Abstract
In this article, we consider the periodic problem for bipolar non-isentropic Euler–Maxwell equations with damping terms in plasmas. By means of an induction argument on the order of the time-space derivatives of solutions in energy estimates, the global smooth solution with small amplitude was established close to a non-constant steady-state solution with asymptotic stability property. Furthermore, we obtain the global stability of solutions with exponential decay in time near the non-constant steady-states for bipolar non-isentropic Euler–Poisson equations. This phenomenon on the charge transport shows the essential relation and difference between the bipolar non-isentropic and the bipolar isentropic Euler–Maxwell/Poisson equations.
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References
Chen F.: Introduction to Plasma Physics and Controlled Fusion, vol. 1. Plenum Press, New York (1984)
Chen G.Q., Jerome J.W., Wang D.H.: Compressible Euler–Maxwell equations. Transp. Theory Stat. Phys. 29, 311–331 (2000)
Degond P., Deluzet F., Savelief D.: Numerical approximation of the Euler–Maxwell model in the quasineutral limit. J. Comput. Phys. 231, 1917–1946 (2012)
Duan R.J.: Global smooth flows for the compressible Euler–Maxwell system: the relaxation case. J. Hyperbolic Differ. Equ. 8, 375–413 (2011)
Duan R.J., Liu Q.Q., Zhu C.J.: The Cauchy problem on the compressible two-fluids Euler–Maxwell equations. SIAM J. Math. Anal. 44, 102–133 (2012)
Evans L.C.: Partial Differential Equations, Graduate Studies in Mathematics, 19. American Mathematical Society, Providence, RI (1998)
Feng Y.H., Peng Y.J., Wang S.: Stability of non-constant equilibrium solutions for two-fluid Euler–Maxwell systems. Nonlinear Anal. Real World 26, 372–390 (2015)
Feng Y.H., Wang S., Kawashima S.: Global existence and asymptotic decay of solutions to the non-isentropic Euler–Maxwell system. Math. Models Methods Appl. Sci. 24, 2851–2884 (2014)
Feng, Y.H., Wang, S., Li, X.: Stability of non-constant steady-state solutions for non-isentropic Euler–Maxwell system with a temperature damping term. Math. Methods Appl. Sci. 39, 2514–2528 (2016)
Germain, P., Masmoudi, N.: Global Existence for the Euler–Maxwell System. (2011). arXiv:1107.1595
Guo, Y., Ionescu, A.D., Pausader, B.: Global Solutions of the Euler–Maxwell Two-Fluid System in 3D. (2013). arXiv:1303.1060
Guo Y., Strauss W.: Stability of semiconductor states with insulating and contact boundary conditions. Arch. Ration. Mech. Anal. 179, 1–30 (2005)
Kato T.: The Cauchy problem for quasi-linear symmetric hyperbolic systems. Arch. Ration. Mech. Anal. 58, 181–205 (1975)
Klainerman S., Majda A.: Singular limits of quasilinear hyperbolic systems with large parameters and the incompressible limit of compressible fluids. Commun. Pure Appl. Math. 34, 481–524 (1981)
Majda A.: Compressible Fluid Flow and Systems of Conservation Laws in Several Space Variables. Springer, New York (1984)
Markowich P.A., Ringhofer C.A., Schmeiser C.: Semiconductor Equations. Springer, New York (1990)
Matsumura A., Nishida T.: The initial value problem for the equation of motion of compressible viscous and heat-conductive fluids. Proc. Jpn. Acad. Ser. A 55, 337–342 (1979)
Matsumura A., Nishida T.: The initial value problem for the equation of motion of viscous and heat-conductive gases. J. Math. Kyoto Univ. 20, 67–104 (1980)
Peng Y.J., Wang S.: Convergence of compressible Euler–Maxwell equations to incompressible Euler equations. Commun. Part. Differ. Equ. 33, 349–376 (2008)
Peng Y.J., Wang S., Gu G.L.: Relaxation limit and global existence of smooth solutions of compressible Euler–Maxwell equations. SIAM J. Math. Anal. 43, 944–970 (2011)
Peng Y.J.: Global existence and long-time behavior of smooth solutions of two-fluid Euler–Maxwell equations. Ann. Inst. Henri Poincare Anal. 29, 737–759 (2012)
Peng Y.J.: Stability of non-constant equilibrium solutions for Euler–Maxwell equations. J. Math. Pures Appl. 103, 39–67 (2015)
Rishbeth H., Garriott O.K.: Introduction to Ionospheric Physics. Academic Press, London (1969)
Ueda Y., Kawashima S.: Decay property of regularity-loss type for the Euler–Maxwell system. Methods Appl. Anal. 18, 245–267 (2011)
Ueda Y., Wang S., Kawashima S.: Dissipative structure of the regularity-loss type and time asymptotic decay of solutions for the Euler–Maxwell system. SIAM J. Math. Anal. 44, 2002–2017 (2012)
Wang S., Feng Y.H., Li X.: The asymptotic behavior of globally smooth solutions of bipolar non-isentropic compressible Euler–Maxwell system for plasma. SIAM J. Math. Anal. 44, 3429–3457 (2012)
Wang S., Feng Y.H., Li X.: The asymptotic behavior of globally smooth solutions of non-isentropic Euler–Maxwell equations for plasmas. Appl. Math. Comput 231, 299–306 (2014)
Wen H.Y., Zhu C.J.: Global symmetric classical solutions of the full compressible Navier–Stokes equations with vacuum and large initial data. J. Math. Pures Appl. 102, 498–545 (2014)
Xu J.: Global classical solutions to the compressible Euler–Maxwell equations. SIAM J. Math. Anal. 43, 2688–2718 (2011)
Yong W.A.: Entropy and global existence for hyperbolic balance laws. Arch. Ration. Mech. Anal. 172, 247–266 (2004)
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Li, X., Wang, S. & Feng, YH. Stability of non-constant steady-state solutions for bipolar non-isentropic Euler–Maxwell equations with damping terms. Z. Angew. Math. Phys. 67, 133 (2016). https://doi.org/10.1007/s00033-016-0728-x
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DOI: https://doi.org/10.1007/s00033-016-0728-x