arXiv subjects

Igor Honoré

Publications and source records attributed to Igor Honoré.

7 recordsLinked to original sources

arXiv2025

When Pythagoras meets Navier-Stokes

In this article, we develop a new method, based on a time decomposition of a Cauchy problem elaborated in [6], to retrieve the well-known $L^\infty ([0,T],L^2(\mathbb{R}^d,\mathbb{R}^d))$ control of the solution of the incompressible Navier-Stokes equation in $\mathbb{R}^d$. We precisely explain how the Pythagorean theorem in $L^2(\mathbb{R}^d,\mathbb{R}^d)$ allows to get the proper energy estimate; however such an argument does not work anymore in $L^p(\mathbb{R}^d,\mathbb{R}^d)$, $p \neq 2$. We also deduce, by similar arguments, an already known $L^\infty ([0,T],L^1(\mathbb{R}^3,\mathbb{R}^3))$ control of vorticity for $d=3$.

math.AP↗

arXiv2024

Second derivatives of solutions to the 3D incompressible Navier-Stokes equation in Lebesgue spaces

We obtain new controls for the Leray solutions $u$ of the incompressible Navier-Stokes equation in $\mathbb{R}^3$. Specifically, we estimate $u$, $\nabla u$, and $\nabla^2 u$ in suitable Lebesgue spaces $L^{\tilde r}_TL^r$, $r <+ \infty$ with some constraints on $\tilde r>0$. Our method is based on a Duhamel formula around a perturbed heat equation, allowing to thoroughly exploit the well-known energy estimates which balances the potential singularities. We also perform a new Bihari-LaSalle argument in this context. Eventually, we adapt our strategy to prove that $\sup_{t \in [0,T]} \int_{0}^t (t-s)^{-θ} \|\nabla^k u(s,\cdot)\|_{L^r} ds<+ \infty$, for all $θ< \frac{3-kr}{2r}$, $k \in [0,2]$, and $1<r<\frac{3}{k}$.

math.AP↗

arXiv2024

Transport equations in Hölder space by vanishing viscosity and applications

We obtain a sharp limit Hölder continuity of the solution for the transport equations thanks to a vanishing viscosity analysis. We also derive the same control for parabolic equations and for inviscid Burgers' equation. Eventually, under a structural hypothesis on the coefficients, we provide existence and uniqueness of a Hölder continuous solution.

math.AP↗

arXiv2022

Parabolic bootstrap for some non-linear equations

We obtain the well-posedness and Schauder estimates for a class of system of linear, quasi-linear and non-linear second order partial differential equations. We deduce existence and uniqueness of a global smooth solution of a non-linear and non-local equation that we call "semi" incompressible Navier Stokes equation in R 3 .

math.AP↗

arXiv2020

Sharp Schauder Estimates for some Degenerate Kolmogorov Equations

We provide here some sharp Schauder estimates for degenerate PDEs of Kolmogorov type when the coefficients lie in some suitable anisotropic H{ö}lder spaces and the first order term is non-linear and unbounded. We proceed through a perturbative approach based on forward parametrix expansions. Due to the low regularizing properties of the degenerate variables, for the procedure to work, we heavily exploit duality results between appropriate Besov spaces. Our method can be seen as constructive and provides, even in the non-degenerate case, an alternative approach to Schauder estimates.

math.AP↗

arXiv2020

Strong regularization by Brownian noise propagating through a weak H{ö}rmander structure

We establish strong uniqueness for a class of degenerate SDEs of weak H{ö}rmander type under suitable H{ö}lder regularity conditions for the associated drift term. Our approach relies on the Zvonkin transform which requires to exhibit good smoothing properties of the underlying parabolic PDE with rough, here H{ö}lder, drift coefficients and source term. Such regularizing effects are established through a perturbation technique (forward parametrix approach) which also heavily relies on appropriate duality properties on Besov spaces. For the method employed, we exhibit some sharp thresholds on the H{ö}lder exponents for the strong uniqueness to hold.

math.PR↗

arXiv2018

Non-Asymptotic Gaussian Estimates for the Recursive Approximation of the Invariant Measure of a Diffusion

We obtain non-asymptotic Gaussian concentration bounds for the difference between the invariant measure $ν$ of an ergodic Brownian diffusion process and the empirical distribution of an approximating scheme with decreasing time step along a suitable class of (smooth enough) test functions f such that f -- $ν$(f) is a coboundary of the infinitesimal generator. We show that these bounds can still be improved when the (squared) Fr{ö}benius norm of the diffusion coefficient lies in this class. We apply these bounds to design computable non-asymptotic confidence intervals for the approximating scheme. As a theoretical application, we finally derive non-asymptotic deviation bounds for the almost sure Central Limit Theorem.

math.PR↗