Erika Ushikoshi | Searcharxiv

arXiv2023

Helmholtz-Weyl decomposition on a time dependent domain for time periodic Navier-Stokes flows with large flux

We consider the Helmholtz-Weyl decomposition on a time dependent bounded domain $Ω(t)$ in $\mathbb{R}^3$. Especially, we investigate the domain dependence of each component in the decomposition, namely, the harmonic vector fields (i.e., $\mathrm{div}$ and $\mathrm{rot}$ free vectors), vector potentials, and scalar potentials equipped with suitable boundary conditions, when $Ω(t)$ moves along to $t\in\mathbb{R}$. As an application, we construct a time periodic solution of the incompressible Navier-Stokes equations for some large boundary data with non-zero fluxes.

math.AP↗

arXiv2018

The first Hadamard variation of Neumann-Poincaré eigenvalues on the sphere

The Neumann-Poincaré operator on the sphere has $\frac{1}{2(2k+1)}$, $k=0,1,2,\ldots$, as its eigenvalues and the corresponding multiplicity is $2k+1$. We consider the bifurcation of eigenvalues under deformation of domains, and show that Frechét derivative of the sum of the bifurcations is zero. We then discuss the connection of this result with some conjectures regarding the Neumann-Poincaré operator.

math.SP↗