LARGE TIME BEHAVIOR OF SOLUTIONS TO THE ISENTROPIC COMPRESSIBLE FLUID MODELS OF KORTEWEG TYPE IN R3

We consider the long-time behavior and optimal decay rates of global strong solutions to the isentropic compressible Navier-Stokes-Korteweg system in R-3. When the regular initial data belong to the Sobolev space Hl+1(R-3) boolean AND (B) over dot(1,infinity)(-s)(R-3) x H-l(R-3) boolean AND (B) over dot(1,infinity)(-s)(R-3) with l >= 3 and s is an element of [0,1], we show that the density and momentum of the system converges to its equilibrium state at the rates (1+t)(-3/4 -s/ 2) in the L-2-norm or (1+t)(-3/2 -s/2) in the L-infinity-norm, respectively, which are proved to be optimal for the compressible Navier-Stokes-Korteweg system.