July & August  2021, 20(7&8): 2519-2533. doi: 10.3934/cpaa.2021083

On the stability of two-dimensional nonisentropic elastic vortex sheets

1. 

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

2. 

Institute of Applied Mathematics, Academy of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100190, China

3. 

School of Mathematical Sciences, Beijing Normal University and Laboratory of Mathematics and Complex Systems, Ministry of Education, Beijing 100875, China

* Corresponding author

Dedicated to Professor Shuxing Chen on the Occasion of His 80th Birthday

Received  February 2021 Revised  April 2021 Published  July & August 2021 Early access  May 2021

Fund Project: R. M. Chen is supported in part by the NSF grant DMS-1907584. F. Huang was supported in part by National Center for Mathematics and Interdisciplinary Sciences, AMSS, CAS and NSFC Grant No. 11371349 and 11688101. D. Wang was supported in part by NSF grant DMS-1907519. D. Yuan was supported by China Scholarship Council No.201704910503, NSFC Grant No.12001045 and China Postdoctoral Science Foundation No.2020M680428

We are concerned with the stability of vortex sheet solutions for the two-dimensional nonisentropic compressible flows in elastodynamics. This is a nonlinear free boundary hyperbolic problem with characteristic discontinuities, which has extra difficulties when considering the effect of entropy. The addition of the thermal effect to the system makes the analysis of the Lopatinski$ \breve{{\mathrm{i}}} $ determinant extremely complicated. Our results are twofold. First, through a qualitative analysis of the roots of the Lopatinski$ \breve{{\mathrm{i}}} $ determinant for the linearized problem, we find that the vortex sheets are weakly stable in some supersonic and subsonic regions. Second, under the small perturbation of entropy, the nonlinear stability can be adapted from the previous two-dimensional isentropic elastic vortex sheets [6] by applying the Nash-Moser iteration. The two results confirm the strong elastic stabilization of the vortex sheets. In particular, our conditions for the linear stability (1) ensure that a stable supersonic regime as well as a stable subsonic one always persist for any given nonisentropic configuration, and (2) show how the stability condition changes with the thermal fluctuation. The existence of the stable subsonic bubble, a phenomenon not observed in the Euler flow, is specially due to elasticity.

Citation: Robin Ming Chen, Feimin Huang, Dehua Wang, Difan Yuan. On the stability of two-dimensional nonisentropic elastic vortex sheets. Communications on Pure and Applied Analysis, 2021, 20 (7&8) : 2519-2533. doi: 10.3934/cpaa.2021083
References:
[1]

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.

[2]

G. Q. ChenP. Secchi and T. Wang, Nonlinear stability of relativistic vortex sheets in three dimensional Minkowski spacetime, Arch. Ration. Mech. Anal., 232 (2019), 591-695.  doi: 10.1007/S00205-018-1330-5.

[3]

G. Q. ChenP. Secchi and T. Wang, Stability of multidimensional thermoelastic contact discontinuities, Arch. Ration. Mech. Anal., 237 (2020), 1271-1323.  doi: 10.1007/s00205-020-01531-5.

[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. ChenJ. Hu and D. Wang, Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients, Math. Ann., 376 (2020), 863-912.  doi: 10.1007/s00208-018-01798-w.

[6]

R. M. ChenJ. HuD. WangT. Wang and D. Yuan, Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics, J. Differ. Equ., 269 (2020), 6899-6940.  doi: 10.1016/j.jde.2020.05.003.

[7]

S. X. Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boudary, Translated from Chin. Ann. Math., 3(2) (1982), 222-232. Front. Math. China., 2(1) (2007), 87-102. doi: 10.1007/s11464-007-0006-5.

[8]

S. X. Chen, Study of Multidimensional Systems of Conservation Laws: Problems, Difficulties and Progress. Proceedings of the International Congress of Mathematicians 2010, 4(2015).

[9]

J. F. Coulombel and A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara., 50 (2004), 79-90. 

[10]

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.

[11]

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.

[12]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Third Edition, Grundlehren der Mathematischen Wissenschaften (Fundatmental Principles of Mathematical Sciences), Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[13]

C. Hao and D. Wang, A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics, J. Differ. Equ., 261 (2016), 712-737.  doi: 10.1016/j.jde.2016.03.025.

[14]

R. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334. 

[15]

X. Hu and Y. Huang, Well-posedness of the free boundary problem for incompressible elastodynamics, J. Differ. Equ., 266 (2019), 7844-7889.  doi: 10.1016/j.jde.2018.12.018.

[16] S. B. Gavage and D. Serre, First Order Systems of Hyperbolic Partial Differential Equations with Applications, The Clarendon Press, Oxford University Press, Oxford, 2007. 
[17]

F. HuangD. Wang and D. Yuan, Nonlinear stability and existence of vortex sheets for invisicd liquid-gas two-phase flow, Discrete Contin. Dyn. Syst.-A, 39 (2019), 3535-3575.  doi: 10.3934/dcds.2019146.

[18]

H. LiW. Wang and Z. Zhang, Well-posedness of the free boundary problem in incompressible elastodynamics, J. Differ. Equ., 267 (2019), 6604-6643.  doi: 10.1016/j.jde.2019.07.001.

[19]

J. W. Miles, On the reflection of sound at an interface of relative motion, J. Acoust. Soc. Am., 29 (1957), 226-228.  doi: 10.1121/1.1908836.

[20]

J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid. Mech., 4 (1958), 538-552.  doi: 10.1017/S0022112058000653.

[21]

A. MorandoP. Secchi and P. Trebeschi, On the evolution equation of compressible vortex sheets, Math. Nachr., 293 (2020), 945-969.  doi: 10.1002/mama.201800162.

[22]

A. MorandoY. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann., 378 (2020), 1471-1504.  doi: 10.1007/s00208/019-01920-6.

[23]

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.

[24]

A. MorandoP. Trebeschi and T. Wang, Two-dimensional vortex sheets for the nonisentropic Euler equations: nonlinear stability, J. Differ. Equ., 266 (2019), 5397-5430.  doi: 10.1016/j.jde.2018.10.029.

[25]

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.

[26] D. Serre, Systems of Conservation Laws.2. Geometric Structure, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000. 
[27]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. 

[28]

Y. Trakhinin, Well-posedness of the free boundary problem in compressible elastodynamics, J. Differ. Equ., 264 (2018), 1661-1715.  doi: 10.1016/j.jde.2017.10.005.

[29]

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

[30]

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.

[31]

Y. G. Wang and F. Yu, Structural stability of supersonic contact disconitnuities in three-dimensonal compressible steady flows, SIAM J. Math. Anal., 47 (2015), 1291-1329.  doi: 10.1137/140976169.

show all references

References:
[1]

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.

[2]

G. Q. ChenP. Secchi and T. Wang, Nonlinear stability of relativistic vortex sheets in three dimensional Minkowski spacetime, Arch. Ration. Mech. Anal., 232 (2019), 591-695.  doi: 10.1007/S00205-018-1330-5.

[3]

G. Q. ChenP. Secchi and T. Wang, Stability of multidimensional thermoelastic contact discontinuities, Arch. Ration. Mech. Anal., 237 (2020), 1271-1323.  doi: 10.1007/s00205-020-01531-5.

[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. ChenJ. Hu and D. Wang, Linear stability of compressible vortex sheets in 2D elastodynamics: variable coefficients, Math. Ann., 376 (2020), 863-912.  doi: 10.1007/s00208-018-01798-w.

[6]

R. M. ChenJ. HuD. WangT. Wang and D. Yuan, Nonlinear stability and existence of compressible vortex sheets in 2D elastodynamics, J. Differ. Equ., 269 (2020), 6899-6940.  doi: 10.1016/j.jde.2020.05.003.

[7]

S. X. Chen, Initial boundary value problems for quasilinear symmetric hyperbolic systems with characteristic boudary, Translated from Chin. Ann. Math., 3(2) (1982), 222-232. Front. Math. China., 2(1) (2007), 87-102. doi: 10.1007/s11464-007-0006-5.

[8]

S. X. Chen, Study of Multidimensional Systems of Conservation Laws: Problems, Difficulties and Progress. Proceedings of the International Congress of Mathematicians 2010, 4(2015).

[9]

J. F. Coulombel and A. Morando, Stability of contact discontinuities for the nonisentropic Euler equations, Ann. Univ. Ferrara., 50 (2004), 79-90. 

[10]

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.

[11]

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.

[12]

C. M. Dafermos, Hyperbolic Conservation Laws in Continuum Physics. Third Edition, Grundlehren der Mathematischen Wissenschaften (Fundatmental Principles of Mathematical Sciences), Springer-Verlag, Berlin, 2010. doi: 10.1007/978-3-642-04048-1.

[13]

C. Hao and D. Wang, A priori estimates for the free boundary problem of incompressible neo-Hookean elastodynamics, J. Differ. Equ., 261 (2016), 712-737.  doi: 10.1016/j.jde.2016.03.025.

[14]

R. Hersh, Mixed problems in several variables, J. Math. Mech., 12 (1963), 317-334. 

[15]

X. Hu and Y. Huang, Well-posedness of the free boundary problem for incompressible elastodynamics, J. Differ. Equ., 266 (2019), 7844-7889.  doi: 10.1016/j.jde.2018.12.018.

[16] S. B. Gavage and D. Serre, First Order Systems of Hyperbolic Partial Differential Equations with Applications, The Clarendon Press, Oxford University Press, Oxford, 2007. 
[17]

F. HuangD. Wang and D. Yuan, Nonlinear stability and existence of vortex sheets for invisicd liquid-gas two-phase flow, Discrete Contin. Dyn. Syst.-A, 39 (2019), 3535-3575.  doi: 10.3934/dcds.2019146.

[18]

H. LiW. Wang and Z. Zhang, Well-posedness of the free boundary problem in incompressible elastodynamics, J. Differ. Equ., 267 (2019), 6604-6643.  doi: 10.1016/j.jde.2019.07.001.

[19]

J. W. Miles, On the reflection of sound at an interface of relative motion, J. Acoust. Soc. Am., 29 (1957), 226-228.  doi: 10.1121/1.1908836.

[20]

J. W. Miles, On the disturbed motion of a plane vortex sheet, J. Fluid. Mech., 4 (1958), 538-552.  doi: 10.1017/S0022112058000653.

[21]

A. MorandoP. Secchi and P. Trebeschi, On the evolution equation of compressible vortex sheets, Math. Nachr., 293 (2020), 945-969.  doi: 10.1002/mama.201800162.

[22]

A. MorandoY. Trakhinin and P. Trebeschi, Structural stability of shock waves in 2D compressible elastodynamics, Math. Ann., 378 (2020), 1471-1504.  doi: 10.1007/s00208/019-01920-6.

[23]

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.

[24]

A. MorandoP. Trebeschi and T. Wang, Two-dimensional vortex sheets for the nonisentropic Euler equations: nonlinear stability, J. Differ. Equ., 266 (2019), 5397-5430.  doi: 10.1016/j.jde.2018.10.029.

[25]

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.

[26] D. Serre, Systems of Conservation Laws.2. Geometric Structure, Oscillations, and Initial-Boundary Value Problems, Cambridge University Press, Cambridge, 2000. 
[27]

Y. Trakhinin, The existence of current-vortex sheets in ideal compressible magnetohydrodynamics, Arch. Ration. Mech. Anal., 191 (2009), 245-310. 

[28]

Y. Trakhinin, Well-posedness of the free boundary problem in compressible elastodynamics, J. Differ. Equ., 264 (2018), 1661-1715.  doi: 10.1016/j.jde.2017.10.005.

[29]

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

[30]

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.

[31]

Y. G. Wang and F. Yu, Structural stability of supersonic contact disconitnuities in three-dimensonal compressible steady flows, SIAM J. Math. Anal., 47 (2015), 1291-1329.  doi: 10.1137/140976169.

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