Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density
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| Format: | Preprint |
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2026
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| _version_ | 1866912882930745344 |
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| author | Wang, Shuai Zhong, Xin |
| author_facet | Wang, Shuai Zhong, Xin |
| contents | We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_06303 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density Wang, Shuai Zhong, Xin Analysis of PDEs We establish the global existence of a class of weak solutions to the isentropic compressible Navier-Stokes and magnetohydrodynamic (MHD) equations on the whole plane under a suitably small initial energy. The solutions constructed here admit far-field vacuum and unbounded densities. Moreover, they possess an intermediate regularity regime between the finite-energy weak solutions of Lions-Feireisl and the framework of Hoff. This particularly extends our previous half-plane case with Dirichlet boundary conditions (arXiv:2601.11852) to the whole-plane MHD coupling, and we also generalize the works of Hoff (Comm. Pure Appl. Math. 55 (2002), pp. 1365-1407) and Suen and Hoff (Arch. Ration. Mech. Anal. 205 (2012), pp. 27-58) by allowing vacuum states and unbounded density. Our analysis lies in a new perspective that exploits the spatial integrability of the density and the resulting integrability of the pressure, together with the specific structure of the MHD system. |
| title | Global weak solutions with higher regularity to the two-dimensional isentropic compressible Navier-Stokes and magnetohydrodynamic equations with far-field vacuum and unbounded density |
| topic | Analysis of PDEs |
| url | https://arxiv.org/abs/2602.06303 |