June  2019, 39(6): 3535-3575. doi: 10.3934/dcds.2019146

Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow

1. 

Institute of Applied Mathematics, AMSS, Academia Sinica, Beijing 100190, China

2. 

Department of Mathematics, University of Pittsburgh, Pittsburgh, PA 15260, USA

3. 

School of Mathematical Sciences, University of Chinese Academy of Sciences, Beijing 100049, China

4. 

Institute of Applied Mathematics, AMSS, CAS, Beijing 100190, China

* Corresponding author: Difan Yuan

Received  September 2018 Published  February 2019

We are concerned with the vortex sheet solutions for the inviscid two-phase flow in two dimensions. In particular, the nonlinear stability and existence of compressible vortex sheet solutions under small perturbations are established by using a modification of the Nash-Moser iteration technique, where a priori estimates for the linearized equations have a loss of derivatives. Due to the jump of the normal derivatives of densities of liquid and gas, we obtain the normal estimates in the anisotropic Sobolev space, instead of the usual Sobolev space. New ideas and techniques are developed to close the energy estimates and derive the tame estimates for the two-phase flows.

Citation: Feimin Huang, Dehua Wang, Difan Yuan. Nonlinear stability and existence of vortex sheets for inviscid liquid-gas two-phase flow. Discrete & Continuous Dynamical Systems - A, 2019, 39 (6) : 3535-3575. doi: 10.3934/dcds.2019146
References:
[1]

S. Alinhac, Existence d'ondes de rarefaction pour des systemes quasi-lineaires hyperboliques multidimensionnels, Comm. Partial Differential Equation, 14 (1989), 173-230. doi: 10.1080/03605308908820595. Google Scholar

[2]

G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal, 187 (2008), 369-408. doi: 10.1007/s00205-007-0070-8. Google Scholar

[3]

G.-Q. Chen, P. Secchi and T. Wang, Nonlinear stability of relativistic vortex sheets in three dimensional Minkowski spacetime, preprint, arXiv: 1707.02672, 2017.Google Scholar

[4]

R. M. ChenJ. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math., 311 (2017), 18-60. doi: 10.1016/j.aim.2017.02.014. Google Scholar

[5]

R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in 2D elastodynamics: Variable coefficients, Math. Ann. (to appear), arXiv: 1804.07850, 2018.Google Scholar

[6]

R. M. Chen, J. Hu and D. Wang, Nonlinear stability and existence of compressible vortex sheets in two-dimensional elastodynamics, preprint, 2018.Google Scholar

[7]

S.-X. Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary, Front. Math. China, 2 (2007), 87-102. doi: 10.1007/s11464-007-0006-5. Google Scholar

[8]

J. F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pure Appl., 84 (2005), 786-818. doi: 10.1016/j.matpur.2004.10.005. Google Scholar

[9]

J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012. doi: 10.1512/iumj.2004.53.2526. Google Scholar

[10]

J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Ec. Norm.Super., 41 (2008), 85-139. doi: 10.24033/asens.2064. Google Scholar

[11]

S. Evje, Global weak solutions for a compressible gas-liquid model with well-deformation interaction, J. Diff. Eqs., 251 (2011), 2352-2386. doi: 10.1016/j.jde.2011.07.013. Google Scholar

[12]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM. J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. Google Scholar

[13]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Diff. Eqs., 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032. Google Scholar

[14]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867. Google Scholar

[15]

J. A. Fejer and W. Miles, On the stability of a plane vortex sheet with respect to three dimensional disturbances, J. Fluid Mech., 15 (1963), 335-336. doi: 10.1017/S002211206300029X. Google Scholar

[16]

J. Francheteau and G. Metivier, Existence of weak shocks for multidimenional hyperbolic quasilinear systems, Asterisque., 268 (2000), 1-198. Google Scholar

[17]

H. A. Friis and S. Evje, Global weak solutions for a gas-liquid model with external forces general pressure law, SIAM J. Appl. Math., 71 (2011), 409-442. doi: 10.1137/100813336. Google Scholar

[18]

H. A. Friis and S. Evje, Well-posedness of a compressible gas-liquid model with a friction term important for well control operations, SIAM J. Appl. Math., 71 (2011), 2014-2047. doi: 10.1137/110835499. Google Scholar

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[20]

C.-C. Hao and H.-L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332. doi: 10.1137/110851602. Google Scholar

[21]

F. Huang, J. Kuang, D. Wang and W. Xiang, Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle, J. Differential Equations, 266 (2019), 4337-4376, arXiv: 1804.04769, 2018. doi: 10.1016/j.jde.2018.09.036. Google Scholar

[22]

Q.-S. Jiu and Z.-P. Xin, On strong convergence to 3-D axisymmetric vortex sheets, J. Diff. Eqs., 223 (2006), 33-50. doi: 10.1016/j.jde.2005.04.001. Google Scholar

[23]

Q.-S. Jiu and Z.-P. Xin, On strong convergence to 3-D steady vortex sheets, J. Diff. Eqs., 239 (2007), 448-470. doi: 10.1016/j.jde.2007.05.008. Google Scholar

[24]

P. D. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. Google Scholar

[25]

R. L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci., 24 (2006), 481-491. doi: 10.1155/S0161171200002970. Google Scholar

[26]

A. Morando, P. Secchi and P. Trebeschi, On the evolution equation of compressible vortex sheets, preprint, arXiv: 1806.06740, 2018.Google Scholar

[27]

A. Morando and P. Trebeschi, Two-dimensional vortex sheets for the nonisentropic Euler equations: linear stability, J. Hyperbolic Differ. Equ., 5 (2008), 487-518. doi: 10.1142/S021989160800157X. Google Scholar

[28]

A. Morando, P. Trebeschi and T. Wang, Two-dimensional vortex sheets for the nonisentropic Euler equations: nonlinear stability, preprint, arXiv: 1808.09290, 2018.Google Scholar

[29]

A. MorandoY. Trakhinin and P. Trebeschi, Local existence of MHD contact discontinuities, Arch. Ration. Mech. Anal., 9 (2016), 289-313. doi: 10.3934/dcdss.2016.9.289. Google Scholar

[30]

M. OhnoY. Shizuta and T. Yanagisawa, The trace theorem on anisotropic Sobolev spaces, Tohoku Math. J., 46 (1994), 393-401. doi: 10.2748/tmj/1178225719. Google Scholar

[31]

S. Pai, Two-Phase Flows, Vieweg Tracts in Pure Appl. Phys., vol. 3, Vieweg, Braunschweig, 1977. Google Scholar

[32]

J. B. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861. Google Scholar

[33]

L. Ruan and Y. Trakhinin, Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results, preprint, arXiv: 1810.04386, 2018.Google Scholar

[34]

L. RuanD. WangS. Weng and C. Zhu, Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: linear stability, Commun. Math. Sci., 14 (2016), 735-776. doi: 10.4310/CMS.2016.v14.n3.a7. Google Scholar

[35]

P. Secchi, Well-posedness for a mixed problem for the equations of ideal magneto-hydrodynamics, Arch. Math., 64 (1995), 237-245. doi: 10.1007/BF01188574. Google Scholar

[36]

P. Secchi, Some properties of anisotropic Sobolev spaces, Arch. Math. (Basel), 75 (2000), 207-216. doi: 10.1007/s000130050494. Google Scholar

[37]

Y. Trakhinin, Existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331-366. doi: 10.1007/s00205-005-0364-7. Google Scholar

[38]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. doi: 10.1007/s00205-008-0124-6. Google Scholar

[39]

C. Wang and Z. Zhang, A new proof of Wu's Theorem on vortex sheets, Sci. China. Math., 55 (2012), 1449-1462. doi: 10.1007/s11425-012-4421-z. Google Scholar

[40]

Y.-G. Wang and F. Yu, Stability of contact discontinuities in three-dimensional compressible steady flows, J. Diff. Eqs., 255 (2013), 1278-1356. doi: 10.1016/j.jde.2013.05.014. Google Scholar

[41]

Y.-G. Wang and F. Yu, Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 208 (2013), 341-389. doi: 10.1007/s00205-012-0601-9. Google Scholar

[42]

Y.-G. Wang and H. Yuan, Weak stability of transonic contact discontinuities in three-diemensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388. doi: 10.1007/s00033-014-0404-y. Google Scholar

[43]

S. Wu, Mathematical analysis of vortex sheets, Commun. Pure Appl. Math., 59 (2006), 1065-1206. doi: 10.1002/cpa.20110. Google Scholar

show all references

References:
[1]

S. Alinhac, Existence d'ondes de rarefaction pour des systemes quasi-lineaires hyperboliques multidimensionnels, Comm. Partial Differential Equation, 14 (1989), 173-230. doi: 10.1080/03605308908820595. Google Scholar

[2]

G.-Q. Chen and Y.-G. Wang, Existence and stability of compressible current-vortex sheets in three-dimensional magnetohydrodynamics, Arch. Ration. Mech. Anal, 187 (2008), 369-408. doi: 10.1007/s00205-007-0070-8. Google Scholar

[3]

G.-Q. Chen, P. Secchi and T. Wang, Nonlinear stability of relativistic vortex sheets in three dimensional Minkowski spacetime, preprint, arXiv: 1707.02672, 2017.Google Scholar

[4]

R. M. ChenJ. Hu and D. Wang, Linear stability of compressible vortex sheets in two-dimensional elastodynamics, Adv. Math., 311 (2017), 18-60. doi: 10.1016/j.aim.2017.02.014. Google Scholar

[5]

R. M. Chen, J. Hu and D. Wang, Linear stability of compressible vortex sheets in 2D elastodynamics: Variable coefficients, Math. Ann. (to appear), arXiv: 1804.07850, 2018.Google Scholar

[6]

R. M. Chen, J. Hu and D. Wang, Nonlinear stability and existence of compressible vortex sheets in two-dimensional elastodynamics, preprint, 2018.Google Scholar

[7]

S.-X. Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boundary, Front. Math. China, 2 (2007), 87-102. doi: 10.1007/s11464-007-0006-5. Google Scholar

[8]

J. F. Coulombel, Well-posedness of hyperbolic initial boundary value problems, J. Math. Pure Appl., 84 (2005), 786-818. doi: 10.1016/j.matpur.2004.10.005. Google Scholar

[9]

J. F. Coulombel and P. Secchi, The stability of compressible vortex sheets in two space dimensions, Indiana Univ. Math. J., 53 (2004), 941-1012. doi: 10.1512/iumj.2004.53.2526. Google Scholar

[10]

J. F. Coulombel and P. Secchi, Nonlinear compressible vortex sheets in two space dimensions, Ann. Sci. Ec. Norm.Super., 41 (2008), 85-139. doi: 10.24033/asens.2064. Google Scholar

[11]

S. Evje, Global weak solutions for a compressible gas-liquid model with well-deformation interaction, J. Diff. Eqs., 251 (2011), 2352-2386. doi: 10.1016/j.jde.2011.07.013. Google Scholar

[12]

S. Evje, Weak solutions for a gas-liquid model relevant for describing gas-kick in oil wells, SIAM. J. Math. Anal., 43 (2011), 1887-1922. doi: 10.1137/100813932. Google Scholar

[13]

S. Evje and K. H. Karlsen, Global existence of weak solutions for a viscous two-phase model, J. Diff. Eqs., 245 (2008), 2660-2703. doi: 10.1016/j.jde.2007.10.032. Google Scholar

[14]

S. Evje and K. H. Karlsen, Global weak solutions for a viscous liquid-gas model with singular pressure law, Commun. Pure Appl. Anal., 8 (2009), 1867-1894. doi: 10.3934/cpaa.2009.8.1867. Google Scholar

[15]

J. A. Fejer and W. Miles, On the stability of a plane vortex sheet with respect to three dimensional disturbances, J. Fluid Mech., 15 (1963), 335-336. doi: 10.1017/S002211206300029X. Google Scholar

[16]

J. Francheteau and G. Metivier, Existence of weak shocks for multidimenional hyperbolic quasilinear systems, Asterisque., 268 (2000), 1-198. Google Scholar

[17]

H. A. Friis and S. Evje, Global weak solutions for a gas-liquid model with external forces general pressure law, SIAM J. Appl. Math., 71 (2011), 409-442. doi: 10.1137/100813336. Google Scholar

[18]

H. A. Friis and S. Evje, Well-posedness of a compressible gas-liquid model with a friction term important for well control operations, SIAM J. Appl. Math., 71 (2011), 2014-2047. doi: 10.1137/110835499. Google Scholar

[19] S. B. Gavage and D. Serre, First Order Systems of Hyperbolic Partial Differential Equations With Applications, The Clarendon Press, Oxford University Press, Oxford, 2007.
[20]

C.-C. Hao and H.-L. Li, Well-posedness for a multidimensional viscous liquid-gas two-phase flow model, SIAM J. Math. Anal., 44 (2012), 1304-1332. doi: 10.1137/110851602. Google Scholar

[21]

F. Huang, J. Kuang, D. Wang and W. Xiang, Stability of supersonic contact discontinuity for two-dimensional steady compressible Euler flows in a finite nozzle, J. Differential Equations, 266 (2019), 4337-4376, arXiv: 1804.04769, 2018. doi: 10.1016/j.jde.2018.09.036. Google Scholar

[22]

Q.-S. Jiu and Z.-P. Xin, On strong convergence to 3-D axisymmetric vortex sheets, J. Diff. Eqs., 223 (2006), 33-50. doi: 10.1016/j.jde.2005.04.001. Google Scholar

[23]

Q.-S. Jiu and Z.-P. Xin, On strong convergence to 3-D steady vortex sheets, J. Diff. Eqs., 239 (2007), 448-470. doi: 10.1016/j.jde.2007.05.008. Google Scholar

[24]

P. D. Lax, Hyperbolic systems of conservation laws Ⅱ, Commun. Pure Appl. Math., 10 (1957), 537-566. doi: 10.1002/cpa.3160100406. Google Scholar

[25]

R. L. Mishkov, Generalization of the formula of Faa di Bruno for a composite function with a vector argument, Int. J. Math. Math. Sci., 24 (2006), 481-491. doi: 10.1155/S0161171200002970. Google Scholar

[26]

A. Morando, P. Secchi and P. Trebeschi, On the evolution equation of compressible vortex sheets, preprint, arXiv: 1806.06740, 2018.Google Scholar

[27]

A. Morando and P. Trebeschi, Two-dimensional vortex sheets for the nonisentropic Euler equations: linear stability, J. Hyperbolic Differ. Equ., 5 (2008), 487-518. doi: 10.1142/S021989160800157X. Google Scholar

[28]

A. Morando, P. Trebeschi and T. Wang, Two-dimensional vortex sheets for the nonisentropic Euler equations: nonlinear stability, preprint, arXiv: 1808.09290, 2018.Google Scholar

[29]

A. MorandoY. Trakhinin and P. Trebeschi, Local existence of MHD contact discontinuities, Arch. Ration. Mech. Anal., 9 (2016), 289-313. doi: 10.3934/dcdss.2016.9.289. Google Scholar

[30]

M. OhnoY. Shizuta and T. Yanagisawa, The trace theorem on anisotropic Sobolev spaces, Tohoku Math. J., 46 (1994), 393-401. doi: 10.2748/tmj/1178225719. Google Scholar

[31]

S. Pai, Two-Phase Flows, Vieweg Tracts in Pure Appl. Phys., vol. 3, Vieweg, Braunschweig, 1977. Google Scholar

[32]

J. B. Rauch and F. Massey, Differentiability of solutions to hyperbolic initial boundary value problems, Trans. Amer. Math. Soc., 189 (1974), 303-318. doi: 10.2307/1996861. Google Scholar

[33]

L. Ruan and Y. Trakhinin, Elementary symmetrization of inviscid two-fluid flow equations giving a number of instant results, preprint, arXiv: 1810.04386, 2018.Google Scholar

[34]

L. RuanD. WangS. Weng and C. Zhu, Rectilinear vortex sheets of inviscid liquid-gas two-phase flow: linear stability, Commun. Math. Sci., 14 (2016), 735-776. doi: 10.4310/CMS.2016.v14.n3.a7. Google Scholar

[35]

P. Secchi, Well-posedness for a mixed problem for the equations of ideal magneto-hydrodynamics, Arch. Math., 64 (1995), 237-245. doi: 10.1007/BF01188574. Google Scholar

[36]

P. Secchi, Some properties of anisotropic Sobolev spaces, Arch. Math. (Basel), 75 (2000), 207-216. doi: 10.1007/s000130050494. Google Scholar

[37]

Y. Trakhinin, Existence of compressible current-vortex sheets: Variable coefficients linear analysis, Arch. Ration. Mech. Anal., 177 (2005), 331-366. doi: 10.1007/s00205-005-0364-7. Google Scholar

[38]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. doi: 10.1007/s00205-008-0124-6. Google Scholar

[39]

C. Wang and Z. Zhang, A new proof of Wu's Theorem on vortex sheets, Sci. China. Math., 55 (2012), 1449-1462. doi: 10.1007/s11425-012-4421-z. Google Scholar

[40]

Y.-G. Wang and F. Yu, Stability of contact discontinuities in three-dimensional compressible steady flows, J. Diff. Eqs., 255 (2013), 1278-1356. doi: 10.1016/j.jde.2013.05.014. Google Scholar

[41]

Y.-G. Wang and F. Yu, Stabilization effect of magnetic fields on two-dimensional compressible current-vortex sheets, Arch. Ration. Mech. Anal., 208 (2013), 341-389. doi: 10.1007/s00205-012-0601-9. Google Scholar

[42]

Y.-G. Wang and H. Yuan, Weak stability of transonic contact discontinuities in three-diemensional steady non-isentropic compressible Euler flows, Z. Angew. Math. Phys., 66 (2015), 341-388. doi: 10.1007/s00033-014-0404-y. Google Scholar

[43]

S. Wu, Mathematical analysis of vortex sheets, Commun. Pure Appl. Math., 59 (2006), 1065-1206. doi: 10.1002/cpa.20110. Google Scholar

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