The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in \(R^ 3\)
We consider the motion of an ideal incompressible fluid past a finite number of isolated rigid bodies \(O_ 1,...,O_ m\) in \({\mathbb{R}}^ 3.\) The velocity \(v=(v\) 1(x,t),v 2(x,t),v 3(x,t)) and the (scalar) pressure \(p=p(x,t)\) of the fluid motion are governed by the Euler equation in \(\Omega ={\mathbb{R}}\) \(3\setminus (O_ 1\cup...\cup O_ m)\) \[ (1)\quad \partial v/\partial t+(v\cdot \nabla)v+\nabla p=f,\quad div v=0,\quad t\in [0,T] \] subject to the following conditions at infinity and on the boundary \(S=\partial \Omega\) \[ (2)\quad \lim_{| x| \to \infty}v(x,t)=v_{\infty},\quad v\cdot n|_ S=0,\quad t\in [0,T], \] satisfying the initial condition (3) \(v(x,0)=v_ 0(x)\), where \(f=f(x,t)\) is a given external force vector, \(v_ 0(x)\) is a given initial velocity and \(v_{\infty}\) is a given constant vector. The purpose of the present paper is to prove the existence of a solution \(\{\) v,p\(\}\) of (1), (2) and (3) which satisfies the asymptotic condition at infinity that v converges to \(v_{\infty}\) faster than \(| x|^{-(1+\delta)}\) for a certain \(\delta\geq 0\).
scientific article; zbMATH DE number 4044308