Global Existence of Solutions for a Nonstrictly Hyperbolic System
We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded L∞ estimates z(ρδ,ε,uδ,ε)≤B(x) and w(ρδ,ε,uδ,ε)≤β when a(x) is increasing (similarly, w(ρδ,ε, uδ,ε)≤B(x...
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| Format: | Article |
| Language: | English |
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Wiley
2014-01-01
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| Series: | Abstract and Applied Analysis |
| Online Access: | http://dx.doi.org/10.1155/2014/691429 |
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| author | De-yin Zheng Yun-guang Lu Guo-qiang Song Xue-zhou Lu |
| author_facet | De-yin Zheng Yun-guang Lu Guo-qiang Song Xue-zhou Lu |
| author_sort | De-yin Zheng |
| collection | DOAJ |
| description | We obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded L∞ estimates z(ρδ,ε,uδ,ε)≤B(x) and w(ρδ,ε,uδ,ε)≤β when a(x) is increasing (similarly, w(ρδ,ε, uδ,ε)≤B(x) and z(ρδ,ε,uδ,ε)≤β when a(x) is decreasing) for the ε-viscosity and δ-flux approximation solutions of nonhomogeneous, resonant system without the restriction z0(x)≤0 or w0(x)≤0 as given in Klingenberg and Lu (1997), where z and w are Riemann invariants of nonhomogeneous, resonant system; B(x)>0 is a uniformly bounded function of x depending only on the function a(x) given in nonhomogeneous, resonant system, and β is the bound of B(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions. |
| format | Article |
| id | doaj-art-3962c8e8c20c49a29a3fe317b4e21cfe |
| institution | Kabale University |
| issn | 1085-3375 1687-0409 |
| language | English |
| publishDate | 2014-01-01 |
| publisher | Wiley |
| record_format | Article |
| series | Abstract and Applied Analysis |
| spelling | doaj-art-3962c8e8c20c49a29a3fe317b4e21cfe2025-02-03T01:33:06ZengWileyAbstract and Applied Analysis1085-33751687-04092014-01-01201410.1155/2014/691429691429Global Existence of Solutions for a Nonstrictly Hyperbolic SystemDe-yin Zheng0Yun-guang Lu1Guo-qiang Song2Xue-zhou Lu3Department of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaDepartment of Mathematics, Hangzhou Normal University, Hangzhou 310036, ChinaCollege of Health Administration, Anhui Medical University, Hefei 230032, ChinaLaboratoire Ondes and Milieux Complexes UMR 6294, CNRS-Universite du Havre, 76058 Le Havre, FranceWe obtain the global existence of weak solutions for the Cauchy problem of the nonhomogeneous, resonant system. First, by using the technique given in Tsuge (2006), we obtain the uniformly bounded L∞ estimates z(ρδ,ε,uδ,ε)≤B(x) and w(ρδ,ε,uδ,ε)≤β when a(x) is increasing (similarly, w(ρδ,ε, uδ,ε)≤B(x) and z(ρδ,ε,uδ,ε)≤β when a(x) is decreasing) for the ε-viscosity and δ-flux approximation solutions of nonhomogeneous, resonant system without the restriction z0(x)≤0 or w0(x)≤0 as given in Klingenberg and Lu (1997), where z and w are Riemann invariants of nonhomogeneous, resonant system; B(x)>0 is a uniformly bounded function of x depending only on the function a(x) given in nonhomogeneous, resonant system, and β is the bound of B(x). Second, we use the compensated compactness theory, Murat (1978) and Tartar (1979), to prove the convergence of the approximation solutions.http://dx.doi.org/10.1155/2014/691429 |
| spellingShingle | De-yin Zheng Yun-guang Lu Guo-qiang Song Xue-zhou Lu Global Existence of Solutions for a Nonstrictly Hyperbolic System Abstract and Applied Analysis |
| title | Global Existence of Solutions for a Nonstrictly Hyperbolic System |
| title_full | Global Existence of Solutions for a Nonstrictly Hyperbolic System |
| title_fullStr | Global Existence of Solutions for a Nonstrictly Hyperbolic System |
| title_full_unstemmed | Global Existence of Solutions for a Nonstrictly Hyperbolic System |
| title_short | Global Existence of Solutions for a Nonstrictly Hyperbolic System |
| title_sort | global existence of solutions for a nonstrictly hyperbolic system |
| url | http://dx.doi.org/10.1155/2014/691429 |
| work_keys_str_mv | AT deyinzheng globalexistenceofsolutionsforanonstrictlyhyperbolicsystem AT yunguanglu globalexistenceofsolutionsforanonstrictlyhyperbolicsystem AT guoqiangsong globalexistenceofsolutionsforanonstrictlyhyperbolicsystem AT xuezhoulu globalexistenceofsolutionsforanonstrictlyhyperbolicsystem |