Piotr Kokocki
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Piotr Kokocki has a total of 6 co-authors. Here are the top co-authors by number of shared publications:
- Aleksander Ćwiszewski
Faculty of Mathematics and Computer Science, Nicolaus Copernicus University, Toruń, Poland
4 shared publications - Tomasz Cieślak
Institute of Mathematics Polish Academy of Sciences 00-656 Warszawa, Poland
4 shared publications - Wojciech S. Ożański
Department of Mathematics, Florida State University, Tallahassee, USA
4 shared publications Kamil Dunst
2 shared publications- Przemysław Kosewski1 shared publication
Tomasz Cieślak
1 shared publication
Latest Publications
- Open Access
Uniqueness problem for Prandtl spirals
In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching condition and an appropriate decay condition at the origin of the spiral, then the velocity field is uniquely determined. We subsequently extend the result to the case of fields with vorticity composed of unions of concentric logarithmic spirals. As a by-product, we derive an alternative way of deriving formula for the velocity corresponding to the Prandtl spirals. The proof relies on an approach that is of independent interest. We construct an explicit conformal map from the exterior of a logarithmic spiral onto a strip. This transformation reduces the problem to establishing the uniqueness of a holomorphic function defined on the strip, under non-standard boundary and decay conditions.
Open AccessUniqueness problem for Prandtl spirals
In this paper, we study the problem of uniqueness of a divergence-free velocity field with vorticity given by the Prandtl spiral. We show that if the class of admissible velocities is restricted to those satisfying the velocity matching condition and an appropriate decay condition at the origin of the spiral, then the velocity field is uniquely determined. We subsequently extend the result to the case of fields with vorticity composed of unions of concentric logarithmic spirals. As a by-product, we derive an alternative way of deriving formula for the velocity corresponding to the Prandtl spirals. The proof relies on an approach that is of independent interest. We construct an explicit conformal map from the exterior of a logarithmic spiral onto a strip. This transformation reduces the problem to establishing the uniqueness of a holomorphic function defined on the strip, under non-standard boundary and decay conditions.
- 23 February 2024
- Published by Springer Nature in Journal of Mathematical Fluid Mechanics
- Vol. 26 (2) , 1-27
Linear Instability of Symmetric Logarithmic Spiral Vortex Sheets
- 23 February 2024
- Published by Springer Nature in Journal of Mathematical Fluid Mechanics
- Vol. 26 (2) , 1-27
Linear Instability of Symmetric Logarithmic Spiral Vortex Sheets
- 22 February 2024
- Published by Elsevier in Journal of Differential Equations
- Vol. 389, 508-539
Well-posedness of logarithmic spiral vortex sheets
- 22 February 2024
- Published by Elsevier in Journal of Differential Equations
- Vol. 389, 508-539
Well-posedness of logarithmic spiral vortex sheets
- 9 September 2023
- Published by Elsevier in Nonlinear Analysis
- Vol. 237
Standing waves for Schrödinger equations with Kato–Rellich potentials
- 9 September 2023
- Published by Elsevier in Nonlinear Analysis
- Vol. 237
Standing waves for Schrödinger equations with Kato–Rellich potentials
- Open Access
Standing Waves for Schrödinger Equations with Kato-Rellich potentials
We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski.
Open AccessStanding Waves for Schrödinger Equations with Kato-Rellich potentials
We show the existence of standing waves for the nonlinear Schrödinger equation with Kato-Rellich type potential. We consider both resonant with the nonlinearity satisfying one of Landesman-Lazer type or sign conditions and non-resonant case where the linearization at infinity has zero kernel. The approach relies on the geometric and topological analysis of the parabolic semiflow associated to the involved elliptic problem. Tail estimates techniques and spectral theory of unbounded linear operators are used to exploit subtle compactness properties necessary for use of the Conley index theory due to Rybakowski.
- Open Access
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.Open AccessLinear 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. - Open Access
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 (Vortr\"age 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 $0$.Open AccessWell-posedness of logarithmic spiral vortex sheets
We consider a family of 2D logarithmic spiral vortex sheets which include the celebrated spirals introduced by Prandtl (Vortr\"age 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 $0$. - 1 January 2021
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 53 (4) , 4727-4743
Double Spiral Singularities for a Flow of Regular Planar Curves
In this paper, we study the singularity formation for the geometric flow of complex curves $z_t = -z_{xxx} + \frac{3}{2}\overline z_{x} z_{xx}^2,$ which was derived in [R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett., 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents, and furthermore we establish its asymptotic expansion in the vicinity of the singularity.- 1 January 2021
- Published by Society for Industrial & Applied Mathematics (SIAM) in SIAM Journal on Mathematical Analysis
- Vol. 53 (4) , 4727-4743
Double Spiral Singularities for a Flow of Regular Planar Curves
In this paper, we study the singularity formation for the geometric flow of complex curves $z_t = -z_{xxx} + \frac{3}{2}\overline z_{x} z_{xx}^2,$ which was derived in [R. E. Goldstein and D. M. Petrich, Phys. Rev. Lett., 69 (1992), pp. 555--558] while considering the vortex patch dynamics for the incompressible 2D Euler equation. We prove that arbitrary curve, consisting of two rotating logarithmic spirals, is a finite time singularity developed by a smooth solution of the flow. We provide exact construction of the solution in the terms of appropriate Painlevé II transcendents, and furthermore we establish its asymptotic expansion in the vicinity of the singularity.- Open Access
On global solutions of defocusing mKdV equation with specific initial data of critical regularity
We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlev\'e equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the matrix-valued function satisfying the associated Riemann-Hilbert problem can be represented by means of a local parametrix around the origin, whose existence can be proved by a vanishing lemma. The aim of this paper is to construct the explicit form of this parametrix and apply it to obtain improved asymptotic relations for the real and purely imaginary Ablowitz-Segur solutions of the inhomogeneous Painlev\'e II equation.
Open AccessOn global solutions of defocusing mKdV equation with specific initial data of critical regularity
We study the asymptotic behavior of the Ablowitz-Segur solutions for the second Painlev\'e equation using the Riemann-Hilbert approach and methods based on asymptotic expansions of classical special functions. Recent results show that the matrix-valued function satisfying the associated Riemann-Hilbert problem can be represented by means of a local parametrix around the origin, whose existence can be proved by a vanishing lemma. The aim of this paper is to construct the explicit form of this parametrix and apply it to obtain improved asymptotic relations for the real and purely imaginary Ablowitz-Segur solutions of the inhomogeneous Painlev\'e II equation.
- Open Access
On global dynamics of reaction--diffusion systems at resonance
In this paper we use the homotopy invariants methods to study the global dynamics of the reaction-diffusion systems that are at resonance at infinity. Considering degrees of the resonance for the nonlinear perturbation we establish Landesman-Lazer type conditions and use them to express the Rybakowski-Conley index of the invariant set consisting of all bounded solutions. Obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations.
Open AccessOn global dynamics of reaction--diffusion systems at resonance
In this paper we use the homotopy invariants methods to study the global dynamics of the reaction-diffusion systems that are at resonance at infinity. Considering degrees of the resonance for the nonlinear perturbation we establish Landesman-Lazer type conditions and use them to express the Rybakowski-Conley index of the invariant set consisting of all bounded solutions. Obtained results are applied to study the existence of solutions connecting stationary points for the system of nonlinear heat equations.