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 and Continuous Dynamical Systems, 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[16]

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

[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.

[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.

[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.

[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.

[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.

[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.

[24]

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

[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.

[26]

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

[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.

[28]

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

[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.

[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.

[31]

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

[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.

[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.

[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.

[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.

[36]

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

[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.

[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.

[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.

[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.

[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.

[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.

[43]

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

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