Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data☆☆Partially supported by the National Natural Science Foundation of China under contracts 10471138, 10601059, and 10526039 (Tianyuan Jijin); NSFC–NSAF Grant No. 10676037; 973 project of China, Grant No. 2006CB805902; Zheng Ge Ru Funds, Grants from RGC of HKSAR CUHK4028/04P and CUHK4299/02P.

AbstractThis paper concerns the large time behavior of strong and classical solutions to the two-dimensional Stokes approximation equations for the compressible flows. We consider the unique global strong solution or classical solution to the two-dimensional Stokes approximation equations for the compressible flows with large external potential force, together with a Navier-slip boundary condition, for arbitrarily large initial data. Under the conditions that the corresponding steady state exists uniquely with the steady state density away from vacuum, we prove that the density is bounded from above independently of time, consequently, it converges to the steady state density in Lp and the velocity u converges to the steady state velocity in W1,p for any 1⩽p<∞ as time goes to infinity; furthermore, we show that if the initial density contains vacuum at least at one point, then the derivatives of the density must blow up as time goes to infinity