Incompressible Euler equations in 3D bounded domains in a critical space

Abstract

We consider the 3D incompressible Euler equations in bounded domains $\hat{I}\text{c◯}$ with smooth boundary $\partial \hat{I}\text{c◯}$. Based on the paper by Iwabuchi, Matsuyama and Taniguchi (2019), we define the Besov space $B_{p,q}^s(A)$ by means of the Stokes operator $A$ with the Neumann boundary condition on $\partial \hat{I}\text{c◯}$, and prove unique local existence theorem of strong solution for the initial data in the critical Besov space $B_{2,1}^{\frac{5}{2}}(A)$. Our proof relies on the method of vanishing viscosity. The commutator estimate plays an essential role for derivation of energy bounds which hold uniformly with respect to viscosity constants.