Systems of partial differential equations of mixed hyperbolic–parabolic type
We prove the global existence and uniqueness of solutions of certain mixed hyperbolic–parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consi...
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| Veröffentlicht in: | Journal of Differential Equations Jg. 204; H. 1; S. 163 - 201 |
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| Hauptverfasser: | , |
| Format: | Journal Article |
| Sprache: | Englisch |
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Elsevier Inc
10.09.2004
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| ISSN: | 0022-0396, 1090-2732 |
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| Abstract | We prove the global existence and uniqueness of solutions of certain mixed hyperbolic–parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier–Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new. |
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| AbstractList | We prove the global existence and uniqueness of solutions of certain mixed hyperbolic–parabolic systems of partial differential equations in one space dimension with initial data that is assumed to be pointwise bounded with possibly large oscillation and with small total energy. The systems we consider are general enough to include the Navier–Stokes equations of compressible flow, the equations of compressible MHD, models of chemical combustion, and others. In particular, the application of our results to the MHD system gives an existence result which is new. |
| Author | Tsyganov, Eugene Hoff, David |
| Author_xml | – sequence: 1 givenname: Eugene surname: Tsyganov fullname: Tsyganov, Eugene organization: Max Planck Institute for Mathematics in the Sciences, Inselstrasse 22-26, D-04103 Leipzig, Germany – sequence: 2 givenname: David surname: Hoff fullname: Hoff, David email: hoff@indiana.edu organization: Department of Mathematics, Indiana University, Bloomington, IN 47405-7106, USA |
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| Cites_doi | 10.1006/jdeq.2001.4111 10.1016/0022-0396(92)90042-L 10.1007/s002050050055 10.1007/s00205-002-0233-6 10.1080/03605300008821583 |
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| Keywords | 35A05 Compressible magnetohydrodynamics Compressible Navier–Stokes equations 35M10 76W05 76N10 Systems of mixed hyperbolic–parabolic type |
| Language | English |
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| References | Racke (BIB8) 1992 Chen, Hoff, Trivisa (BIB2) 2003; 166 Cabannes (BIB1) 1970 Chen, Hoff, Trivisa (BIB3) 2000; 25 Hoff (BIB5) 1992; 95 S. Kawashima, Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph.D. Thesis, Kyoto University, 1983. Hoff (BIB6) 1997; 139 Williams (BIB9) 1965 Chen, Wang (BIB4) 2002; 182 Hoff (10.1016/j.jde.2004.03.010_BIB6) 1997; 139 Chen (10.1016/j.jde.2004.03.010_BIB4) 2002; 182 Williams (10.1016/j.jde.2004.03.010_BIB9) 1965 10.1016/j.jde.2004.03.010_BIB7 Chen (10.1016/j.jde.2004.03.010_BIB3) 2000; 25 Cabannes (10.1016/j.jde.2004.03.010_BIB1) 1970 Chen (10.1016/j.jde.2004.03.010_BIB2) 2003; 166 Hoff (10.1016/j.jde.2004.03.010_BIB5) 1992; 95 Racke (10.1016/j.jde.2004.03.010_BIB8) 1992 |
| References_xml | – year: 1965 ident: BIB9 publication-title: Combustion Theory – year: 1970 ident: BIB1 publication-title: Theoretical Magneto-fluid Dynamics – volume: 166 start-page: 321 year: 2003 end-page: 358 ident: BIB2 article-title: Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data publication-title: Arch. Rational Mech. Anal – reference: S. Kawashima, Systems of a hyperbolic–parabolic composite type, with applications to the equations of magnetohydrodynamics, Ph.D. Thesis, Kyoto University, 1983. – volume: 25 start-page: 2233 year: 2000 end-page: 2257 ident: BIB3 article-title: Global solutions of the compressible Navier–Stokes equations with large discontinuous initial data publication-title: Comm. Partial Differential Equations – volume: 182 start-page: 344 year: 2002 end-page: 376 ident: BIB4 article-title: Global solutions of nonlinear magnetohydrodynamics with large initial data publication-title: J. Differential Equations – volume: 139 start-page: 303 year: 1997 end-page: 354 ident: BIB6 article-title: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids publication-title: Arch. Rational Mech. Anal – volume: 95 start-page: 33 year: 1992 end-page: 74 ident: BIB5 article-title: Global well-posedness of the Cauchy problem for the Navier–Stokes equations of nonisentropic flow with discontinuous initial data publication-title: J. Differential Equations – year: 1992 ident: BIB8 publication-title: Lectures on Nonlinear Evolution Equations – volume: 182 start-page: 344 year: 2002 ident: 10.1016/j.jde.2004.03.010_BIB4 article-title: Global solutions of nonlinear magnetohydrodynamics with large initial data publication-title: J. Differential Equations doi: 10.1006/jdeq.2001.4111 – volume: 95 start-page: 33 year: 1992 ident: 10.1016/j.jde.2004.03.010_BIB5 article-title: Global well-posedness of the Cauchy problem for the Navier–Stokes equations of nonisentropic flow with discontinuous initial data publication-title: J. Differential Equations doi: 10.1016/0022-0396(92)90042-L – volume: 139 start-page: 303 year: 1997 ident: 10.1016/j.jde.2004.03.010_BIB6 article-title: Discontinuous solutions of the Navier–Stokes equations for multidimensional flows of heat-conducting fluids publication-title: Arch. Rational Mech. Anal doi: 10.1007/s002050050055 – volume: 166 start-page: 321 year: 2003 ident: 10.1016/j.jde.2004.03.010_BIB2 article-title: Global solutions to a model for exothermically reacting, compressible flows with large discontinuous initial data publication-title: Arch. Rational Mech. Anal doi: 10.1007/s00205-002-0233-6 – year: 1970 ident: 10.1016/j.jde.2004.03.010_BIB1 – volume: 25 start-page: 2233 year: 2000 ident: 10.1016/j.jde.2004.03.010_BIB3 article-title: Global solutions of the compressible Navier–Stokes equations with large discontinuous initial data publication-title: Comm. Partial Differential Equations doi: 10.1080/03605300008821583 – year: 1965 ident: 10.1016/j.jde.2004.03.010_BIB9 – ident: 10.1016/j.jde.2004.03.010_BIB7 – year: 1992 ident: 10.1016/j.jde.2004.03.010_BIB8 |
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| Title | Systems of partial differential equations of mixed hyperbolic–parabolic type |
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