Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

Raffaele Folino; Ramón G. Plaza; Delyan Zhelyazov

doi:10.3934/cpaa.2022133

Article Contents

2022, Volume 21, Issue 12: 4019-4040. Doi: 10.3934/cpaa.2022133

This issue Previous Article A note on the Gagliardo-Nirenberg inequality in a bounded domain Next Article On the almost sure scattering for the energy-critical cubic wave equation with supercritical data

Spectral stability of small-amplitude dispersive shocks in quantum hydrodynamics with viscosity

1.
Departamento de Matemáticas y Mecánica, Instituto de Investigaciones en Matemáticas Aplicadas y en Sistemas
2.
Universidad Nacional Autónoma de México, Circuito Escolar s/n, Ciudad Universitaria C.P. 04510, Ciudad de México, Mexico

^* Corresponding author: Ramón G. Plaza
^* Corresponding author: Ramón G. Plaza

Received: March 2022

Revised: July 2022

Early access: September 2022

Published: December 2022

The work of D. Zhelyazov was supported by a Post-doctoral Fellowship by the Dirección General de Asuntos del Personal Académico (DGAPA), UNAM. The work of R. G. Plaza was partially supported by DGAPA-UNAM, program PAPIIT, grant IN-104922

Abstract / Introduction Full Text(HTML) Figure(1) Related Papers Cited by

Abstract

A compressible viscous-dispersive Euler system in one space dimension in the context of quantum hydrodynamics is considered. The dispersive term is due to quantum effects described through the Bohm potential and the viscosity term is of linear type. It is shown that small-amplitude viscous-dispersive shock profiles for the system under consideration are spectrally stable, proving in this fashion a previous numerical observation by Lattanzio et al. [28,29]. The proof is based on spectral energy estimates which profit from the monotonicty of the profiles in the small-amplitude regime.

Keywords:

Mathematics Subject Classification: Primary: 76Y05, 35Q35; Secondary: 35B35, 35P15.

Citation:

Full Text(HTML)

Figure 1. Fredholm borders $ \Sigma_\pm $ as the complex roots $ \lambda = \lambda_+(\xi) $ (orange) and $ \lambda = \lambda_-(\xi) $ (blue), with $ \xi \in \mathbb{R} $, of the dispersion relation (3.4). Here $ P^+ = 0.519 $, $ P^- = P^+ + \varepsilon $, $ \varepsilon = 0.2 $, $ J^+ = -0.418 $, $ \gamma = 1.5 $, $ \mu = 0.1 $ and $ k = 0.5 $. The essential spectrum of $ {\mathcal{L}} $ is sharply bounded to the left of these curves in the complex plane (color online)

Download: Full-size image PowerPoint slide

Related Papers

Cited by

References

[1]	P. Antonelli and P. Marcati, On the finite energy weak solutions to a system in quantum fluid dynamics, Comm. Math. Phys., 287 (2009), 657-686. doi: 10.1007/s00220-008-0632-0.
[2]	B. Barker, J. Humpherys, K. Rudd and K. Zumbrun, Stability of viscous shocks in isentropic gas dynamics, Comm. Math. Phys., 281 (2008), 231-249. doi: 10.1007/s00220-008-0487-4.
[3]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. Ⅰ, Phys. Rev. (2), 85 (1952), 166-179. doi: 10.1103/PhysRev.85.166.
[4]	D. Bohm, A suggested interpretation of the quantum theory in terms of "hidden" variables. Ⅱ, Phys. Rev. (2), 85 (1952), 180-193. doi: 10.1103/PhysRev.85.180.
[5]	D. Bohm, B. J. Hiley and P. N. Kaloyerou, An ontological basis for the quantum theory, Phys. Rep., 144 (1987), 321-375. doi: 10.1016/0370-1573(87)90024-X.
[6]	F. Dalfovo, S. Giorgini, L. P. Pitaevskii and S. Stringari, Theory of Bose-Einstein condensation in trapped gases, Rev. Mod. Phys., 71 (1999), 463-512. doi: 10.1103/RevModPhys.71.463.
[7]	P. Degond, F. Méhats and C. Ringhofer, Quantum hydrodynamic models derived from the entropy principle, in Nonlinear partial differential equations and related analysis (eds. G.-Q. Chen, G. Gasper and J. Jerome), vol. 371 of Contemp. Math., Amer. Math. Soc., Providence, RI, (2005), 107–131. doi: 10.1090/conm/371/06850.
[8]	A. Diaw and M. S. Murillo, A viscous quantum hydrodynamics model based on dynamic density functional theory, Sci. Rep., 7 (2017), 15532. doi: 10.1038/s41598-017-14414-9.
[9]	D. K. Ferry and J.-R. Zhou, Form of the quantum potential for use in hydrodynamic equations for semiconductor device modeling, Phys. Rev. B, 48 (1993), 7944-7950. doi: 10.1103/PhysRevB.48.7944.
[10]	I. M. Gamba, A. Jüngel and A. Vasseur, Global existence of solutions to one-dimensional viscous quantum hydrodynamic equations, J. Differ. Equ., 247 (2009), 3117-3135. doi: 10.1016/j.jde.2009.09.001.
[11]	C. L. Gardner, The quantum hydrodynamic model for semiconductor devices, SIAM J. Appl. Math., 54 (1994), 409-427. doi: 10.1137/S0036139992240425.
[12]	I. Gasser, Traveling wave solutions for a quantum hydrodynamic model, Appl. Math. Lett., 14 (2001), 279-283. doi: 10.1016/S0893-9659(00)00149-X.
[13]	J. Goodman, Nonlinear asymptotic stability of viscous shock profiles for conservation laws, Arch. Ration. Mech. Anal., 95 (1986), 325-344. doi: 10.1007/BF00276840.
[14]	J. Goodman, Remarks on the stability of viscous shock waves, in Viscous profiles and numerical methods for shock waves (Raleigh, NC, 1990) (ed. M. Shearer), Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, (1991), 66-72.
[15]	J. Grant, Pressure and stress tensor expressions in the fluid mechanical formulation of the Bose condensate equations, J. Phys. A: Math. Nucl. Gen., 6 (1973), L151-L153. doi: 10.1088/0305-4470/6/11/001.
[16]	F. Graziani, Z. Moldabekov, B. Olson and M. Bonitz, Shock physics in warm dense matter: A quantum hydrodynamics perspective, Contrib. to Plasma Phys., 62 (2022), e202100170. doi: 10.1002/ctpp.202100170.
[17]	A. V. Gurevich and L. P. Pitayevskiǐ, Nonstationary structure of a collisionless shock wave, Sov. Phys. JETP, 38 (1974), 590-604.
[18]	D. Henry, Geometric Theory of Semilinear Parabolic Equations, no. 840 in Lecture Notes in Mathematics, Springer-Verlag, New York, 1981. doi: 10.1007/BFb0089647.
[19]	M. A. Hoefer and M. J. Ablowitz, Interactions of dispersive shock waves, Phys. D, 236 (2007), 44-64. doi: 10.1016/j.physd.2007.07.017.
[20]	M. A. Hoefer, M. J. Ablowitz, I. Coddington, E. A. Cornell, P. Engels and V. Schweikhard, Dispersive and classical shock waves in Bose-Einstein condensates and gas dynamics, Phys. Rev. A, 74 (2006), 023623. doi: 10.1103/PhysRevA.74.023623.
[21]	J. Humpherys, On the shock wave spectrum for isentropic gas dynamics with capillarity, J. Differ. Equ., 246 (2009), 2938-2957. doi: 10.1016/j.jde.2008.07.028.
[22]	C. K. R. T. Jones, Geometric singular perturbation theory, in Dynamical Systems (Montecatini Terme, 1994) (ed. R. Johnson), vol. 1609 of Lecture Notes in Math., Springer, Berlin, (1995), 44-118. doi: 10.1007/BFb0095239.
[23]	A. Jüngel and J. P. Milišić, Physical and numerical viscosity for quantum hydrodynamics, Commun. Math. Sci., 5 (2007), 447-471.
[24]	T. Kapitula and K. Promislow, Spectral and Dynamical Stability of Nonlinear Waves, vol. 185 of Applied Mathematical Sciences, Springer-Verlag, New York, 2013. doi: 10.1007/978-1-4614-6995-7.
[25]	T. Kato, Perturbation Theory for Linear Operators, 2$^{nd}$ edition. Classics in Mathematics, Springer-Verlag, New York, 1980. doi: 10.1007/978-3-642-66282-9.
[26]	I. M. Khalatnikov, An Introduction to the Theory of Superfluidity, Advanced Book Classics, Addison-Wesley Publishing Company, Advanced Book Program, Redwood City, CA, 1989. doi: 10.1201/9780429502897.
[27]	L. Landau, Theory of the superfluidity of Helium Ⅱ, Phys. Rev., 60 (1941), 356-358. doi: 10.1103/PhysRev.60.356.
[28]	C. Lattanzio, P. Marcati and D. Zhelyazov, Dispersive shocks in quantum hydrodynamics with viscosity, Phys. D, 402 (2020), 132222. doi: 10.1016/j.physd.2019.132222.
[29]	C. Lattanzio, P. Marcati and D. Zhelyazov, Numerical investigations of dispersive shocks and spectral analysis for linearized quantum hydrodynamics, Appl. Math. Comput., 385 (2020), 1-13. doi: 10.1016/j.amc.2020.125450.
[30]	C. Lattanzio and D. Zhelyazov, Spectral analysis of dispersive shocks for quantum hydrodynamics with nonlinear viscosity, Math. Models Methods Appl. Sci., 31 (2021), 1719-1747. doi: 10.1142/S0218202521500378.
[31]	C. Lattanzio and D. Zhelyazov, Traveling waves for quantum hydrodynamics with nonlinear viscosity, J. Math. Anal. Appl., 493 (2021), 17 pp. doi: 10.1016/j.jmaa.2020.124503.
[32]	E. Madelung, Quantentheorie in hydrodynamischer Form, Z. Physik, 40 (1927), 322-326. doi: 10.1007/BF01400372.
[33]	A. Matsumura and K. Nishihara, On the stability of travelling wave solutions of a one-dimensional model system for compressible viscous gas, Japan J. Appl. Math., 2 (1985), 17-25. doi: 10.1007/BF03167036.
[34]	S. R. Z. Sagdeev, Kollektivnye protsessy i udarnye volny vrazrezhennol plazme (Collective processes and shock waves in a tenuous plasma), in Voprosy teorii plazmy (Problems of Plasma Theory) (ed. M. A. Leontovich), vol. 5 of Atomizdat, Moscow, (1964). English transl.: Reviews of Plasma Physics (Consultants Bureau, New York, 1966), 23 pp. doi: 10.1016/0029-5582(67)91024-3.
[35]	W. Zhang, X. Li and Y. Yong, Asymptotic stability of monotone increasing traveling wave solutions for viscous compressible fluid equations with capillarity term, J. Math. Anal. Appl., 434 (2016), 401-412. doi: 10.1016/j.jmaa.2015.09.020.
[36]	D. Zhelyazov, Existence of standing and traveling waves in quantum hydrodynamics with viscosity, preprint, arXiv: 2202.07573.