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Finally, we prove a blow up criterion for the full compressible Navier-Stokes equations just in terms of the gradient of the velocity.In this thesis, the author study the blowup of solutions for strong and classical solutions to the compressible Navier-Stokes equations. In the first part, we prove a blow up criterion for strong solutions to the compressible Navier-Stokes equations, similar to the Beal-Kato-Majda criterion for the ideal incompressible flows.The same criterion for classical solutions to the compressible Navier-Stokes equations is established in the second part of this thesis. In addition, initial vacuum is allowed in both cases.Huang, Xiangdi.Adviser: Zhouping Xin.Source: Dissertation Abstracts International, Volume: 73-09(E), Section: B.Thesis (Ph.D.)--Chinese University of Hong Kong, 2009.Includes bibliographical references (leaves 90-96).Electronic reproduction. Hong Kong : Chinese University of Hong Kong, [2012] System requirements: Adobe Acrobat Reader. Available via World Wide Web.Electronic reproduction. Ann Arbor, MI : ProQuest Information and Learning Company, [200-] System requirements: Adobe Acrobat Reader. Available via World Wide Web.Abstract also in Chinese