We consider the motion of a viscous incompressible fluid in a container with moving walls, in other words we have to deal not with a space-time cylinder but with a non-cylinder domain in \({\mathbb{R}}^ 3\times [0,T]\). To be more precise, we consider a domain \[ \Omega_ T=\cup_{0\leq t\leq T}\Omega (t)x\{t\}, \] where each \(\Omega\) (t) is a bounded open set of \(R^ 3\) and \(T>0\) is a finite number. The motion of the fluid in \(\Omega_ T\) is governed by the following equations \[ (1)\quad \partial_ tu-\mu \Delta u+u\cdot \nabla u=f-\nabla p;\quad \nabla \cdot u=0\quad in\quad \Omega_ T, \] where \(\partial_ t=\partial /\partial_ t\), \(u=u(t)=u(x,t)=(u_ 1(x,t),u_ 2(x,t),u_ 3(x,t))\) is the velocity, \(p=p(t)=p(x,t)\) the pressure, \(f=f(t)=f(x,t)=(f_ 1(x,t),f_ 2(x,t),f_ 3(x,t))\) the external force, and \(\mu\) the viscosity. We take the motion of the fluid at \(t=0\) to be known, hence \(u(x,0)=u_ 0\) is a prescribed vector field on \(\Omega\) (0). Let \[ \Gamma_ T=\cup_{0\leq t\leq T}\Gamma (t)x\{t\}, \] where \(\Gamma\) (t) is the boundary of \(\Omega\) (t). We shall assume \(u_{\Gamma}=0\). The case of nonzero boundary data makes the problem more complicated but it would not appear to have any serious difficulty. The classical formulation of the problem is the velocity u and the pressure p have to satisfy (1) and the initial-boundary conditions \[ (2)\quad u=0\quad on\quad \Gamma;\quad u(x,0)=u_ 0\quad in\quad \Omega (0). \] For the cylindrical case: i.e. \(\Omega (t)=\Omega (0)\) for all \(t>0\) there exists a very extensive literature, whereas in the non-cylindrical case the theory is much less developed and it has been considered the last years only. In particular the paper of \textit{H. Fujita}, \textit{N. Sauer} [J. Fac. Sci., Univ. Tokyo, Sect. I A 17, 403-420 (1970; Zbl 0206.397)] generalizes the fundamental existence theorem of Hopf using a kind of penalty method. Other papers consider problem (1), (2) reducing it to one in a cylindrical domain. In this paper we improve the approach developed by the author [J. Fac. Sci., Univ. Tokyo, Sect. I A 32, 213-221 (1985; Zbl 0576.76028)] to show that certain other important results which are known in the cylindrical case carry over in a modified form to the non-cylindrical case as the regularity, and the decay. Section 2 is devoted to the notations, and to the statement of problems. Section 3 to the preliminaries. In Sect. 4 we shall prove the regularity of weak solutions. In Sect. 5 we shall show that a weak solution became smooth after some definite time \(T_ 0\), and we discuss the decay problem.