2024 / December Volume 19 No.4
Compressible Navier-Stokes equation with BV initial data PART II. Global stability
| Published Date |
2024 / December
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| Title | Compressible Navier-Stokes equation with BV initial data PART II. Global stability |
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| Pagination | 251-364 |
| Abstract | In the previous work (Arch. Ration. Mech. Anal. 245: 375-477, 2022), we studied the well-posedness of weak solutions to the 1-D full compressible Navier-Stokes equation with initial data of small total variation. The local existence, the regularity, and the uniqueness in certain function spaces of the weak solution have been established. Moreover, the global existence is also announced there. In this paper, we continue to investigate the global stability and the time asymptotic behavior of the weak solution, and provide the details of the proof. The main step is to construct the "effective Green's functions". The first is the heat kernel with BV coefficients for a short time. The second is the Green’s function around a constant state for a long time. The former one captures the quasi-linear nature of the system, while the latter respects the dissipative structure. Then the weak solution is written into an integral system in terms of this "effective Green's function", and the time asymptotic behavior is established based on a priori estimate.
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| DOI | |
| AMS Subject Classification |
76N06, 35Q35
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| Received |
2024-10-22
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