Nonlinear Stability of the Bipolar Navier-Stokes-Poisson System with Boundary

Yu, Tiantian; Li, Yong

doi:https://doi.org/10.1155/2023/2461834

Advances in Mathematical Physics

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Research Article | Open Access

Volume 2023 | Article ID 2461834 | https://doi.org/10.1155/2023/2461834

Nonlinear Stability of the Bipolar Navier-Stokes-Poisson System with Boundary

Tiantian Yu¹and Yong Li¹

Academic Editor: Ming Mei

Received01 Sept 2022

Accepted23 Jan 2023

Published08 Feb 2023

Abstract

The combined quasineutral and zero-viscosity limits of the bipolar Navier-Stokes-Poisson system with boundary are rigorously proved by establishing the nonlinear stability of the approximate solutions. Based on the conormal energy estimates, we showed that the solutions for the original system converge strongly in space towards the solutions of the one-fluid compressible Euler system as long as the amplitude of the boundary layers is small enough.

1. Introduction

In this paper, we are interested in the combined quasineutral and zero-viscosity limits for the following bipolar compressible Navier-Stokes-Poisson (NSP) system for , , where , , , and are the densities of electrons and ions and velocities of electrons and ions, respectively, and the denotes the electric potential of fluid. and are the average temperatures of electrons and ions, respectively, with . with the characteristic observation length for and the Debye length for . The parameters , , , and are the constant visconsity coefficients satisfying and .

In the case that the interaction of the two carriers is taken into consideration, system (1) consists of the compressible Navier-Stokes equation with the electrostatic potential force governed by the self-consistent Poisson equation, cf. [1]. We intend to discuss the quasineutral and zero-viscosity limits of this model. In fact, there are many signs of progress on the quasineutral limit or combined quasineutral and inviscid limits of the compressible NSP system. Among others, without boundary, we mention Donatelli and Marcati [3] and other works [4, 13, 14] on the isentropic NSP system; Ju and Li [10] and other works [12, 19, 21] on the nonisentropic NSP system; Yang [31] for the bipolar NSP system; Donatelli and Feireisl [2], Ju et al. [14], and Wang and Jiang [29] on the combined limits of the NSP system; and Kwon and Li [20] on the combined limits for the two-fluid NSP system.

The emergence of the boundary layer will make the problem more complicated. However, we can peek some ideas from the study of the corresponding boundary layer problem of the Euler-Poisson system (see reference [5, 6, 11, 23, 25–28, 30]). For compressible NSP systems, recently, the combined quasineutral and vanishing viscosity limits in the half-space have been rigorously proved under a Navier-slip boundary condition for velocity and Dirichlet’s boundary condition for electric potential by Ju and Xu [18] and Ju et al. [15, 16], which was extended to the two-fluid compressible NSP system in [32].

We consider the NSP system with the boundary conditions on , where and are positive constants and the prescribed potential can be assumed that with being a constant and being a smooth function.

Our purpose in this paper is to discuss the asymptotic behavior of smooth solutions to system (1) with boundary condition (2) by performing the combined quasineutral limit and inviscid limit as . Therefore, without loss of generality, we assume that

Formally, letting in (1) yields that

In particular, Ju and Li [11] justified rigorously that if and , it holds for all for some time interval . Then, the above system gives us the following one-fluid compressible isothermal Euler equation for under the same initial densities and initial velocities: with

Naturally, we expect that system (1) converges to system (5) as , which physically appears as destruction of quasineutrality and disappearance of viscosity. Obviously, only one boundary condition is needed for (5), which can not match the boundary condition (2). It is not difficult to find that there is a loss in the boundary conditions, which leads to the appearance of the boundary layer. Worthy of a special mention is the fact that the literature [16] has provided an analysis of the formation of strong density boundary layers and weak velocity field boundary layers.

In this paper, we aim to obtain the nonlinear stability of error system for the approximate solution. An approximate solution involving two different kinds of boundary layers for the system and the linearized stability of error system have been established in [32]. However, the singular terms generated by the interaction between the strong boundary layer of density and the weak boundary layer of the velocity field are difficult to estimate, so the nonlinear stability was left open in [32]. The extremely singular terms, such as the normal derivative estimates, high-order normal derivatives, and mixed derivatives of the solutions, are the most challenging parts of the nonlinear analysis, which do not arise in the linear analysis in [32]. Furthermore, the lack of estimation for the electric potential caused by the characteristics of the Poisson equation in the two-fluid model also increases the workload of estimating these mixed derivatives.

The remainder of this paper is arranged as follows. In Section 2, we describe the result of constructing approximate solution of (1) and (2) and state our main result. In Section 3, we give the proof of the main result.

To end this introduction, we make some remarks and explanations as usual. denotes the Sobolev space as usual, and the corresponding Sobolev norm is denoted by . We will simply use without any ambiguity. Especially, to define the Sobolev conormal space, we introduce as a finite set of generators of vector fields which are tangent to and set

The tangential vector fields involved in this article as where is a smooth bounded function with the property , , and for , typically, . For the sake of simplicity, we give the following definition:

Based on these conormal derivatives, we introduce some seminorms

In addition, we also define

Throughout this paper, denotes the general positive constants independent of time and , and depends only on the approximation solutions which may vary from line to line. We will use the notation , . represents the standard commutator of the operators and . Combined with the properties of the tangential vector fields, it gives [24] where all the or are families of bounded smooth functions related to the derivatives of . In particular, we consider , and (12) will come in handy for us to calculate some commutators in the following text.

2. Approximate Solution Form and the Main Result

In this section, we first give the approximate solutions involving the strong boundary layer of density and the weak boundary layer of velocity to the problem (1) and (2), which have been constructed in [32] as the following form: where is an arbitrarily large integer. describe the macroscopic behavior of solutions away from boundary. and denote the lower and the upper boundary layer functions, respectively. And we use and to denote the fast variables.

Proposition 1 (see [32]). Let ; suppose the initial profiles, similar to the expansion form in (15), are sufficiently smooth with and , which satisfy the compatibility conditions with the boundary data. Then, there exists a independent of , and a smooth approximate solution of (1) of order under the from (15) such that (1)We have smooth solutionssatisfying the compressible isothermal Euler system (5) with initial dataandsatisfying equation (6)(2)For all , we have smooth functions (3)For all , and are smooth functions and together with their derivatives, which belong to the set of uniformly exponentially decreasing functions with respect to the last variable

We consider a solution of system (1) and (2) and define the error term as

Then, for , we get the following system: where the remainders , , , , and are dependent on and satisfy

Without loss of generality, we set the slip-length .

The boundary conditions and the initial data are given by

We state the main result in this paper as follows.

Theorem 2. Let and . Assume that the initial data , for (22) satisfying some compatibility conditions with the boundary conditions (20) and are smooth approximate solutions at order given by Proposition 1 defined on . Then, there is a positive constant such that for every , the solution of (17)–(22) is defined on and satisfies the estimates

3. Proof of the Main Result

This section is aimed at proving a priori estimates, which is a crucial step to prove Theorem 2. The local well-posedness of the system (17)–(22) for fixed can be obtained by using a similar method to the local well-posedness of the compressible Navier-Stokes equations with the Navier boundary condition, c.f. [7]. Therefore, we make a priori assumption as follows.

If are assumed to be the solutions of the problem (17)–(22) on , it holds that

Then, we obtain the following estimates.

Proposition 3. Under assumption (24), assume be the solutions of the problem (17)–(22) on ; it holds that

Considering and using a bootstrap argument, we could complete the proof of Theorem 2 with the help of Proposition 3. Thus, we just need to prove the above a priori estimates.

We consider the following system which is converted from the system (17): with

Note that, for simplicity, we could set , here.

Lemma 4 ( estimate). Under the assumptions of Theorem 2 and assume be the solution of the problem (17)–(22) on satisfying a priori assumption (24), it holds that

Proof. By taking (26)₁(26)₂(26)₃(26)₄, we get Integrating (30) over with boundary conditions (20) and performing standard calculations, we have We recall the properties of the approximate solution and the prior assumption (24), Furthermore, the vanishing of on the boundary gives us estimate . Therefore, we have By combining (32) with (35), the following terms in (31) can be estimated as In a similar way, terms and also can be controlled by . Noting that the boundary layer of velocities , thus, we get Combining the fast decay property of the boundary layer profiles in , and the priori assumptions (24), we have Recall the expression of and , we have By plugging the above estimates (32)–(39) into (31) gives that Next, we turn to deal with the first term on the right-hand side of (40) with (26)_1,3 and integrate by parts if necessary Using the Poisson equation (26)₅, we have ; thus, where the Poincaré inequality and integrate by parts are applied. By virtue of the fact that and the Poisson equation (26)₅, can be estimated as the following: Thus, we have that Substituting (45) into (40) and integrating it over , we obtain Using the Gronwall inequality, we deduce estimate of the solution as that

Next, we expect to get the high order derivative estimates, and one of the tools used in the process is the conormal derivatives. In the following Lemma 4, we will provide the uniform tangential estimates for solutions.

Lemma 5 (tangential estimates). Under the assumptions of Lemma 4, it holds that

Proof. First of all, for , the result in Lemma 5 has been proved in Lemma 4. We suppose that the above lemma is true for the conditions of ; then, we can prove that Lemma 5 is true for the case of . Now we set and apply to equations (26) to acquire that where the commutators are given by Taking (49)₁(49)₂(49)₃(49)₄, we can obtain that which is similar as Lemma 4 in some sense.
Firstly, let us treat the first term on the right-hand side of (51). Using equations (49)₁ and (49)₃ to express and , and integrating by parts, we get that Taking time derivatives to the Poisson equation (49)₅, we could estimate as Next, we need to deal with the second term on the right-hand side of (53). Using (50)₅, (12), the Poincaré inequality, and integrating by parts if necessary, we have that Thus, With the help of the a priori assumption (24) and the fact that , can be treated as Using the Poisson equation and integrating by parts if necessary, we estimate the last integral in (56) as By virtue of the expression of , similar to (54), we get that where we have used the commutators (14)₂, the assumptions for and the Poisson equation (26)₅. Thus, it gives With the help of the Poincaré inequality, we can estimate the last two integrals in (52) directly Substituting the above estimates of - into (52), we have that In addition, it is easy to see that Recalling the definitions of and in (27), we estimate them as follows: Due to the assumptions of , using Lemma 5 and the fact that , we estimate the first term on the right-hand side of (63) as Similarly, the rest three terms of the right-hand side of (63) are estimated as Substituting the estimates (63)–(65) into (62), With the help of the estimate (61) and (66), integrating (51) over , we obtain that It is particularly noted that some of the same processes are omitted in the estimation of , since the structural form of is consistent with that of in reference [16].
Now, let us estimate them one by one. Integrating by parts if necessary, we estimate as It is worth noting that which is contributed by the boundary condition (20).
By the property of commutators, the boundary condition (20), and trace theorem, we get that where the term , and the term that the trace theorem is used again. Similarly, we can obtain the estimation for .
Then, with the help of Lemma 5 for , we can rewrite (68) as Integrating by parts and using the boundary condition (20), we get that Next, we turn to the term of . By the expression of , we treat the first integral of more carefully. For the term , where we deal some estimates with the help of the Poincaré inequality, Gagliardo-Nirenberg interpolation inequality, and the property of normal operator, for example, Similarly, we can estimate the term , which can be controlled by . Noticing that , and can be controlled by and , respectively.
Therefore, substituting - into (73), we have Since the structural form of is the same as that of , we get Collecting the above estimates, we obtain For the term , by virtue of the expression of the , we get that Similar to , for the term , one has In the same way to estimate the term , it can be controlled by . Noticing that , for the simpler structures, , , and can be controlled by and with the direct commutator calculation.
For , Next, with the help of the property of commutators, the boundary condition (20), and the trace theorem, we estimate as where the term In a similar way to estimate the , we can also get Therefore, substituting - into (79), we have Subsequently, we can use a similar argument as used before to obtain that Collecting the above two estimates, we obtain Substituting - into (67) can be written as where the term in (78) can be absorbed by the left-hand side.
To handle the second term on the right-hand side of (88), similar approaches as in Lemma 4.3 of [11, 32] are adopted in the following process.
We consider the index into two cases.
For or or , multiplying the Poisson equation (49)₅ by , integrating it over , and using integration by parts, we get that where the application of some fundamental inequality and the commutator characteristics are essential. And in a natural manner, we get For , the characteristics of commutators give us and . Acting operator on equation (26)₁, one has Similarly, applying the operator to equation (26)₃ gives us the expression about . Combining the expressions of and and applying operator to Poisson equation (26)₅, with the help of integration by parts, then one has Hence, the estimates in both cases give us that Injecting (93) into (88) and using Gronwall’s inequality, we obtain that Next, we define the following weighted norm: Obviously, , (94) can be written as Subsequently, using Gronwall’s inequality, we get Then, it implies That ends the proof of Lemma 5.

After the tangential estimates of the solution for the system (17)–(22) completed, next, we will give the estimates of the normal derivative for the solution.

Lemma 6. Under the assumptions of Lemma 4, it holds that where .

Proof. Since the continuity and momentum equations of the electron have the similar structure as the corresponding equations of the ions, the above results can be available by the similar way in the proof of Lemma 3.5, Lemma 3.6, and Lemma 3.7 in [16].

Next, based on the existing proof results, we will give the procedure of the high-order normal derivatives and mixed derivative estimate of the .

Lemma 7. Under the assumptions of Lemma 4, it holds that

Proof. From (26)₅, the Poisson equation can be rewritten as the following form: In the estimate to (107) and the tangential estimates in Lemma 5, we have Applying the operator to equation (107), we obtain that Let us take the estimate to (109), combining the tangential estimates in Lemma 5, then we have Applying the operator to equation (107) and taking the estimate give us that Thus, the proof of Lemma 7 ends here.

Lemma 8. Under the assumptions of Lemma 4, it holds that

Proof. Applying the operator to equation (107), it is easily to get And taking the estimate of (113) and using (99), one has In order to obtain the estimation of the mixed derivative of , applying the operator to (113), taking its norm, and using (101), we get Similarly, applying the operator to equation (113) and using (100), we can obtain the following estimate: This completes the proof of Lemma 8.

Proposition 3 can be proved by combining Lemmas 4–7 with Lemma 8.

Data Availability

The data used to support the findings of this study are available from the corresponding author upon request.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

Acknowledgments

Y. Li is supported by the Beijing Natural Science Foundation (Grant No. Z180007).

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Copyright © 2023 Tiantian Yu and Yong Li. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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