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[J]. Electronic Research Archive, 2021, 29(2): 2077-2099. doi: 10.3934/era.2020106
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.
[1] |
The null condition for quasilinear wave equations in two space dimensions. I. Invent. Math. (2001) 145: 597-618. ![]() |
[2] |
The null condition for quasilinear wave equations in two space dimensions. II. Amer. J. Math. (2001) 123: 1071-1101. ![]() |
[3] |
Dynamique des gaz à masse totale finie. Asymptotic Anal. (1990) 3: 215-220. ![]() |
[4] |
Interaction of rarefaction waves in jet stream. J. Differential. Equations (2010) 248: 2931-2954. ![]() |
[5] |
Interaction of rarefaction waves and vacuum in a convex duct. Arch. Ration. Mech. Anal. (2014) 213: 423-446. ![]() |
[6] |
D. Christodoulou, The Formation of Shocks in 3-Dimensional Fluids, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2007. doi: 10.4171/031
![]() |
[7] | R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, NY, 1948. |
[8] |
Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum. Comm. Pure Appl. Math. (2011) 64: 328-366. ![]() |
[9] |
Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum. Arch. Ration. Mech. Anal. (2012) 206: 515-616. ![]() |
[10] |
Global smooth solutions to Euler equations for a perfect gas. Indiana Univ. Math. J. (1998) 47: 1397-1432. ![]() |
[11] |
Expanding large global solutions of the equations of compressible fluid mechanics. Invent. Math. (2018) 214: 1205-1266. ![]() |
[12] |
Well-posedness for compressible Euler equations with physical vacuum singularity. Comm. Pure Appl. Math. (2009) 62: 1327-1385. ![]() |
[13] |
Well-posedness of compressible Euler equations in a physical vacuum. Comm. Pure Appl. Math. (2015) 68: 61-111. ![]() |
[14] | S. Klainerman, The null condition and global existence to nonlinear wave equations, in Nonlinear Systems of Partial Differential Equations in Applied Mathematics, Part 1, Lectures in Appl. Math., 23, Amer. Math. Soc., Providence, RI, 1986,293–326. |
[15] | T. T. Li, Global Classical Solutions for Quasilinear Hyperbolic Systems, RAM: Research in Applied Mathematics, 32, Masson, Paris; John Wiley & Sons, Ltd., Chichester, 1994. |
[16] | A necessary and sufficient condition for the global existence of smooth solutions to Cauchy problems for first-order quasilinear hyperbolic systems. Acta Math. Sinica (1985) 28: 606-613. |
[17] |
Compressible flow with damping and vacuum. Japan J. Indust. Appl. Math. (1996) 13: 25-32. ![]() |
[18] |
Vacuum states for compressible flow. Discrete Contin. Dynam. Systems (1998) 4: 1-32. ![]() |
[19] |
Compressible flow with vacuum and physical singularity. Methods Appl. Anal. (2000) 7: 495-509. ![]() |
[20] |
Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity. Invent. Math. (2018) 214: 1-169. ![]() |
[21] |
Formation of singularities in compressible fluids in two-space dimensions. Proc. Amer. Math. Soc. (1989) 107: 705-714. ![]() |
[22] |
Solutions classiques globales des équations d'Euler pour un fluide parfait compressible. Ann. Inst. Fourier (Grenoble) (1997) 47: 139-153. ![]() |
[23] |
Formation of singularities in three-dimensional compressible fluids. Commun. Math. Phys. (1985) 101: 475-485. ![]() |
[24] |
Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum. Arch. Ration. Mech. Anal. (2017) 225: 141-176. ![]() |
[25] |
Global smooth supersonic flows in infinite eapanding nozzles. SIAM J. Math. Anal. (2015) 47: 3151-3211. ![]() |
[26] |
Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density. Comm. Pure Appl. Math. (1998) 51: 229-240. ![]() |
[27] |
On global multidimensional supersonic flows with vacuum states at infinity. Arch. Ration. Mech. Anal. (2015) 218: 1189-1238. ![]() |
[28] |
The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D irrotational Euler equations. Phys. Scr. (2018) 93: 1-35. ![]() |
[29] |
On global smooth solutions of 3-D compressible Euler equations with vanishing density, in infinitely expanding balls. Discrete Contin. Dyn. Syst. A (2020) 40: 2213-2265. ![]() |
[30] |
A necessary and sufficient condition for global existence of classical solutions to Cauchy problem of quasilinear hyperbolic systems in diagonal form. Acta Math. Sci. Ser. B (Engl. Ed.) (2000) 20: 571-576. ![]() |