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).
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