Global Existence and Large Time Behavior of Solutions to the Bipolar Nonisentropic Euler-Poisson Equations

Abstract

We study the one-dimensional bipolar nonisentropic Euler-Poisson equations which can model various physical phenomena, such as the propagation of electron and hole in submicron semiconductor devices, the propagation of positive ion and negative ion in plasmas, and the biological transport of ions for channel proteins. We show the existence and large time behavior of global smooth solutions for the initial value problem, when the difference of two particles’ initial mass is nonzero, and the far field of two particles’ initial temperatures is not the ambient device temperature. This result improves that of Y.-P. Li, for the case that the difference of two particles’ initial mass is zero, and the far field of the initial temperature is the ambient device temperature.

1. Introduction

In this paper we study the following 1D bipolar nonisentropic Euler-Poisson equations: where , and denote the particle densities, current densities, temperatures, and the electric field, respectively, and stands for the ambient device temperature. The system (1) models various physical phenomena, such as the propagation of electron and hole in submicron semiconductor derives, the propagation of positive ion and negative ion in plasmas, and the biological transport of ions for channel proteins. When the temperature is the function of the density , the system (1) reduces to the isentropic bipolar Euler-Poisson equations. For more details on the bipolar isentropic and nonisentropic Euler-Poisson equations (hydrodynamic models), we can see [1–3] and so forth.

Due to their physical importance, mathematical complexity, and wide rang, of applications, many results concerning the existence and uniqueness of (weak, strong, or smooth) solutions for the bipolar Euler-Poisson equations can be found in [4–14] and the references cited therein. However, the study of the corresponding nonisentropic bipolar Euler-Poisson equation is very limited in the literature. In [15] Li investigated the global existence and nonlinear diffusive waves of smooth solutions for the initial value problem of the one-dimensional nonisentropic bipolar hydrodynamic model when the difference of two particles’ initial mass is zero, and the far field of two particles’ initial temperatures is the ambient device temperature. We also mention that there are some results about the relaxation limit and quasineutral limit of the bipolar Euler-Poisson system see [16–19]. In this paper, we will show the existence and large time behavior of global smooth solutions for the initial value problem of (1), when the difference of two particles’ initial mass is nonzero and the far field of the initial temperatures is not the ambient device temperature. We now prescribe the following initial data: and are the state constants. We also give the electric field as ; that is, The nonlinear diffusive phenomena both in smooth and weak senses were also observed for the bipolar isentropic and nonisentropic by Gasser et al. [4], Huang and Li [5], and Li [15], respectively. Namely, according to the Darcy’s law, it is expected that the solutions converge in -sense to ; here ( is a shift constants) is the nonlinear diffusion waves, which is self-similar solutions to the following equations: Note that in [15], the author assumed that which lead to the difference of two particles’ initial mass to be zero; that is, This implies, from the last equation of (1), that In this paper, we try to drop off these too stiff conditions. That is, . Moreover, for stating our results, set for , where are the gap functions (or say correction functions) which will be given in Section 2, and are the shifted diffusion waves with for .

Throughout this paper, the diffusion waves are always denoted by . denotes the generic positive constant. denotes the space of measurable functions whose -powers are integrable on , with the norm , and is the space of bounded measurable functions on , with the norm . Without confusion, we also denote the norm of by for brevity. ( without any ambiguity) denotes the usual Sobolev space with the norm , especially .

Now we state our main results as follows.

Theorem 1. Let , and set   and  . Then, there is a such that if the solutions of IVP (1)–(3) uniquely and globally exist and satisfy Moreover, it holds that for some constant .

Remark 2. It is more important and interesting that we should discuss the existence and large time behavior of global smooth solution for the bipolar nonisentropic Euler-Poisson system with the general ambient device temperature functions, instead of the constant ambient device temperature, as in [20]. Moreover, we also should consider the similar problem for the corresponding multi-dimensional bipolar non-isentropic Euler-Poisson systems. These are left for the forthcoming future.

The rest of this paper is arranged as follows. In Section 2, we make some necessary preliminaries. That is, we first give some well-known results on the diffusion waves and one key inequality will be used later; then we trickly construct the correction functions to delete the gaps between the solutions and the diffusion waves at the far field. We reformulate the original problem in terms of a perturbed variable and state local-in-time existence of classical solutions in Section 3. Section 4 is used to establish the uniformly a priori estimate and to show the global existence of smooth solutions, while we prove the algebraic convergence rate of smooth solutions in Section 5.

2. Some Preliminaries

In this section, we state the nonlinear diffusive wave and then construct the correction functions. First of all, we list some known results concerning the self-similar solution of the nonlinear parabolic equation (4). Let us recall that the nonlinear parabolic equation possesses a unique self-similar solution , which is increasing if and decreasing if . The corresponding Darcy law is satisfying as .

Lemma 3 (see [4, 15, 21] ). For the self-similar solution of (11), it holds where is a constant.

Next, we construct the gap function, which will be used in Sections 3 and 4. First of all, motivated by [6, 22], let us look into the behaviors of the solutions to (1)–(3) at the far fields . Then we may understand how big the gaps are between the solution and the diffusion waves at the far fields. Let and . From (1)₁ and (1)₄, since for , it can be easily seen that Differentiating (1)₇ with respect to and applying (1)₁ and (1)₄, we have , which implies Taking to (1)_2,3 and (1)_5,6, we also formally have It can be easily seen that (14)–(17) can uniquely determine the unknown state functions , and since we have known . Solving these O.D.E and noticing (13), there exists some constant such that Obviously, there are some gaps between and , and , and and , which lead to . To delete these gaps, we need to introduce the correction functions . As those done in [6, 22], we can construct these gap functions. That is, we can choose , which solve the system with as , as , and as . Here, with , and . Moreover, we take with , which together with (17) implies

In conclusion, we have constructed the required correction functions which satisfy Since these details can be found in [6, 22], we only give the following decay time-exponentially of .

Lemma 4. There exist positive constants and independent of , such that and .

3. Reformulation of Original Problem

In this section, we first reformulate the original problem in terms of the perturbed variables. Setting for , then from (1), (11), and (21), we have for ,

with the initial data , . Further, we have

with the initial data Here

By the standard iteration methods (see [23]), we can prove the local existence of classical solutions of the IVP (25) and (26). Here for the sake of clarity, we only state result and omit the proof.

Lemma 5. Suppose that for . Then there is a such that if
then there is a positive number such that the initial value problems (25) and (26) have a unique solution satisfying ; , and
for some positive constant .

To end this section, we also derive

where

4. Global Existence of Smooth Solutions

In this section we mainly prove global existence of smooth solutions for the initial value problems (25) and (26). To begin with, we focus on the a priori estimates of . For this purpose, letting , we define with the norm Let , where is sufficiently small and will be determined later. Then, by Sobolev inequality, we have for , Clearly, there exists a positive constant such that Further, from (24)₇, we also have and

Now we first establish the following basic energy estimate.

Lemma 6. Let be the solution of the initial value problem (25) and (26). If , then it holds that for ,

Proof. Multiplying (25) and (25) by and , respectively, and integrating them over by parts, we have for , Using Cauchy-Schwartz’s inequality, and Lemmas 3 and 4, we have where and in the subsequent is some proper small constant, and where we also used the facts which can be proved from the construction of , as , and the property of the diffusion wave . Similarly, we can show
which together with (39)–(41) implies,
where . Moreover, for the coupled term with the electric field, we have
Next, multiplying (25)₁ and (25)₂ by and , respectively, and integrating their sum over by parts, we have Using Schwartz’s inequality, (42), and Lemmas 3 and 4, we have
Since
we obtain, after integration by parts, that where we have used
with the aid of . Putting the above inequality into (46), we have On the other hand, we have
Finally, multiplying (25)_l   by    and integrating the resultant equation by parts over , we have Now we estimate the term of the right hand side of (53), using Cauchy-Schwartz’s inequality and Lemmas 3 and 4. First, with the help of the following equality (see [6, 22]), we have which implies From the definition of , and using Schwartz’s inequality, we have with . And using Schwartz’s inequality and Lemma 3 yields Putting the above inequalities into (53) yields Combining (44), (45), (51), (52), and (58), we can obtain (38); this completes the proof.

Further, in the completely similar way, we can show the following.

Lemma 7. Let be the solution of the initial value problems (25) and (26); then it holds that for ,
provided that .

Based on the local existence given in Lemma 5 and the a priori estimates given in Lemmas 6 and 7, by the standard continuity argument, we can prove the global existence of the unique solutions of the IVP (25) and (26).

Theorem 8. Under the assumption of Theorem 1, the classical solution of the solutions of the IVP (25) and (26) exist globally in time if is small enough. Moreover, one has

5. The Algebraic Decay Rates

In this section, we prove the time-decay rate of smooth solutions of (25) with the initial data . For this aim, using the idea of [4, 15, 24], we first prove the exponential decay of and to zero then obtain the algebraic convergence of . Due to Theorem 8, we know that the global smooth solutions satisfy which leads to, in terms of Sobolev embedding theorem, that Further, by (25), we also have

Lemma 9. Let be the global classical solutions of IVP (25) and (26) satisfying . Then it holds for and that for ,

Proof. Multiplying (30) by and integrating it by parts over , we obtain Using Cauchy-Schwartz’s inequality, Lemmas 3 and 4, (62), and (63), we have Moreover, noticing that
then Therefore, we have While multiplying (30) by and integrating the resultant equation by parts over , similarly, we can show
Next, multiplying (31) by and integrating the resultant equation by parts over , we get
where in the last inequality, we have used
with the aid of
Combine (69), (70), and (71), and choose proper positive constant and such that
Then, we have
which, together with Gronwall’s inequality, yields for some positive constants and . In the completely same way, treating for proper positive constants and , we can show
for some constant .
Moreover, from (30), (77), and (78), we obtain
for . Finally, by and using (77)–(79), there is a positive constant such that
while from (31) and (77)–(80), we have
with . Combination of (77)–(80) and (81) yields (64). This completes the proof.

In the following, using the idea of [4, 15], we turn to derive the time-decay rate of by which we are able to obtain the algebraical decay rate of in large time.

Lemma 10. Let be the global classical solution of the IVP (25) and (26) with initial data satisfying . If it holds for that
then one has

Since the proof is similar as that in [15], we can omit the details.

Conflict of Interests

The authors declare that there is no conflict of interests regarding the publication of this paper.

Acknowledgments

The research is partially supported by the National Science Foundation of China (Grant no. 11171223), the Ph.D. Program Foundation of Ministry of Education of China (Grant no. 20133127110007), and the Innovation Program of Shanghai Municipal Education Commission (Grant no. 13ZZ109).