arXiv subjects

Matthieu Pageard

Publications and source records attributed to Matthieu Pageard.

4 recordsLinked to original sources

arXiv2026

Remarks on some quasi-linear fourth order parabolic equations arising in mathematical physics

This note proves existence and uniqueness of strong solutions to a broad class of fourth-order quasi-linear parabolic equations in the class of Wiener spaces. It improves upon previous results in the literature, devoted to some special cases falling within the general framework developed here, in which uniqueness held true in a smaller class than the space where existence was proven. Gain of analyticity, as well as generalisations to higher-order equations and to propagation of higher regularities, are also discussed.

math.AP↗

arXiv2026

Solutions of the Navier-Stokes equations with forced rapid space-time decay

We study the pointwise decay properties of solutions to the incompressible Navier-Stokes equations, both in the space and time variables. It is well known that generic global solutions on $\mathbb{R}^n$ do not decay faster at infinity than $|x|^{-(n+1)}$ and $t^{-(n+1)/2}$ in the pointwise sense. In this paper, we address the control problem of constructing an external forcing and a solution to the Navier-Stokes equations whose space-time decay properties go beyond these limiting rates. A distinctive feature of the forcing term is that its spatial profile can be fixed once and for all, independently of the initial data of the problem, and localized in an arbitrarily small region of $\mathbb{R}^n$. Only the temporal profile of the external force displays a dependency on the initial datum.

math.AP↗

arXiv2025

Non-Algebraic Decay for Solutions to the Navier-Stokes Equations

Around forty years ago, Michael Wiegner provided, in a seminal paper, sharp algebraic decay rates for solutions of the Navier--Stokes equations, showing that these solutions behave asymptotically like the solutions of the heat equation with the same data as $t\to+\infty$, in the $L^2$-norm, up to some critical decay rate. In the present paper, we close a gap that appears in the conclusion of Wiegner's theorem in the 2D case, for solutions with non-algebraic decay rate.

math.AP↗

arXiv2025

Well-posedness for 2D non-homogeneous incompressible fluids with general density-dependent odd viscosity

We study the initial value problem for a system of equations describing the motion of two-dimensional non-homogeneous incompressible fluids exhibiting odd (non-dissipative) viscosity effects. We consider the complete odd viscous stress tensor with a general density-dependent viscosity coefficient $f(\rho)$. Under suitable assumptions, we prove the local existence and uniqueness of strong solutions in $H^s(\mathbb{R}^2)$ $(s>2)$, for a class of viscosity coefficients covering the particular case $f(\rho)=a\rho^\alpha+b$ for any $(a,b,\alpha)\in\mathbb{R}^3$, generalising the result of Fanelli, Granero-Belinch\'on and Scrobogna, devoted to the case $f(\rho)=\rho$. Additionally, we are able to do so without requiring the initial density variation to belong to $L^2(\mathbb{R}^2)$. As a major step of the proof, we exhibit an effective velocity for this sytem, generalising the so-called "Els\"asser formulation" recently derived by Fanelli and Vasseur.

math.AP↗