LAUSR

For simplicity we assume d 2 {2, 3}; higher dimensions can be treated in the same way. The main goal of this paper is to establish the immediate gain of space-time analyticity for solutions to (1.1)–(1.2), using a direct energy-type method, in the case of a domain with curved boundary. Our main result is Theorem 2.8 below, which shows that from a Sobolev smooth initial datum the solution instantaneously becomes space-time analytic, with analyticity radius which is uniform up to the curved analytic boundary. The direct energy-type approach utilized in this paper was presented in [33] for the Stokes system and in [12] for the Navier-Stokes equations on the half space. This method is robust and easily expendable to the case of non-analytic Gevrey-classes, jointly in space-time, provided the boundary belongs to the same Gevrey class. Analyticity and Gevrey-class regularity have proven to be important for studying the vanishing viscosity problem for the Navier-Stokes equations in bounded domains [45, 46, 37, 30, 14, 50, 41, 19], and for establishing nonlinear inviscid damping near the Couette flow [3, 4, 5]. Moreover, the analyticity radius provides a measure of the minimal scale in a turbulent flow [24]. Analyticity and Gevrey-class regularity for the Navier-Stokes equation is a classical subject [13, 49]. Initially, interior analyticity for the Navier-Stokes system in d 2 space dimensions was proven by Kahane [25], using an iteration of high order Sobolev norms. The problem of interior space-time analyticity was then addressed by Masuda [39], and then by Kato-Masuda [26], assuming that the external force is analytic. Analyticity up to the boundary of the domain was established by Komatsu in [28, 29], based on earlier work by Kinderlehrer and Nirenberg [27] for parabolic type equations. Subsequently, Giga [20] developed a semigroup approach for analyticity up to the boundary for the Navier-Stokes system. On the other hand, in the absence of boundaries, Foias and Temam [18] introduced an alternative approach to analyticity and Gevrey-class regularity which is based on L2 energy estimates and Fourier analysis (cf. [17, 13] for an earlier energy approach for the time analyticity). This method has proven to be a powerful tool to establish analyticity as well as to estimate the analyticity or the Gevrey radius. The