GLOBAL REGULAR SOLUTIONS FOR ONE-DIMENSIONAL DEGENERATE COMPRESSIBLE NAVIER-STOKES EQUATIONS WITH LARGE DATA AND FAR FIELD VACUUM

In this paper, the Cauchy problem for the one-dimensional isentropic compressible Navier-Stokes equations (CNS) is considered. When the viscosity mu(rho) depends on the density rho in a sublinear power law (rho(delta) with 0 < delta <= 1), based on an elaborate analysis of the intrinsic singular structure of this degenerate system, we prove the global-in-time well-posedness of regular solutions with conserved total mass, momentum, and finite total energy in some inhomogeneous Sobolev spaces. Moreover, the solutions we obtained satisfy that rho keeps positive for all point x is an element of R but decays to zero in the far field, which is consistent with the facts that the total mass of the whole space is conserved, and CNS is a model of nondilute fluids where rho is bounded below away from zero. The key to the proof is the introduction of a well-designed reformulated structure by introducing some new variables and initial compatibility conditions, which, actually, can transfer the degeneracies of the time evolution and the viscosity to the possible singularity of some special source terms. Then, combined with the BD entropy estimates and transport properties of the so-called effective velocity v = u + phi + (rho)(x) (u is the velocity of the fluid, and :phi(rho) is a function of rho defined by phi'(rho) = mu(rho)/rho(2)), one can obtain the required uniform a priori estimates of corresponding solutions. Moreover, in contrast to the classical theory in the case of the constant viscosity, one can show that the L-infinity norm of u of the global regular solution we obtained does not decay to zero as time t goes to infinity. It is worth pointing out that the well-posedness theory established here can be applied to the viscous Saint-Venant system for the motion of shallow water.