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

We consider the 3D incompressible Euler equations in bounded domains $\mathrm{\Omega }$ with smooth boundary $\mathrm{\partial }\mathrm{\Omega }$. 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 $\mathrm{\partial }\mathrm{\Omega }$, 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.