Existence and stability of solutions to the compressible Euler equations with an outer force |
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| Affiliation: | 1. School of Mathematics and Statistics, Beijing Institute of Technology, Beijing 100081, PR China;2. School of Mathematics and Statistics Science, Ludong University, Yantai 264025, PR China;1. Eindhoven University of Technology, Department of Mechanical Engineering, P.O. Box 513, 5600 MB Eindhoven, The Netherlands;2. IBM Research, Server 3, IBM Tech. Campus, Damastown, Dublin 15, Ireland;1. Department of Mathematics, University of Houston, Houston, TX 77204, United States;2. Center for Numerical Porous Media (NumPor), King Abdullah University of Science and Technology (KAUST), Thuwal 23955-6900, Saudi Arabia;3. Department of Mathematical Sciences, Worcester Polytechnic Institute, 100 Institute Rd., Worcester, MA 01609, United States |
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| Abstract: | We study the compressible Euler equation with an outer force. The global existence theorem has been proved in many papers, provided that the outer force is bounded. However, the stability of their solutions has not yet been obtained until now. Our goal in this paper is to prove the existence of a global solution without such an assumption as boundedness. Moreover, we deduce a uniformly bounded estimate with respect to the time. This yields the stability of the solution.When we prove the global existence, the most difficult point is to obtain the bounded estimate for approximate solutions. To overcome this, we employ an invariant region, which depends on both space and time variables. To use the invariant region, we introduce a modified difference scheme. To prove their convergence, we apply the compensated compactness framework. |
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| Keywords: | The compressible Euler equation Outer force The compensated compactness Difference scheme |
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