Convergence rate of solutions toward stationary solutions to the bipolar Navier-Stokes-Poisson equations in a half line
© Zhou and Li; licensee Springer 2013
Received: 2 November 2012
Accepted: 26 April 2013
Published: 14 May 2013
In this paper, we show the convergence rate of a solution toward the stationary solution to the initial boundary value problem for the one-dimensional bipolar compressible Navier-Stokes-Poisson equations. For the supersonic flow at spatial infinity, if an initial perturbation decays with the algebraic or the exponential rate in the spatial asymptotic point, the solution converges to the corresponding stationary solution with the same rate in time as time tends to infinity. For the transonic flow at spatial infinity, the solution converges to the stationary solution in time with the lower rate than that of the initial perturbation in the spatial. These results are proved by the weighted energy method.
Keywordsconvergence rate Navier-Stokes-Poisson equation stationary wave weighted energy method
in a one-dimensional half space . Here the unknown functions are the densities , the velocities (), and the electron field E. () is the pressure depending only on the density. () is viscosity coefficient. Throughout this paper, we assume that two fluids of electrons and ions have the same equation of state with for and , and also they have the same viscosity coefficients . The bipolar Navier-Stokes-Poisson system is used to simulate the transport of charged particles (e.g., electrons and ions). It consists of the compressible Navier-Stokes equation of two-fluid under the influence of the electro-static potential force governed by the self-consisted Poisson equation. Note that when we only consider one particle in the fluids, we also have the unipolar Navier-Stokes-Poisson equations. For more details, we can refer to [1–4].
Recently, some important progress was made for the compressible unipolar Navier-Stokes-Poisson system. The local and/or global existence of a renormalized weak solution for the Cauchy problem of the multi-dimensional compressible Navier-Stokes-Poisson system were proved in [5–7]. Chan  also considered the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in . Hao and Li  established the global strong solutions of the initial value problem for the multi-dimensional compressible Navier-Stokes-Poisson system in a Besov space. The global existence and -decay rate of the smooth solution of the initial value problem for the compressible Navier-Stokes-Poisson system in were achieved by Li and his collaborators in [10, 11]. The pointwise estimates of the smooth solutions for the three-dimensional isentropic compressible Navier-Stokes-Poisson equation were obtained in . The quasineutral limit of the compressible Navier-Stokes-Poisson system was studied in [13–15]. However, the results about the bipolar Navier-Stokes-Poisson equations are very few. Lastly, Li et al.  showed the global existence and asymptotic behavior of smooth solutions for the initial value problem of the bipolar Navier-Stokes-Poisson equations. Duan and Yang  studied the unique existence and asymptotic stability of a stationary solution for the initial boundary value problem, and they showed that the large-time behavior of solutions for the bipolar Navier-Stokes-Poisson equations coincided with the one for the single Navier-Stokes system in the absence of the electric field. The consistency is also observed and proved between the bipolar Euler-Poisson system and the single damped Euler equation; for example, see [18–20] and the references therein.
In this paper, we are mainly concerned with the decay rate of solutions to (1.1)-(1.4) toward the stationary solution . Now we state the main result in the following theorem.
- (i)When , in addition, the initial data also satisfies , , , , for a certain positive constant α, then the solution to (1.1)-(1.3) satisfies the decay estimate(1.10)
- (ii)When , and there exists a positive constant such that if the initial data also satisfies for a certain constant α satisfying , where is a constant defined by
where , and δ are defined in Section 2, and .
Notations Throughout this paper, denotes the generic positive constant independent of time. () denotes the space of measurable functions with the finite norm , and is the space of bounded measurable functions on ℝ with the norm . We use to denote the -norm. () stands for the space of -functions f whose derivatives (in the sense of distribution) () are also -functions with the norm . Moreover, () denotes the space of the k-times continuously differentiable functions on the interval with values in .
The rest of the paper is organized as follows. In Section 2, we review the results of the stationary solution and the non-stationary solutions, then we reformulate our problem. Finally, we give the a priori estimates for the cases and in Section 3 and 4, respectively.
2 Stationary solution and global existence of non-stationary solution
and denote the Mach number at infinity . Then one has the following lemma.
As to the stability of the stationary solution of (1.1)-(1.4), Duan and Yang showed the following results in .
Lemma 2.2 (see )
The following lemma, concerning the existence of the solution locally in time, is proved by the standard iteration method. Hence we omit the proof.
Lemma 2.3 If the initial data satisfies (1.8) and , there exists a positive constant T such that the initial boundary value problem (2.7)-(2.9) has a unique solution . Moreover, if the initial data satisfies (1.8), (1.9) and and , there exists a unique solution in .
3 A priori estimates for
- (i)(Algebraic decay) Suppose that is a solution to (2.7)-(2.9) for certain positive constants α and T. Then there exist positive constants and C such that if , then the solution satisfies the estimate(3.1)
- (ii)(Exponential decay) Suppose that is a solution to (2.7)-(2.9) for certain positive constants ζ and T. Then there exist positive constants , C, β (<ζ) and α satisfying such that if , then the solution satisfies the estimate(3.2)
For the sake of clarity, we divide the proof of Proposition 4.1 into the following lemmas. We first derive the basic energy estimate.
Therefore, integrating (3.7) over , substituting the above inequalities (3.8)-(3.12) into the resultant equality and then taking suitably small, we obtain the desired estimate (3.3). □
in the resultant equality, and take ε suitably small. These computations together with (3.3), (3.13) and (3.23) give the desired estimate (3.24). □
Here, we have used the Poincaré-type inequality (3.23) again. Thus, taking δ, β and α suitably small, we obtain the desired a priori estimate (3.2). □
4 A priori estimate for
In order to prove Proposition 4.1, we need to get a lower estimate for and the Mach number on the stationary solution defined by .
Lemma 4.2 (see )
Based on Lemma 4.2, we obtain the weighted estimate of .
for and .
Finally, integrate (4.5) over , substitute (4.6)-(4.13) in the resultant equality, and take and δ suitably small. This procedure yields the desired estimate (4.2) for .
Here, we have used the fact that holds. Therefore, we obtain the estimate (4.2) for the case of . □
In order to complete the proof of Proposition 4.1, we need to obtain the weighted estimate of .
where ε is an arbitrary positive constant. We note that the third term on the right-hand side of the above inequality is estimated by applying the Poincaré-type inequality (3.23) for the case of .
Finally, adding (4.16) to (4.18) and taking suitably small give the desired estimate (4.14). □
By the same inductive argument as in deriving (4.1), we can prove Proposition 4.1, which immediately yields the decay estimate (1.12).
The research of Li is partially supported by the National Science Foundation of China (Grant No. 11171223) and the Innovation Program of Shanghai Municipal Education Commission (Grant No. 13ZZ109).
- Besse C, Claudel J, Degond P, et al.: A model hierarchy for ionospheric plasma modeling. Math. Models Methods Appl. Sci. 2004, 14: 393–415. 10.1142/S0218202504003283MathSciNetView ArticleGoogle Scholar
- Degond P: Mathematical modelling of microelectronics semiconductor devices. AMS/IP Stud. Adv. Math. 15. In Some Current Topics on Nonlinear Conservation Laws. Am. Math. Soc., Providence; 2000:77–110.Google Scholar
- Jüngel A Progress in Nonlinear Differential Equations and Their Applications. In Quasi-hydrodynamic Semiconductor Equations. Birkhäuser, Basel; 2001.View ArticleGoogle Scholar
- Sitnko A, Malnev V: Plasma Physics Theory. Chapman & Hall, London; 1995.Google Scholar
- Ducomet B: A remark about global existence for the Navier-Stokes-Poisson system. Appl. Math. Lett. 1999, 12: 31–37.MathSciNetView ArticleGoogle Scholar
- Ducomet B: Local and global existence for the coupled Navier-Stokes-Poisson problem. Q. Appl. Math. 2003, 61: 345–361.Google Scholar
- Zhang Y-H, Tan Z: On the existence of solutions to the Navier-Stokes-Poisson equations of a two-dimensional compressible flow. Math. Methods Appl. Sci. 2007, 30: 305–329. 10.1002/mma.786MathSciNetView ArticleGoogle Scholar
- Chan D:On the nonexistence of global weak solutions to the Navier-Stokes-Poisson equations in . Commun. Partial Differ. Equ. 2010, 35: 535–557. 10.1080/03605300903473418View ArticleGoogle Scholar
- Hao C-C, Li H-L: Global existence for compressible Navier-Stokes-Poisson equations in three and higher dimensions. J. Differ. Equ. 2009, 246: 4791–4812. 10.1016/j.jde.2008.11.019MathSciNetView ArticleGoogle Scholar
- Li H-L, Matsumura A, Zhang G-J:Optimal decay rate of the compressible Navier-Stokes-Poisson system in . Arch. Ration. Mech. Anal. 2010, 196: 681–713. 10.1007/s00205-009-0255-4MathSciNetView ArticleGoogle Scholar
- Zhang G-J, Li H-L, Zhu C-J:Optimal decay rate of the non-isentropic compressible Navier-Stokes-Poisson system in . J. Differ. Equ. 2011, 250: 866–891. 10.1016/j.jde.2010.07.035MathSciNetView ArticleGoogle Scholar
- Wang W-K, Wu Z-G: Pointwise estimates of solution for the Navier-Stokes-Poisson equations in multi-dimensions. J. Differ. Equ. 2010, 248: 1617–1636. 10.1016/j.jde.2010.01.003View ArticleGoogle Scholar
- Donatelli D, Marcati P: A quasineutral type limit for the Navier-Stokes-Poisson system with large data. Nonlinearity 2008, 21: 135–148. 10.1088/0951-7715/21/1/008MathSciNetView ArticleGoogle Scholar
- Ju Q-C, Li F-C, Li Y, Wang S: Rate of convergence from the Navier-Stokes-Poisson system to the incompressible Euler equations. J. Math. Phys. 2009., 50: Article ID 013533Google Scholar
- Wang S, Jiang S: The convergence of the Navier-Stokes-Poisson system to the incompressible Euler equations. Commun. Partial Differ. Equ. 2006, 31: 571–591. 10.1080/03605300500361487View ArticleGoogle Scholar
- Li H-L, Yang T, Zou C: Time asymptotic behavior of the bipolar Navier-Stokes-Poisson system. Acta Math. Sci., Ser. B 2009, 29: 1721–1736. 10.1016/S0252-9602(10)60013-6MathSciNetView ArticleGoogle Scholar
- Duan R-J, Yang X-F: Stability of rarefaction wave and boundary layer for outflow problem on the two-fluid Navier-Stokes-Poisson equations. Commun. Pure Appl. Anal. 2013, 12: 985–1014.MathSciNetView ArticleGoogle Scholar
- Gasser I, Hsiao L, Li H-L: Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors. J. Differ. Equ. 2003, 192: 326–359. 10.1016/S0022-0396(03)00122-0MathSciNetView ArticleGoogle Scholar
- Huang F-M, Li Y-P: Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discrete Contin. Dyn. Syst. 2009, 24: 455–470.MathSciNetView ArticleGoogle Scholar
- Huang F-M, Mei M, Wang Y: Large time behavior of solution to n -dimensional bipolar hydrodynamic model for semiconductors. SIAM J. Math. Anal. 2011, 43: 1595–1630. 10.1137/100810228MathSciNetView ArticleGoogle Scholar
- Kawashima S, Nishibata S, Zhu P: Asymptotic stability of the stationary solution to the compressible Navier-Stokes-Poisson equations in half space. Commun. Math. Phys. 2003, 240: 483–500.MathSciNetView ArticleGoogle Scholar
- Kawashima S, Zhu P: Asymptotic stability of nonlinear wave for the compressible Navier-Stokes-Poisson equations in half space. J. Differ. Equ. 2008, 224: 3151–3179.MathSciNetView ArticleGoogle Scholar
- Nakamura T, Nishibata S, Yuge T: Convergence rate of solutions toward stationary solutions to the compressible Navier-Stokes equation in a half line. J. Differ. Equ. 2007, 241: 94–111. 10.1016/j.jde.2007.06.016MathSciNetView ArticleGoogle Scholar
- Nikkuni Y, Kawashima S: Stability of stationary solutions to the half-space problem for the discrete Boltzmann equation with multiple collisions. Kyushu J. Math. 2000, 54: 233–255. 10.2206/kyushujm.54.233MathSciNetView ArticleGoogle Scholar
- Matsumura A, Nishihara K: Asymptotic stability of traveling waves for scalar viscous conservation laws with non-convex nonlinearity. Commun. Math. Phys. 1994, 165: 83–96. 10.1007/BF02099739MathSciNetView ArticleGoogle Scholar
This article is published under license to BioMed Central Ltd. This is an Open Access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.