American Institute of Mathematical Sciences

June  2021, 29(2): 2077-2099. doi: 10.3934/era.2020106

The global supersonic flow with vacuum state in a 2D convex duct

 School of Mathematical Sciences and Mathematical Institute, Nanjing Normal University, Nanjing 210023, China

* Corresponding author: Gang Xu

Received  June 2020 Revised  August 2020 Published  September 2020

Fund Project: The third author is supported by NSFC grants No.11571141, No.11971237 and the Natural Science Foundation of the Jiangsu Higher Education Institutions of China (No.19KJA320001)

This paper concerns the motion of the supersonic potential flow in a two-dimensional expanding duct. In the case that two Riemann invariants are both monotonically increasing along the inlet, which means the gases are spread at the inlet, we obtain the global solution by solving the problem in those inner and border regions divided by two characteristics in $(x, y)$-plane, and the vacuum will appear in some finite place adjacent to the boundary of the duct. In addition, we point out that the vacuum here is not the so-called physical vacuum. On the other hand, for the case that at least one Riemann invariant is strictly monotonic decreasing along some part of the inlet, which means the gases have some local squeezed properties at the inlet, we show that the $C^1$ solution to the problem will blow up at some finite location in the non-convex duct.

Citation: Jintao Li, Jindou Shen, Gang Xu. The global supersonic flow with vacuum state in a 2D convex duct. Electronic Research Archive, 2021, 29 (2) : 2077-2099. doi: 10.3934/era.2020106
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References:
Supersonic flow in 2D convex duct
A global smooth solution with vacuum in 2D convex duct
Inner regions and border regions
Goursat problem in inner region
The case that $y(x_{N-1}) = f(x_{N-1})$
The case that $y(x_{N-1}) = -f(x_{N-1})$
Solution in border region
Blowup in 2D straight duct
Solution in ${\Omega}_{vac}$
The regularity near vacuum boundary
The case without vacuum
The image in the $(u, v)-$plane
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