arXiv subjects

Francois Vigneron

Publications and source records attributed to Francois Vigneron.

8 recordsLinked to original sources

arXiv2025

Factorization of the quadratic Misiurewicz-Thurston polynomials

This note provides the complete factorization of the Misiurewicz-Thurston polynomial $q_{\ell,n}=p_{\ell+n}(z) - p_\ell(z)$ over $\mathbb{C}$, which plays a central role in the study of the Mandelbrot set, where \[ p_0(z) = 0, \qquad p_{n+1}(z) = p_n(z)^2 + z. \] The roots can be classified into two categories. First, there are hyperbolic points $\operatorname{hyp}(k)$ for any divisor $k$ of $n$, which are parameters whose critical orbits are of exact period $k$. Those are roots of $q_{\ell,n}$ with multiplicity $\left\lfloor \frac{\ell -1}{k} \right\rfloor + 2$. Next are the points $\operatorname{mis}(j,k)$ for $2\leq j\leq \ell$ whose critical orbits are pre-periodic of exact period $k$ with an exact pre-period $j$. Those are simple roots of $q_{\ell,n}$.

math.DS↗

arXiv2024

How to split a tera-polynomial

This article presents a new algorithm to compute all the roots of two families of polynomials that are of interest for the Mandelbrot set $\mathcal{M}$ : the roots of those polynomials are respectively the parameters $c\in\mathcal{M}$ associated with periodic critical dynamics for $f_c(z)=z^2+c$ (hyperbolic centers) or with pre-periodic dynamics (Misiurewicz-Thurston parameters). The algorithm is based on the computation of discrete level lines that provide excellent starting points for the Newton method. In practice, we observe that these polynomials can be split in linear time of the degree. This article is paired with a code library [Mandel] that implements this algorithm. Using this library and about 723 000 core-hours on the HPC center Rom\'eo (Reims), we have successfully found all hyperbolic centers of period $\leq 41$ and all Misiurewicz-Thurston parameters whose period and pre-period sum to $\leq 35$. Concretely, this task involves splitting a tera-polynomial, i.e. a polynomial of degree $\sim10^{12}$, which is orders of magnitude ahead of the previous state of the art. It also involves dealing with the certifiability of our numerical results, which is an issue that we address in detail, both mathematically and along the production chain. The certified database is available to the scientific community. For the smaller periods that can be represented using only hardware arithmetic (floating points FP80), the implementation of our algorithm can split the corresponding polynomials of degree $\sim10^{9}$ in less than one day-core. We complement these benchmarks with a statistical analysis of the separation of the roots, which confirms that no other polynomial in these families can be split without using higher precision arithmetic.

math.NA↗

arXiv2021

Global well-posedness and long-time asymptotics of a general nonlinear non-local Burgers Equation

This paper is concerned with the study of a nonlinear non-local equation that has a commutator structure. The equation reads $\partial_t u-F(u) (-Δ)^{s/2} u+(-Δ)^{s/2} (uF(u))=0$, $x\in \mathbb{T}^d$, with s $\in$ (0, 1]. We are interested in solutions stemming from periodic positive bounded initial data. The given function $F \in C^\infty (R^+)$ must satisfy $F' > 0$ a.e. on (0, +$\infty$). For instance, all the functions $F (u) = u^n$ with n $\in$ N * are admissible non-linearities. We construct global classical solutions starting from smooth positive data, and global weak solutions starting from positive data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time asymptotics of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

math.AP↗

arXiv2019

On the Wiener-Khinchin transform of functions that behave as approximate power-laws. Applications to fluid turbulence

As we all know, the Fourier transform is continuous in the weak sense of tempered distribution; this ensures the weak stability of Fourier pairs. This article investigates a stronger form of stability of the pair of homogeneous profiles $(|x|^{-α}, c_d |ξ|^{d-α})$ on $\mathbb{R}^d$. It encompasses, for example, the case where the homogeneous profiles exist only on a large but finite range. In this case, we provide precise error estimates in terms of the size of the tails outside the homogeneous range. We also prove a series of refined properties of the Fourier transform on related questions including criteria that ensure an approximate homogeneous behavior asymptotically near the origin or at infinity. The sharpness of our results is checked with numerical simulations. We also investigate how these results consolidate the mathematical foundations of turbulence theory.

math-ph↗

arXiv2017

Existence of Solutions of a Non-Linear Eigenvalue Problem with a Variable Weight

We study the non-linear minimization problem on $H^1_0(Ω)\subset L^q$ with $q=\frac{2n}{n-2}$, $α>0$ and $n\geq4$~: \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ωa(x,u)|\nabla u|^2 - λ\int_Ω |u|^2.\] where $a(x,s)$ presents a global minimum $α$ at $(x_0,0)$ with $x_0\inΩ$. In order to describe the concentration of $u(x)$ around $x_0$, one needs to calibrate the behaviour of $a(x,s)$ with respect to $s$. The model case is \[\inf_{\substack{u\in H^1_0(Ω) \|u\|_{L^q}=1}}\int_Ω(α+|x|^β|u|^k)|\nabla u|^2 - λ\int_Ω |u|^2.\] In a previous paper dedicated to the same problem with $λ=0$, we showed that minimizers exist only in the range $β kn/q + 2$, minimizers do exist.

math.AP↗

arXiv2015

Global Well-Posedness Of A Non-Local Burgers Equation: The Periodic Case

This paper is concerned with the study of a non-local Burgers equation for positive bounded periodic initial data. The equation reads $$ u_t - u |\nabla| u + |\nabla|(u^2) = 0. $$ We construct global classical solutions starting from smooth positive data, and global weak solutions starting from data in $L^\infty$. We show that any weak solution is instantaneously regularized into $C^\infty$. We also describe the long-time behavior of all solutions. Our methods follow several recent advances in the regularity theory of parabolic integro-differential equations.

math.AP↗

arXiv2011

Non-Linear Effects in a Yamabe-Type Problem with Quasi-Linear Weight

We study the quasi-linear minimization problem on $H^1_0(Ω)\subset L^q$ with $q=\frac{2n}{n-2}$~: $$\inf_{\|u\|_{L^q}=1}\int_Ω(1+|x|^β|u|^k)|\nabla u|^2.$$ We show that minimizers exist only in the range $β<kn/q$ which corresponds to a dominant non-linear term. On the contrary, the linear influence for $β\geq kn/q$ prevents their existence.

math.AP↗

arXiv2007

New Asymptotic Profiles of Nonstationnary Solutions of the Navier-Stokes System

We show that solutions $u(x,t)$ of the non-stationnary incompressible Navier--Stokes system in $\R^d$ ($d\geq2$) starting from mild decaying data $a$ behave as $|x|\to\infty$ as a potential field: u(x,t) = e^{tΔ}a(x) + γ_d\nabla_x(\sum_{h,k} \frac{δ_{h,k}|x|^2 - d x_h x_k}{d|x|^{d+2}} K_{h,k}(t))+\mathfrak{o}(\frac{1}{|x|^{d+1}}) where $γ_d$ is a constant and $K_{h,k}=\int_0^t(u_h| u_k)_{L^2}$ is the energy matrix of the flow. We deduce that, for well localized data, and for small $t$ and large enough $|x|$, c t |x|^{-(d+1)} \le |u(x,t)|\le c' t |x|^{-(d+1)}, where the lower bound holds on the complementary of a set of directions, of arbitrary small measure on $\mathbb{S}^{d-1}$. We also obtain new lower bounds for the large time decay of the weighted-$L^p$ norms, extending previous results of Schonbek, Miyakawa, Bae and Jin.

math.AP↗