In this paper we prove that if (u, K) is an almost-minimizer of the Griffith functional and K is $\epsilon$-close to a plane in some ball B $\subset$ R N while separating the ball B in two big parts, then K is C 1,$\alpha$ in a slightly smaller ball. Our result contains and generalizes the 2 dimensional result of [4], with a different and more soph...
In 1981, Frisch and Morf [1] postulated the existence of complex singularities in solutions of Navier-Stokes equations. Present progress on this conjecture is hindered by the computational burden involved in simulations of the Euler equations or the Navier-Stokes equations at high Reynolds numbers. We investigate this conjecture in the case of flui...
We present our continuous efforts from a modeling and numerical viewpoint to develop a powerful and flexible mathematical and computational framework called Ocular Mathematical Virtual Simulator (OMVS). The OMVS aims to solve problems arising in biomechanics and hemodynamics within the human eye. We discuss our contribution towards improving the re...
We consider the focusing energy-critical nonlinear wave equation for radially symmetric initial data in space dimensions $D \ge 4$. This equation has a unique (up to sign and scale) nontrivial, finite energy stationary solution $W$, called the ground state. We prove that every finite energy solution with bounded energy norm resolves, continuously i...
In this article, we consider the general task of performing Gaussian process regression (GPR) on pointwise observations of solutions of the 3 dimensional homogeneous free space wave equation.In a recent article, we obtained promising covariance expressions tailored to this equation: we now explore the potential applications of these formulas.We fir...
The main purpose of this work is to obtain a comparison principle for viscosity solutions of a system of elliptic Walsh's spider Hamilton-Jacobi-Bellman equations, possessing a new boundary condition called non linear local-time Kirchhoff 's transmission. The main idea is to build test functions at the neighborhood of the vertex solutions of ODE, w...
We study the free Schrödinger equation on finite metric graphs with infinite ends. We give sufficient conditions to obtain the L^1 (R) → L^∞ (R) time decay rate at least t^{−1/2}. These conditions allow certain metric graphs with circles and/or with commensurable lengths of the bounded edges. Further we study the dynamics of the probability flow be...
In recent years, there has been a spike in interest in multi-phase tissue growth models. Depending on the type of tissue, the velocity is linked to the pressure through Stoke's law, Brinkman's law or Darcy's law. While each of these velocity-pressure relations has been studied in the literature, little emphasis has been placed on the fine relations...
Enlightened by Lemma 1.7 in \cite{LiangLuo2021}, we prove a similar lemma which is based upon oscillatory integrals and Langer's turning point theory. From it we show that the Schr{\"o}dinger equation $${\rm i}\partial_t u = -\partial_x^2 u+x^2 u+\epsilon \langle x\rangle^\mu\sum_{k\in\Lambda}\left(a_k(\omega t)\sin(k|x|^\beta)+b_k(\omega t) \cos(k...
We consider the Schrodinger equation with an external potential and a cubic nonlinearity, in the semiclassical limit. The initial data are sums of WKB states, with smooth phases and smooth, compactly supported initial amplitudes, with disjoint supports. We show that according to the size of the initial data, a superposition principle may or may not...
