On existence of global classical solutions to the 3D compressible MHD equations with vacuum
Abstract In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R 3 × [ 0 , T ] $\mathbb{R}^{3}\times [0, T]$ with a...
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| Format: | Article |
| Language: | English |
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SpringerOpen
2022-01-01
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| Series: | Journal of Inequalities and Applications |
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| Online Access: | https://doi.org/10.1186/s13660-022-02753-9 |
| _version_ | 1819260900881727488 |
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| author | Mingyu Zhang |
| author_facet | Mingyu Zhang |
| author_sort | Mingyu Zhang |
| collection | DOAJ |
| description | Abstract In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R 3 × [ 0 , T ] $\mathbb{R}^{3}\times [0, T]$ with any T ∈ ( 0 , ∞ ) $T\in (0, \infty )$ , provided that the initial magnetic field in the L 3 $L^{3}$ -norm and the initial density are suitably small. Note that the first result is obtained under the condition of ρ 0 ∈ L γ ∩ W 2 , q $\rho _{0}\in L^{\gamma }\cap W^{2, q}$ with q ∈ ( 3 , 6 ) $q\in (3, 6)$ and γ ∈ ( 1 , 6 ) $\gamma \in (1, 6)$ . It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., ρ 0 ∉ L 1 $\rho _{0} \notin L^{1}$ ). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved. |
| first_indexed | 2024-12-23T19:33:15Z |
| format | Article |
| id | doaj.art-42ce2f0cf37c42d3927f701d082a2bec |
| institution | Directory Open Access Journal |
| issn | 1029-242X |
| language | English |
| last_indexed | 2024-12-23T19:33:15Z |
| publishDate | 2022-01-01 |
| publisher | SpringerOpen |
| record_format | Article |
| series | Journal of Inequalities and Applications |
| spelling | doaj.art-42ce2f0cf37c42d3927f701d082a2bec2022-12-21T17:33:51ZengSpringerOpenJournal of Inequalities and Applications1029-242X2022-01-012022112510.1186/s13660-022-02753-9On existence of global classical solutions to the 3D compressible MHD equations with vacuumMingyu Zhang0School of Mathematics and Information Science, Weifang UniversityAbstract In this paper, the existence of global classical solutions is justified for the three-dimensional compressible magnetohydrodynamic (MHD) equations with vacuum. The main goal of this paper is to obtain a unique global classical solution on R 3 × [ 0 , T ] $\mathbb{R}^{3}\times [0, T]$ with any T ∈ ( 0 , ∞ ) $T\in (0, \infty )$ , provided that the initial magnetic field in the L 3 $L^{3}$ -norm and the initial density are suitably small. Note that the first result is obtained under the condition of ρ 0 ∈ L γ ∩ W 2 , q $\rho _{0}\in L^{\gamma }\cap W^{2, q}$ with q ∈ ( 3 , 6 ) $q\in (3, 6)$ and γ ∈ ( 1 , 6 ) $\gamma \in (1, 6)$ . It should be noted that the initial total energy can be arbitrarily large, the initial density allowed to vanish, and the system does not satisfy the conservation law of mass (i.e., ρ 0 ∉ L 1 $\rho _{0} \notin L^{1}$ ). Thus, the results obtained particularly extend the one due to Li–Xu–Zhang (Li et al. in SIAM J. Math. Anal. 45:1356–1387, 2013), where the global well-posedness of classical solutions with small energy was proved.https://doi.org/10.1186/s13660-022-02753-9Compressible magnetohydrodynamic equationsCauchy problemGlobal classical solutionSmall densityVacuum |
| spellingShingle | Mingyu Zhang On existence of global classical solutions to the 3D compressible MHD equations with vacuum Journal of Inequalities and Applications Compressible magnetohydrodynamic equations Cauchy problem Global classical solution Small density Vacuum |
| title | On existence of global classical solutions to the 3D compressible MHD equations with vacuum |
| title_full | On existence of global classical solutions to the 3D compressible MHD equations with vacuum |
| title_fullStr | On existence of global classical solutions to the 3D compressible MHD equations with vacuum |
| title_full_unstemmed | On existence of global classical solutions to the 3D compressible MHD equations with vacuum |
| title_short | On existence of global classical solutions to the 3D compressible MHD equations with vacuum |
| title_sort | on existence of global classical solutions to the 3d compressible mhd equations with vacuum |
| topic | Compressible magnetohydrodynamic equations Cauchy problem Global classical solution Small density Vacuum |
| url | https://doi.org/10.1186/s13660-022-02753-9 |
| work_keys_str_mv | AT mingyuzhang onexistenceofglobalclassicalsolutionstothe3dcompressiblemhdequationswithvacuum |