arXiv subjects

Aurélien Monteillet

Publications and source records attributed to Aurélien Monteillet.

3 recordsLinked to original sources

arXiv2010

Viscosity solutions for a polymer crystal growth model

We prove existence of a solution for a polymer crystal growth model describing the movement of a front $(Γ(t))$ evolving with a nonlocal velocity. In this model the nonlocal velocity is linked to the solution of a heat equation with source $δ_Γ$. The proof relies on new regularity results for the eikonal equation, in which the velocity is positive but merely measurable in time and with Hölder bounds in space. From this result, we deduce \textit{a priori} regularity for the front. On the other hand, under this regularity assumption, we prove bounds and regularity estimates for the solution of the heat equation.

math.AP↗

arXiv2008

Existence of weak solutions for general nonlocal and nonlinear second-order parabolic equations

In this article, we provide existence results for a general class of nonlocal and nonlinear second-order parabolic equations. The main motivation comes from front propagation theory in the cases when the normal velocity depends on the moving front in a nonlocal way. Among applications, we present level-set equations appearing in dislocations' theory and in the study of Fitzhugh-Nagumo systems.

math.AP↗

arXiv2008

Uniqueness Results for Nonlocal Hamilton-Jacobi Equations

We are interested in nonlocal Eikonal Equations describing the evolution of interfaces moving with a nonlocal, non monotone velocity. For these equations, only the existence of global-in-time weak solutions is available in some particular cases. In this paper, we propose a new approach for proving uniqueness of the solution when the front is expanding. This approach simplifies and extends existing results for dislocation dynamics. It also provides the first uniqueness result for a Fitzhugh-Nagumo system. The key ingredients are some new perimeter estimates for the evolving fronts as well as some uniform interior cone property for these fronts.

math.AP↗