We construct the global unique solution to the compressible Euler equations with damping in R 3 . We assume the H 3 norm of the initial data is small, but the higher order derivatives can be arbitrarily large. When theḢ −s norm (0 s < 3/2) orḂ −s 2,∞ norm (0 < s 3/2) of the initial data is finite, by a regularity interpolation trick, we prove the optimal decay rates of the solution. As an immediate byproduct, the L p -L 2 (1 p 2) type of the decay rates follow without requiring that the L p norm of initial data is small.

Abstract. This paper contains some interesting Hardy type inequalities and Rellich type inequalities for the left invariant vector fields on the Heisenberg group.

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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) with l ≥ 3 and s ∈ [0,1], we show that the density and momentum of the system converges to its equilibrium state at the ratesin the L ∞ -norm, respectively, which are proved to be optimal for the compressible Navier-Stokes-Korteweg system.