Alexandre Boritchev
Current Organizations / Institutes
The organizations/institutes from the scholar's latest article
Previous Organizations / Institutes
The organizations / institutes from the scholar's other articles in the past ten years
Research Subjects
Alexandre Boritchev has a total of 9 co-authors. Here are the top co-authors by number of shared publications:
- Grzegorz Karch
Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
3 shared publications - Philippe Laurençot
Laboratoire de Mathématiques (LAMA) UMR 5127 Université Savoie Mont Blanc CNRS F-73000 Chambéry France
3 shared publications - Piotr Biler
Instytut Matematyczny, Uniwersytet Wrocławski, Wrocław, Poland
3 shared publications - Daniel Eceizabarrena
BCAM ‐ Basque Center for Applied Mathematics Bilbao Spain
2 shared publications - Victor Vilaça da Rocha2 shared publications
- Daniel Eceizabarrena
Department of Mathematics and Statistics University of Massachusetts Amherst 01003 Amherst MA United States
1 shared publication
Latest Publications
- 21 November 2022
- Published by Elsevier in Journal of Differential Equations
- Vol. 344, 891-914
Large global solutions of the parabolic-parabolic Keller–Segel system in higher dimensions
- 21 November 2022
- Published by Elsevier in Journal of Differential Equations
- Vol. 344, 891-914
Large global solutions of the parabolic-parabolic Keller–Segel system in higher dimensions
- 1 September 2021
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (9) , 093101
Intermittency of Riemann’s non-differentiable function through the fourth-order flatness
Riemann’s non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch–Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely, the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.
- 1 September 2021
- Published by AIP Publishing in Journal of Mathematical Physics
- Vol. 62 (9) , 093101
Intermittency of Riemann’s non-differentiable function through the fourth-order flatness
Riemann’s non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch–Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely, the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.
- Open Access
Intermittency of Riemann's non-differentiable function through the fourth-order flatness
Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.
Open AccessIntermittency of Riemann's non-differentiable function through the fourth-order flatness
Riemann's non-differentiable function is one of the most famous examples of continuous but nowhere differentiable functions, but it has also been shown to be relevant from a physical point of view. Indeed, it satisfies the Frisch-Parisi multifractal formalism, which establishes a relationship with turbulence and implies some intermittent nature. It also plays a surprising role as a physical trajectory in the evolution of regular polygonal vortices that follow the binormal flow. With this motivation, we focus on one more classic tool to measure intermittency, namely the fourth-order flatness, and we refine the results that can be deduced from the multifractal analysis to show that it diverges logarithmically. We approach the problem in two ways: with structure functions in the physical space and with high-pass filters in the Fourier space.
- 26 April 2021
- Published by Springer Nature in Journal of Dynamics and Differential Equations
- Vol. 34 (4) , 3131-3141
Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation
- 26 April 2021
- Published by Springer Nature in Journal of Dynamics and Differential Equations
- Vol. 34 (4) , 3131-3141
Sharp Sobolev Estimates for Concentration of Solutions to an Aggregation–Diffusion Equation
- 28 September 2020
- Published by Elsevier in Journal of Differential Equations
- Vol. 271, 1092-1108
Concentration phenomena in a diffusive aggregation model
- 28 September 2020
- Published by Elsevier in Journal of Differential Equations
- Vol. 271, 1092-1108
Concentration phenomena in a diffusive aggregation model
- Open Access
Concentration phenomena in a diffusive aggregation model
We consider the drift-diffusion equation $$ u_t-\varepsilon \Delta u+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied.Open AccessConcentration phenomena in a diffusive aggregation model
We consider the drift-diffusion equation $$ u_t-\varepsilon \Delta u+\nabla\cdot(u\nabla K\star u)=0 $$ in the whole space with global-in-time bounded solutions. Mass concentration phenomena for radially symmetric solutions of this equation with small diffusivity are studied. - Open Access
Riemann's non-differentiable function is intermittent
Riemann's non-differentiable function, introduced in the middle of the 19th century as a purely mathematical pathological object, is relevant in the study of the binormal flow, as shown recently by De La Hoz and Vega. From this physical point of view, the function is therefore related to turbulent phenomena. We rigorously study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways: one in the physical space and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0. The regularity of Riemann's non-differentiable function is a classical subject, heavily linked to its small-scale behaviour. However, our subtle asymptotics for a classical hydrodynamical quantity are new and sharp.
Open AccessRiemann's non-differentiable function is intermittent
Riemann's non-differentiable function, introduced in the middle of the 19th century as a purely mathematical pathological object, is relevant in the study of the binormal flow, as shown recently by De La Hoz and Vega. From this physical point of view, the function is therefore related to turbulent phenomena. We rigorously study the fine intermittent nature of this function on small scales. To do so, we define the flatness, an analytic quantity measuring it, in two different ways: one in the physical space and the other one in the Fourier space. We prove that both expressions diverge logarithmically as the relevant scale parameter tends to 0. The regularity of Riemann's non-differentiable function is a classical subject, heavily linked to its small-scale behaviour. However, our subtle asymptotics for a classical hydrodynamical quantity are new and sharp.
- Open Access
- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 38 (5) , 2229-2249
Decaying turbulence for the fractional subcritical Burgers equation
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting:$\begin{equation} \nonumber\frac{\partial u}{\partial t}+(f(u))_x +ν Λ^{α} u = 0, t ≥ 0,\ \ \ \ {\bf{x}} ∈ {\mathbb{T}}^d = ({\mathbb{R}}/{\mathbb{Z}})^d.\end{equation}$Here \begin{document}$ f$\end{document} is strongly convex and satisfies a growth condition, \begin{document}$ Λ = \sqrt{-Δ}, \ ν$\end{document} is small and positive, while \begin{document}$ α ∈ (1,\ 2)$\end{document} is a constant in the subcritical range.For solutions \begin{document}$ u$\end{document} of this equation, we generalise the results obtained for the case \begin{document}$ α = 2$\end{document} (i.e. when \begin{document}$ -Λ^{α}$\end{document} is the Laplacian) in [12]. We obtain sharp estimates for the time-averaged Sobolev norms of \begin{document}$ u$\end{document} as a function of \begin{document}$ ν$\end{document}. These results yield sharp \begin{document}$ν$\end{document}-independent estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation \begin{document}$ \ell$\end{document} in the physical space and the wavenumber \begin{document}$ \bf{k}$\end{document} in the Fourier space. The form of all estimates is the same as in the case \begin{document}$ α = 2$\end{document}; the only thing which changes is that \begin{document}$ ν$\end{document} is replaced by \begin{document}$ ν^{1/(α-1)}$\end{document}.- 1 January 2018
- Published by American Institute of Mathematical Sciences (AIMS) in Discrete and Continuous Dynamical Systems
- Vol. 38 (5) , 2229-2249
Open AccessDecaying turbulence for the fractional subcritical Burgers equation
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting:$\begin{equation} \nonumber\frac{\partial u}{\partial t}+(f(u))_x +ν Λ^{α} u = 0, t ≥ 0,\ \ \ \ {\bf{x}} ∈ {\mathbb{T}}^d = ({\mathbb{R}}/{\mathbb{Z}})^d.\end{equation}$Here \begin{document}$ f$\end{document} is strongly convex and satisfies a growth condition, \begin{document}$ Λ = \sqrt{-Δ}, \ ν$\end{document} is small and positive, while \begin{document}$ α ∈ (1,\ 2)$\end{document} is a constant in the subcritical range.For solutions \begin{document}$ u$\end{document} of this equation, we generalise the results obtained for the case \begin{document}$ α = 2$\end{document} (i.e. when \begin{document}$ -Λ^{α}$\end{document} is the Laplacian) in [12]. We obtain sharp estimates for the time-averaged Sobolev norms of \begin{document}$ u$\end{document} as a function of \begin{document}$ ν$\end{document}. These results yield sharp \begin{document}$ν$\end{document}-independent estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation \begin{document}$ \ell$\end{document} in the physical space and the wavenumber \begin{document}$ \bf{k}$\end{document} in the Fourier space. The form of all estimates is the same as in the case \begin{document}$ α = 2$\end{document}; the only thing which changes is that \begin{document}$ ν$\end{document} is replaced by \begin{document}$ ν^{1/(α-1)}$\end{document}. - 29 September 2017
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 6 (1) , 109-123
Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing
- 29 September 2017
- Published by Springer Nature in Stochastics and Partial Differential Equations: Analysis and Computations
- Vol. 6 (1) , 109-123
Exponential convergence to the stationary measure for a class of 1D Lagrangian systems with random forcing
- Open Access
Decaying turbulence for the fractional subcritical Burgers equation
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +\nu \Lambda^{\alpha} u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is strongly convex and satisfies an additional growth condition, $\Lambda=\sqrt{-\Delta}$, $\nu$ is small and positive, while $\alpha \in (1,\ 2)$ is a constant in the subcritical range. For solutions $u$ of this equation, we generalise the results obtained for the case $\alpha=2$ (i.e. when $-\Lambda^{\alpha}$ is the Laplacian) in [10]. We obtain sharp estimates for the time-averaged Sobolev norms of $u$ as a function of $\nu$. These results yield sharp estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\ell$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. The form of all estimates is the same as in the case $\alpha=2$; the only thing that changes (except implicit constants) is that $\nu$ is replaced by $\nu^{1/(\alpha-1)}$.Open AccessDecaying turbulence for the fractional subcritical Burgers equation
We consider the fractional unforced Burgers equation in the one-dimensional space-periodic setting: $$\partial u/\partial t+(f(u))_x +\nu \Lambda^{\alpha} u= 0, t \geq 0,\ \mathbb{x} \in \mathbb{T}^d=(\mathbb{R}/\mathbb{Z})^d.$$ Here $f$ is strongly convex and satisfies an additional growth condition, $\Lambda=\sqrt{-\Delta}$, $\nu$ is small and positive, while $\alpha \in (1,\ 2)$ is a constant in the subcritical range. For solutions $u$ of this equation, we generalise the results obtained for the case $\alpha=2$ (i.e. when $-\Lambda^{\alpha}$ is the Laplacian) in [10]. We obtain sharp estimates for the time-averaged Sobolev norms of $u$ as a function of $\nu$. These results yield sharp estimates for natural analogues of quantities characterising the hydrodynamical turbulence, namely the averages of the increments and of the energy spectrum. In the inertial range, these quantities behave as a power of the norm of the relevant parameter, which is respectively the separation $\ell$ in the physical space and the wavenumber $\mathbf{k}$ in the Fourier space. The form of all estimates is the same as in the case $\alpha=2$; the only thing that changes (except implicit constants) is that $\nu$ is replaced by $\nu^{1/(\alpha-1)}$.