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Yannick Sire

Publications and source records attributed to Yannick Sire.

At least 19 recordsLinked to original sources
arXiv2026

Giga-Kohn-type results for the fully fractional heat equation

We consider the semilinear fully fractional heat equation \[ (\partial_t-\Delta)^\sigma u = |u|^{p-1}u \quad \text{in } \mathbb{R}^n \times \mathbb{R}_{-}, \qquad 0 < \sigma < 1. \] For $n\leq 2\sigma$ or $1<p\leq \frac{n+2\sigma}{n-2\sigma}$, we generalize the monotonicity formula and Liouville-type theorem when $\sigma=1$ proved by Giga and Kohn. In order to overcome the difficulty that this equation is nonlocal, we give a new interpretation of the classical Giga-Kohn's Pohozaev identity in terms of Hermite expansion. This insight is new and interesting even for $\sigma=1$. We further establish a space-time nonlocal monotonicity formula for the self-similar equation. As far as we are concerned, this is the first monotonicity formula for space-time nonlocal equations without using an extension by Stinga and Torrea.

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arXiv2026

Self-similar blow-up profile for the one-dimensional reduction of generalized SQG with infinite energy

We study the singularity formation mechanisms of the inviscid generalized Surface Quasi-Geostrophic (gSQG) equation on the whole space $\mathbb{R}^2$ and on the upper half-plane $\mathbb{R}^2_+$, allowing infinite energy. In each case, we derive a one-dimensional reduction that captures the leading-order singular behavior of the original 2D system, and use a fixed-point argument to show the existence of finite-time self-similar blow-up solutions for the 1D systems. We also perform numerical simulations for verification and visualization.

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arXiv2026

Fine regularity of fractional harmonic maps and applications

In this paper, we derive several regularity results for harmonic mappings into Euclidean spheres associated with rather general energies related to fractional Sobolev spaces. These maps generalize families of maps introduced by Da Lio, Rivi\`ere and Schikorra and are related to harmonic maps with free boundaries. In our context, there is in general no monotonicity formula, which prevents the use of some classical methods. Despite this limitation, under natural assumptions on a Gagliardo-type energy, we succeed in proving a variety of small energy regularity results and improve on known results, even in the isotropic case for which some monotonicity formula is available. To this end, we exploit recent developments in the regularity theory of nonlocal equations and as a by-product, we explain how these results apply to classes of harmonic maps with free boundary and lead to new potential-theoretic estimates. As another application, we obtain higher differentiability results for the fractional harmonic map heat flow.

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arXiv2026

Global oscillatory solutions for the Yang-Mills heat flow

We investigate the long-time dynamics for the global solution of the $SO(4)$-equivariant Yang-Mills heat flow (YMHF) with structure group $SU(2)$ in space dimension $4$. For a class of initial data with specific decay at spatial infinity, we prove that the long-time dynamics of YMHF can be described by the initial data in a unified manner. As a consequence, the global solutions can exhibit blow-up, blow-down, and more exotically, {\it oscillatory} asymptotic behavior at time infinity. This seems to be the first example of Yang-Mills heat flows with oscillatory behavior as $t\to \infty$.

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arXiv2026

Heat flow of harmonic maps into CAT($0$)-spaces

We introduce a new approach to prove the global existence and uniqueness of suitable weak solutions of the heat flow of harmonic mappings into CAT(0) metric spaces. Our method allows also to prove Lipschitz continuity in spatial variables for such solutions into any CAT$(0)$-space, answering a long-standing open problem in the field. Our approach is based on an elliptic regularization of the gradient flow of the Dirichlet energy and even in the case of smooth Riemannian targets provides a novel viewpoint, together with a new Dynamical Variational Principle and a new proof of the celebrated Eells-Sampson theorem. The spatial Lipschitz regularity for such weak solutions is achieved by fully exploiting the variational structure of the problem at the regularized level and introducing a parabolic frequency function of Almgren-Poon type. Our contribution is the first instance of the use of monotonicity methods for parabolic deformations of maps into singular targets.

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arXiv2026

Liquid crystals and topological vorticity: smoothness of mild solutions

We introduce several new models whose common feature is to take into account effects from topological vorticity. The macroscopic unknown is driven by a dissipative anomalous diffusion (of SQG-type) and is coupled with the orientation of the crystal, moving by the gradient flow of the energy of maps. The main idea of such models is to have a better insight on the vorticity formulation of the Liquid Crystal Flow and to tackle some regularity issues in the associated conserved geometric motions. One of the advantage of the present PDEs is to capture features of the Navier-Stokes equations (or Euler) through a {\sl scalar} unknown, keeping the advection-diffusion structure of the orientation field. We obtain regularity for mild solutions under natural assumptions for the initial data, which are actually near-optimal. Along the way, we also draw some links with natural models of (anti-)ferromagnets previously investigated.

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arXiv2025

Singular set estimates for solutions to elliptic equations in higher co-dimension

Recent advances in quantitative unique continuation properties for solutions to uniformly elliptic, divergence form equations (with Lipschitz coefficients) has led to a good understanding of the vanishing order and size of singular and zero set of solutions. Such estimates also hold at the boundary, provided that the domain is sufficiently regular. In this work, we investigate the boundary behavior of solutions to a class of elliptic equations in the higher co-dimension setting, whose coefficients are neither uniformly elliptic, nor uniformly Lipschitz. Despite these challenges, we are still able to show analogous estimates on the singular set of such solutions near the boundary. Our main technical advance is a variant of the Cheeger-Naber-Valtorta quantitative stratification scheme using cones instead of planes.

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arXiv2024

The Schr\"odinger equation with fractional Laplacian on hyperbolic spaces and homogeneous trees

We investigate dispersive and Strichartz estimates for the Schr\"odinger equation involving the fractional Laplacian in real hyperbolic spaces and their discrete analogues, homogeneous trees. Due to the Knapp phenomenon, the Strichartz estimates on Euclidean spaces for the fractional Laplacian exhibit loss of derivatives. A similar phenomenon appears on real hyperbolic spaces. However, such a loss disappears on homogeneous trees, due to the triviality of the estimates for small times.

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arXiv2024

Conformal deformation of a Riemannian metric via an Einstein-Dirac parabolic flow

We introduce a new parabolic flow deforming any Riemannian metric on a spin manifold by following a constrained gradient flow of the total scalar curvature. This flow is built out of the well-known Dirac-Einstein functional. We prove local well-posedness of smooth solutions. The present contribution is the first installment of more general program on the Einstein-Dirac problem.

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arXiv2024

Extension method in Dirichlet spaces with sub-Gaussian estimates and applications to regularity of jump processes on fractals

We investigate regularity properties of some non-local equations defined on Dirichlet spaces equipped with sub-gaussian estimates for the heat kernel associated to the generator. We prove that weak solutions for homogeneous equations involving pure powers of the generator are actually H\"older continuous and satisfy an Harnack inequality. Our methods are based on a version of the Caffarelli-Silvestre extension method which is valid in any Dirichlet space and our results complement the existing literature on solutions of PDEs on classes of Dirichlet spaces such as fractals.

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arXiv2023

Potential theory for nonlocal drift-diffusion equations

The purpose of this paper is to prove new fine regularity results for nonlocal drift-diffusion equations via pointwise potential estimates. Our analysis requires only minimal assumptions on the divergence free drift term, enabling us to include drifts of critical order belonging merely to BMO. In particular, our results allow to derive new estimates for the dissipative surface quasi-geostrophic equation.

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arXiv2023

Finite-time singularity formation for the heat flow of the $H$-system

We construct the first example of finite time blow-up solutions for the heat flow of the $H$-system, describing the evolution of surfaces with constant mean curvature \begin{equation*} \left\{ \begin{aligned} &u_t = \Delta u - 2u_{x_1}\wedge u_{x_2}~\quad\text{ in }~\mathbb{R}^2\times\mathbb{R}_+,\\ &u(\cdot, 0) = u_0~\qquad\qquad~\text{ in }~\mathbb{R}^2, \end{aligned} \right. \end{equation*} where $u$: $\mathbb{R}^2\times\mathbb{R}_+\to \mathbb{R}^3$. The singularity at finite time forms as a scaled least energy $H$-bubble, denoted as $W$, exhibiting type II blow-up speed. One key observation is that the linearized operators around $W$ projected onto $W^\perp$ and in the $W$-direction are in fact decoupled. On $W^\perp$, the linearization is the linearized harmonic map heat flow, while in the $W$-direction, it is the linearized Liouville-type flow. Based on this, we also prove the non-degeneracy of the $H$-bubbles with any degree.

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arXiv2023

The singular sets of degenerate and nonlocal elliptic equations on Poincar\'e-Einstein manifolds

The main objects of this paper include some degenerate and nonlocal elliptic operators which naturally arise in the conformal invariant theory of Poincar\'e-Einstein manifolds. These operators generally reflect the correspondence between the Riemannian geometry of a complete Poincar\'e-Einstein manifold and the conformal geometry of its associated conformal infinity. In this setting, we develop the quantitative differentiation theory that includes quantitative stratification for the singular set and Minkowski type estimates for the (quantitatively) stratified singular sets. All these, together with a new $\epsilon$-regularity result for degenerate/singular elliptic operators on Poincar\'e-Einstein manifolds, lead to uniform Hausdorff measure estimates for the singular sets. Furthermore, the main results in this paper provide a delicate synergy between the geometry of Poincar\'e-Einstein manifolds and the elliptic theory of associated degenerate elliptic operators.

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arXiv2023

An improved eigenvalue estimate for embedded minimal hypersurfaces in the sphere

Suppose that $\Sigma^n\subset\mathbb{S}^{n+1}$ is a closed embedded minimal hypersurface. We prove that the first non-zero eigenvalue $\lambda_1$ of the induced Laplace-Beltrami operator on $\Sigma$ satisfies $\lambda_1 \geq \frac{n}{2}+ a_n(\Lambda^6 + b_n)^{-1}$, where $a_n$ and $b_n$ are explicit dimensional constants and $\Lambda$ is an upper bound for the length of the second fundamental form of $\Sigma$. This provides the first explicitly computable improvement on Choi & Wang's lower bound $\lambda_1 \geq \frac{n}{2}$ without any further assumptions on $\Sigma$.

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arXiv2023

Non-compactness results for the spinorial Yamabe-type problems with non-smooth geometric data

Let $(M,\textit{g},\sigma)$ be an $m$-dimensional closed spin manifold, with a fixed Riemannian metric $\textit{g}$ and a fixed spin structure $\sigma$; let $\mathbb{S}(M)$ be the spinor bundle over $M$. The spinorial Yamabe-type problems address the solvability of the following equation \[ D_{\textit{g}}\psi = f(x)|\psi|_{\textit{g}}^{\frac2{m-1}}\psi, \quad \psi:M\to\mathbb{S}(M), \ x\in M \] where $D_{\textit{g}}$ is the associated Dirac operator and $f:M\to\mathbb{R}$ is a given function. The study of such nonlinear equation is motivated by its important applications in Spin Geometry: when $m=2$, a solution corresponds to a conformal isometric immersion of the universal covering $\widetilde M$ into $\mathbb{R}^3$ with prescribed mean curvature $f$; meanwhile, for general dimensions and $f\equiv constant\neq0$, a solution provides an upper bound estimate for the B\"ar-Hijazi-Lott invariant. The aim of this paper is to establish non-compactness results related to the spinorial Yamabe-type problems. Precisely, concrete analysis is made for two specific models on the manifold $(S^m,\textit{g})$ where the solution set of the spinorial Yamabe-type problem is not compact: $1).$ the geometric potential $f$ is constant (say $f\equiv1$) with the background metric $\textit{g}$ being a $C^k$ perturbation of the canonical round metric $\textit{g}_{S^m}$, which is not conformally flat somewhere on $S^m$; $2).$ $f$ is a perturbation from constant and is of class $C^2$, while the background metric $\textit{g}\equiv\textit{g}_{S^m}$.

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