New Thought on Matsumura–Nishida Theory in the L_p–L_q Maximal Regularity Framework

Abstract

This paper is devoted to proving the global well-posedness of initial-boundary value problem for Navier–Stokes equations describing the motion of viscous, compressible, barotropic fluid flows in a three dimensional exterior domain with non-slip boundary conditions. This was first proved by an excellent paper due to Matsumura and Nishida (Commun Math Phys 89:445–464, 1983). In [10], they used energy method and their requirement was that space derivatives of the mass density up to third order and space derivatives of the velocity fields up to fourth order belong to L_2 in space-time, detailed statement of Matsumura and Nishida theorem is given in Theorem 1 of Sect. 1 of context. This requirement is essentially used to estimate the L_infty norm of necessary order of derivatives in order to enclose the iteration scheme with the help of Sobolev inequalities and also to treat the material derivatives of the mass density. On the other hand, this paper gives the global wellposedness of the same problem as in [10] in L_p (1 <p le 2) in time and L_2cap L_6 in space maximal regularity class, which is an improvement of the Matsumura and Nishida theory in [10] from the point of view of the minimal requirement of the regularity of solutions. In fact, after changing the material derivatives to time derivatives by Lagrange transformation, enough estimates obtained by combination of the maximal L_p (1 <p le 2) in time and L_2cap L_6 in space regularity and L_p–L_q decay estimate of the Stokes equations with non-slip conditions in the compressible viscous fluid flow case enable us to use the standard Banach’s fixed point argument. Moreover, one of the purposes of this paper is to present a framework to prove the L_p–L_q maximal regularity for parabolic-hyperbolic type equations with non-homogeneous boundary conditions and how to combine the maximal L_p–L_q regularity and L_p–L_q decay estimates of linearized equations to prove the global well-posedness of quasilinear problems in unbounded domains, which gives a new thought of proving the global well-posedness of initial-boundary value problems for systems of parabolic or parabolic-hyperbolic equations appearing in mathematical physics.