Global existence and asymptotic behavior of smooth solutions for a bipolar Euler-Poisson system in the quarter plane
© Li; licensee Springer. 2012
Received: 26 May 2011
Accepted: 16 February 2012
Published: 16 February 2012
In the article, a one-dimensional bipolar hydrodynamic model (Euler-Poisson system) in the quarter plane is considered. This system takes the form of Euler-Poisson with electric field and frictional damping added to the momentum equations. The global existence of smooth small solutions for the corresponding initial-boundary value problem is firstly shown. Next, the asymptotic behavior of the solutions towards the nonlinear diffusion waves, which are solutions of the corresponding nonlinear parabolic equation given by the related Darcy's law, is proven. Finally, the optimal convergence rates of the solutions towards the nonlinear diffusion waves are established. The proofs are completed from the energy methods and Fourier analysis. As far as we know, this is the first result about the optimal convergence rates of the solutions of the bipolar Euler-Poisson system with boundary effects towards the nonlinear diffusion waves.
Mathematics Subject Classification: 35M20; 35Q35; 76W05.
Moreover, we also investigate the time-asymptotic behavior of the solutions to (1.2)-(1.4). Our results discussed below show that even for the case with boundary condition, the solutions of (1.2)-(1.4) can be captured by the corresponding porous equation as in initial data case. For the sake of simplicity, we can assume j+ = 0. This assumption can be removed because of the exponential decay of the momentum at x = ±∞ induced by the linear frictional damping.
Throughout this article C always denotes a harmless positive constant. L p (ℝ+) is the space of square integrable real valued function defined on ℝ+ with the norm and H k (ℝ+) denotes the usual Sobolev space with the norm ∥·∥ k .
Now one of main results in this paper is stated as follows.
where α > 0 and C is positive constant.
Next, with the help of Fourier analysis, we can obtain the following optimal convergence rate.
and it is a technique one. As to more general case, we will discuss it in the forthcoming future. Theorems 1.1 and 1.2 show that the nonlinear diffusive phenomena is maintained in the bipolar Euler-Poisson system with the interaction of two particles and the additional electric field, which indeed implies that this diffusion effect is essentially due to the friction of momentum relaxation.
Using the energy estimates, we can establish a priori estimate, which together with local existence, leads to global existence of the smooth solutions for IBVP (1.2)-(1.4) by standard continuity arguments. In order to obtain the asymptotic behavior and optimal decay rate, noting that E = φ1 - φ2 satisfies the damping "Klein-Gordon" equation (see [14, 15]), we first obtain the exponential decay rate of the electric field E by energy methods. Then, we can establish the algebraical decay rate of the perturbed densities φ1 and φ2. Finally, from the estimates of the wave equation with damping in  and using the idea of , we show the optimal algebraical decay rates of the total perturbed density φ1 + φ2, which together with the exponential decay rate of the difference of two perturbed densities, yields the optimal decay rate. In these procedure, we have overcome the difficulty from the coupling and cancelation interaction between n1 and n2. Finally, it is worth mentioning that similar results about the Euler equations with damping have been extensively studied by many authors, i.e., the authors of [16–19, 21, 22], etc.
The rest of this article is arranged as follows. We first construct the optimal nonlinear diffusion waves and recall some inequalities in Section 2. In Section 3, we reformulate the original problem, and show the main Theorem. Section 4 is to prove an important decay estimate, which has been used to show the main theorem in Section 3.
2 The nonlinear diffusion waves
Note that from the assumptions in Theorem 1.1 and (2.3), there exists δ0 satisfies (2.4).
with the help of the Green function method and energy estimates.
From (2.5) and (2.6), we have
Next, we introduce some inequalities of Sobolev type.
for some constant C > 0.
Finally, for later use, we also need
where δ(x) is the Delta function.
3 Global existence and algebraical decay rate
we can obtain the following estimates by using a similar argument of . Since the proof is tedious but similar as in the previous works, we only list the results and omit its details.
for some positive constant β.
In conclusion, we have
4 The optimal convergence rate
In this section we are going to show the optimal decay rate. First of all, we improve the decay rates in Theorem 3.4 to be optimal as follows.
Based on the above Proposition, we can immediately prove Theorem 1.2.
This prove (1.9).
for k = 0, 1, 2.
Therefore, (4.11), (4.12) and the triangle inequality lead to (4.1).
Combining (4.14) and (4.15), and using the triangle inequality, we can obtain (4.2).
The author is grateful to the anonymous referees for careful reading and valuable comments which led to an important improvement of my original manuscript. The research is partially supported by the National Science Foundation of China (Grant No. 11171223).
- Jüngel A: Quasi-hydrodynamic semiconductor equations. Progress Nonlinear Diff Equ 2001, 41: 191-196.Google Scholar
- Markowich PA, Ringhofev CA, Schmeiser C: Semiconductor Equations. Springer-Verlag, New York; 1990.View ArticleGoogle Scholar
- Sitnko A, Malnev V: Plasma Physics Theory. Chapman & Hall, London; 1995.Google Scholar
- Zhou F, Li YP: Existence and some limits of stationary solutions to a one-dimensional bipolar Euler-Poisson system. J Math Anal Appl 2009, 351: 480-490. 10.1016/j.jmaa.2008.10.032MATHMathSciNetView ArticleGoogle Scholar
- Tsuge N: Existence and uniqueness of stationary solutions to a one-dimensional bipolar hydrodynamic models of semiconductors. Nonlinear Anal TMA 2010, 73: 779-787. 10.1016/j.na.2010.04.015MATHMathSciNetView ArticleGoogle Scholar
- Natalini R: The bipolar hydrodynamic model for semiconductors and the drift-diffusion equation. J Math Anal Appl 1996, 198: 262-281. 10.1006/jmaa.1996.0081MATHMathSciNetView ArticleGoogle Scholar
- Hsiao L, Zhang KJ: The relaxation of the hydrodynamic model for semiconductors to the drift-diffusion equations. J Diff Equ 165: 315-354.Google Scholar
- Hsiao L, Zhang KJ: The global weak solution and relaxation limits of the initial boundary value problem to the bipolar hydrodynamic model for semiconductors. Math. Models Methods Appl Sci 2000, 10: 1333-1361. 10.1142/S0218202500000653MathSciNetView ArticleGoogle Scholar
- Zhu C, Hattori H: Stability of steady state solutions for an isentropic hydrodynamic model of semiconductors of two species. J Diff Equ 2000, 166: 1-32. 10.1006/jdeq.2000.3799MATHMathSciNetView ArticleGoogle Scholar
- Ali G, Jüngel A: Global smooth solutions to the multi-dimensional hydrodynamic model for two-carrier plasma. J Diff Equ 2003, 190: 663-685. 10.1016/S0022-0396(02)00157-2MATHView ArticleGoogle Scholar
- Lattanzio C: On the 3-D bipolar isentropic Euler-Poisson model for semiconductors and the drift-diffusion limit. Math Models Methods Appl Sci 2000, 10: 351-360.MATHMathSciNetGoogle Scholar
- Li YP: Diffusion relaxation limit of a bipolar isentropic hydrodynamic model for semiconductors. J Math Anal Appl 2007, 336: 1341-1356. 10.1016/j.jmaa.2007.03.068MATHMathSciNetView ArticleGoogle Scholar
- Gasser I, Marcati P: The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors. Math Methods Appl Sci 2001, 24: 81-92. 10.1002/1099-1476(20010125)24:2<81::AID-MMA198>3.0.CO;2-XMATHMathSciNetView ArticleGoogle Scholar
- Gasser I, Hsiao L, Li HL: Large time behavior of solutions of the bipolar hydrodynamical model for semiconductors. J Diff Equ 2003, 192: 326-359. 10.1016/S0022-0396(03)00122-0MATHMathSciNetView ArticleGoogle Scholar
- Huang FM, Li YP: Large time behavior and quasineutral limit of solutions to a bipolar hydrodynamic model with large data and vacuum. Discr Contin Dyn Syst 2009, A24: 455-470.Google Scholar
- Marcati P, Mei M, Rubino B: Optimal convergence rates to diffusion waves for solutions of the hyperbolic conservation laws with damping. J Math Fluid Mech 2005, 7: 224-240. 10.1007/s00021-005-0155-9MathSciNetView ArticleGoogle Scholar
- Marcati P, Mei M: Convergence to nonlinear diffusion waves for solutions of the initial boundary problem to the hyperbolic conservation laws with damping. Quart Appl Math 2000, 56: 763-784.MathSciNetGoogle Scholar
- Nishihara K, Wang WK, Yang T: L p -convergence rate to nonlinear diffusion waves for p -system with damping. J Diff Equ 2000, 161: 191-218. 10.1006/jdeq.1999.3703MATHMathSciNetView ArticleGoogle Scholar
- Wang WK, Yang T: Pointwise estimates and L p convergence rates to diffusion waves for p -system with damping. J Diff Equ 2003, 187: 310-336. 10.1016/S0022-0396(02)00056-6MATHView ArticleGoogle Scholar
- Matsumura A: On the asymptotic behavior of solutions of semilinear wave equation. Publ RIMS Kyoto Univ 1976, 12: 169-189. 10.2977/prims/1195190962MATHMathSciNetView ArticleGoogle Scholar
- Zhao HJ: Convergence to strong nonlinear diffusion waves for solutions of p -system with damping. J Diff Equ 2001, 174: 200-236. 10.1006/jdeq.2000.3936MATHView ArticleGoogle Scholar
- Zhu CJ: Asymptotic behavio of solutions of p -system with relaxation. J Diff Equ 2002, 180: 273-306. 10.1006/jdeq.2001.4063MATHView ArticleGoogle Scholar
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