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, doi: 10.3934/era.2020106
References:
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D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344.  Google Scholar

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T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296.  Google Scholar

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T.-P. LiuZ. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.  doi: 10.3934/dcds.1998.4.1.  Google Scholar

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T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.  Google Scholar

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G. Xu and H. Yin, On global multidimensional supersonic flows with vacuum states at infinity, Arch. Ration. Mech. Anal., 218 (2015), 1189-1238.  doi: 10.1007/s00205-015-0878-6.  Google Scholar

[28]

G. Xu and H. Yin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D irrotational Euler equations, Phys. Scr., 93 (2018), 1-35.  doi: 10.1088/1402-4896/aad681.  Google Scholar

[29]

G. Xu and H. Yin, On global smooth solutions of 3-D compressible Euler equations with vanishing density, in infinitely expanding balls, Discrete Contin. Dyn. Syst. A, 40 (2020), 2213-2265.  doi: 10.3934/dcds.2020112.  Google Scholar

[30]

Y. Zheng and F. Liu, 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.), 20 (2000), 571-576.  doi: 10.1016/S0252-9602(17)30669-0.  Google Scholar

show all references

References:
[1]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. I, Invent. Math., 145 (2001), 597-618.  doi: 10.1007/s002220100165.  Google Scholar

[2]

S. Alinhac, The null condition for quasilinear wave equations in two space dimensions. II, Amer. J. Math., 123 (2001), 1071-1101.  doi: 10.1353/ajm.2001.0037.  Google Scholar

[3]

J.-Y. Chemin, Dynamique des gaz à masse totale finie, Asymptotic Anal., 3 (1990), 215-220.  doi: 10.3233/ASY-1990-3302.  Google Scholar

[4]

S. Chen and A. Qu, Interaction of rarefaction waves in jet stream, J. Differential. Equations, 248 (2010), 2931-2954.  doi: 10.1016/j.jde.2010.03.004.  Google Scholar

[5]

S. Chen and A. Qu, Interaction of rarefaction waves and vacuum in a convex duct, Arch. Ration. Mech. Anal., 213 (2014), 423-446.  doi: 10.1007/s00205-014-0738-9.  Google Scholar

[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.  Google Scholar

[7]

R. Courant and K. O. Friedrichs, Supersonic Flow and Shock Waves, Interscience Publishers, Inc., New York, NY, 1948.  Google Scholar

[8]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for moving-boundary 1-D compressible Euler equations in physical vacuum, Comm. Pure Appl. Math., 64 (2011), 328-366.  doi: 10.1002/cpa.20344.  Google Scholar

[9]

D. Coutand and S. Shkoller, Well-posedness in smooth function spaces for the moving-boundary three-dimensional compressible Euler equations in physical vacuum, Arch. Ration. Mech. Anal., 206 (2012), 515-616.  doi: 10.1007/s00205-012-0536-1.  Google Scholar

[10]

M. Grassin, Global smooth solutions to Euler equations for a perfect gas, Indiana Univ. Math. J., 47 (1998), 1397-1432.  doi: 10.1512/iumj.1998.47.1608.  Google Scholar

[11]

M. Hadžić and J. Jang, Expanding large global solutions of the equations of compressible fluid mechanics, Invent. Math., 214 (2018), 1205-1266.  doi: 10.1007/s00222-018-0821-1.  Google Scholar

[12]

J. Jang and N. Masmoudi, Well-posedness for compressible Euler equations with physical vacuum singularity, Comm. Pure Appl. Math., 62 (2009), 1327-1385.  doi: 10.1002/cpa.20285.  Google Scholar

[13]

J. Jang and N. Masmoudi, Well-posedness of compressible Euler equations in a physical vacuum, Comm. Pure Appl. Math., 68 (2015), 61-111.  doi: 10.1002/cpa.21517.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

D. Q. Li and T. H. Qin, A necessary and sufficient condition for the global existence of smooth solutions to Cauchy problems for first-order quasilinear hyperbolic systems, Acta Math. Sinica, 28 (1985), 606-613.   Google Scholar

[17]

T.-P. Liu, Compressible flow with damping and vacuum, Japan J. Indust. Appl. Math., 13 (1996), 25-32.  doi: 10.1007/BF03167296.  Google Scholar

[18]

T.-P. LiuZ. Xin and T. Yang, Vacuum states for compressible flow, Discrete Contin. Dynam. Systems, 4 (1998), 1-32.  doi: 10.3934/dcds.1998.4.1.  Google Scholar

[19]

T.-P. Liu and T. Yang, Compressible flow with vacuum and physical singularity, Methods Appl. Anal., 7 (2000), 495-509.  doi: 10.4310/MAA.2000.v7.n3.a7.  Google Scholar

[20]

J. Luk and J. Speck, Shock formation in solutions to the 2D compressible Euler equations in the presence of non-zero vorticity, Invent. Math., 214 (2018), 1-169.  doi: 10.1007/s00222-018-0799-8.  Google Scholar

[21]

M. A. Rammaha, Formation of singularities in compressible fluids in two-space dimensions, Proc. Amer. Math. Soc., 107 (1989), 705-714.  doi: 10.1090/S0002-9939-1989-0984811-5.  Google Scholar

[22]

D. Serre, Solutions classiques globales des équations d'Euler pour un fluide parfait compressible, Ann. Inst. Fourier (Grenoble), 47 (1997), 139-153.  doi: 10.5802/aif.1563.  Google Scholar

[23]

T. C. Sideris, Formation of singularities in three-dimensional compressible fluids, Commun. Math. Phys., 101 (1985), 475-485.  doi: 10.1007/BF01210741.  Google Scholar

[24]

T. C. Sideris, Global existence and asymptotic behavior of affine motion of 3D ideal fluids surrounded by vacuum, Arch. Ration. Mech. Anal., 225 (2017), 141-176.  doi: 10.1007/s00205-017-1106-3.  Google Scholar

[25]

C. Wang and Z. Xin, Global smooth supersonic flows in infinite eapanding nozzles, SIAM J. Math. Anal., 47 (2015), 3151-3211.  doi: 10.1137/140994289.  Google Scholar

[26]

Z. Xin, Blowup of smooth solutions to the compressible Navier-Stokes equation with compact density, Comm. Pure Appl. Math., 51 (1998), 229-240.  doi: 10.1002/(SICI)1097-0312(199803)51:3<229::AID-CPA1>3.0.CO;2-C.  Google Scholar

[27]

G. Xu and H. Yin, On global multidimensional supersonic flows with vacuum states at infinity, Arch. Ration. Mech. Anal., 218 (2015), 1189-1238.  doi: 10.1007/s00205-015-0878-6.  Google Scholar

[28]

G. Xu and H. Yin, The global existence and large time behavior of smooth compressible fluid in an infinitely expanding ball, I: 3D irrotational Euler equations, Phys. Scr., 93 (2018), 1-35.  doi: 10.1088/1402-4896/aad681.  Google Scholar

[29]

G. Xu and H. Yin, On global smooth solutions of 3-D compressible Euler equations with vanishing density, in infinitely expanding balls, Discrete Contin. Dyn. Syst. A, 40 (2020), 2213-2265.  doi: 10.3934/dcds.2020112.  Google Scholar

[30]

Y. Zheng and F. Liu, 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.), 20 (2000), 571-576.  doi: 10.1016/S0252-9602(17)30669-0.  Google Scholar

Figure 1.  Supersonic flow in 2D convex duct
Figure 2.  A global smooth solution with vacuum in 2D convex duct
Figure 3.  Inner regions and border regions
Figure 4.  Goursat problem in inner region
Figure 5.  The case that $ y(x_{N-1}) = f(x_{N-1}) $
Figure 6.  The case that $ y(x_{N-1}) = -f(x_{N-1}) $
Figure 7.  Solution in border region
Figure 8.  Blowup in 2D straight duct
Figure 9.  Solution in $ {\Omega}_{vac} $
Figure 10.  The regularity near vacuum boundary
Figure 11.  The case without vacuum
Figure 12.  The image in the $ (u, v)- $plane
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