On globally large smooth solutions of full compressible Navier–Stokes equations with moving boundary and temperature-dependent heat-conductivity

• Published in: Nonlinear analysis: real world applications Vol. 64; p. 103430
• Main Author: Ou, Yaobin
• Format: Journal Article
• Language: English
• Published: Elsevier Ltd 01.04.2022
• Subjects: Full Navier–Stokes equations; Global smooth solution; Large initial data; Radial symmetry; Temperature-dependent heat conductivity; Vacuum; Full Navier–Stokes equations; Large initial data; Radial symmetry; Temperature-dependent heat conductivity; Vacuum; Global smooth solution
• ISSN: 1468-1218; 1878-5719
• DOI: 10.1016/j.nonrwa.2021.103430

The global existence and uniqueness of smooth solution to the vacuum free boundary problem of full compressible Navier–Stokes equations with large initial data and radial symmetry is established in this paper, when the fluid connects to vacuum continuously. The main difficulty lies in the fact that, the system is strongly nonlinear with unknown boundary variables, and degenerate near the free boundary separating the fluid and vacuum, thus general theory does not apply to this case. To overcome this trouble, we establish the point-wise estimates on the upper and lower bounds of the deformation variable ηx, the refined estimate for the temperature, and also the uniform-in-time weighted energy estimates for the solutions with high regularity, by careful analysis. Moreover, the expanding rate of the free interface is also proved. The main ingredient of this paper is that the high regularity of the solution is always up to the free boundary. Previous results are only for weak solutions, or the case that the fluids connect to the vacuum with a jump. The assumption on the heat conductivity coefficient κ≈1+θq,q≥2 is also improved to be q>0.