Convergence to equilibria and blowup behavior of global strong solutions to the Stokes approximation equations for two-dimensional compressible flows with large data |
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Authors: | Feimin Huang Jing Li Zhouping Xin |
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Affiliation: | aInstitute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100080, PR China;bInstitute of Mathematical Sciences and Department of Mathematics, The Chinese University of Hong Kong, Shatin, N.T., Hong Kong |
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Abstract: | This 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. |
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Keywords: | Stokes approximation equations Large external potential forces Large-time behavior Uniform upper bound Vacuum Blowup |
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