the Caffarelli, Kohn & Nirenberg's theory is sharp for such solutions. Our approach gives an indication of a number of ideas used by Scheffer. Moreover, it can be used to obtain a stronger result than Scheffer's. Namely, we obtain weak solutions to the Navier-Stokes inequality with both blow-up and a prescribed energy profile.

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arXiv2019

On weak solutions to the Navier-Stokes inequality with internal singularities

We construct weak solutions to the Navier-Stokes inequality, $$ u\cdot \left(\partial_t u -νΔu + (u\cdot \nabla) u +\nabla p \right) \leq 0 $$ in $\mathbb{R}^3$, which blow up at a single point $(x_0,T_0)$ or on a set $S \times \{T_0 \}$, where $S\subset \mathbb{R}^3$ is a Cantor set whose Hausdorff dimension is at least $ξ$ for any preassigned $ξ\in (0,1)$. Such solutions were constructed by Scheffer, Comm. Math. Phys., 1985 & 1987. Here we offer a simpler perspective on these constructions. We sharpen the approach to construct smooth solutions to the Navier-Stokes inequality on the time interval $[0,1]$ satisfying the "approximate equality" $$ \left\| u\cdot \left(\partial_t u-νΔu + (u\cdot \nabla) u +\nabla p \right) \right\|_{L^\infty}\leq \vartheta, $$ and the "norm inflation" $\| u(1) \|_{L^\infty} \geq \mathcal{N} \| u(0) \|_{L^\infty}$ for any preassigned $\mathcal{N}>0$, $\vartheta >0$. Furthermore we extend the approach to construct a weak solution to the Euler inequality $$u\cdot \left(\partial_t u+ (u\cdot \nabla) u +\nabla p \right) \leq 0, $$ which satisfies the approximate equality with $ν=0$ and blows up on the Cantor set $S\times \{T_0 \}$ as above.

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arXiv2018

A sufficient integral condition for local regularity of solutions to the surface growth model

The surface growth model, $u_t + u_{xxxx} + \partial_{xx} u_x^2 =0$, is a one-dimensional fourth order equation, which shares a number of striking similarities with the three-dimensional incompressible Navier--Stokes equations, including the results regarding existence and uniqueness of solutions and the partial regularity theory. Here we show that a weak solution of this equation is smooth on a space-time cylinder $Q$ if the Serrin condition $u_x\in L^{q'}L^q (Q)$ is satisfied, where $q,q'\in [1,\infty ]$ are such that either $1/q+4/q'<1$ or $1/q+4/q'=1$, $q'<\infty$.

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