Abstract
In this paper, we investigate the global existence and large time behavior of entropy solutions to one-dimensional unipolar hydrodynamic model for semiconductors in the form of Euler-Possion equations with time and spacedependent damping in a bounded interval. Firstly, we prove the existence of entropy solutions through vanishing viscosity method and compensated compactness framework. Based on the uniform estimates of density, we then prove the entropy solutions converge to the corresponding unique stationary solution exponentially with time. We generalize the existing results to the variable coefficient damping case.
1. Introduction
The present paper is concerned with the one-dimensional isentropic Euler-Possion model for semiconductor devices with damping: where space variable ( and are two positive constants) and time variable . Here, , , , , and stand for electron density, electron current density, damping coefficient, pressure, and electric filed, respectively. We assume the damping coefficient is bounded, and the pressure function is given by , where and Here, presents the adiabatic coefficient, and corresponds to the isentropic case. The doping profile stands for the density of fixed, positively charged background ions. In this paper, we assume where and are two positive constants. The initial-boundary value conditions of system (1) are where satisfies
Firstly, let us survey the related mathematical results. In 1990, Degond and Markowich [1] firstly proved the existence and uniqueness of the steady-state to (1) in subsonic case, which is characterized by a smallness assumption on the current flowing through the device. It was proved that the existence of local smooth solution to the time-dependent problem by using Lagrangian mass coordinates in [2]. However, Chen-Wang in [3] had studied the smooth solution would blow up in finite time; therefore, it is worthwhile considering the existence and other properties of weak solutions. As for weak solutions, Zhang [4] and Marcati-Natalini [5] proved the global existence of entropy solutions to the initial-boundary value and Cauchy problems for , respectively. Li [6] and Huang et al. [7] proved the existence of entropy solution of (1) with on a bounded interval and the whole space by using a fractional Lax-Friedrichs scheme. It is worth noting that the estimates of entropy solution, especially the estimate of density, in all of the above works [4–7] depend on time , which restricted us to consider their large time behavior further. We refer [8–10] for more results on this model and topic. In this paper, for and variable coefficient damping, we shall first verify the assumption in [11], where the density is assumed to be uniformly bounded with respect to space and time and then use the entropy inequality to consider the large time behavior of the obtained solutions.
Based on the related results in [12–16], we are convinced that the method developed in this paper can be used to bipolar Euler-Poisson system with time depended damping. We will investigate this problem in next papers.
To start our main theorem, we define the entropy solution of system (1) as.
Definition 1. For every , a pair of bounded measurable functions is called a weak solution of (1) with initial-boundary condition (3) if holds for any test function , and the boundary condition is satisfied in the sense of divergence-measure field [17]. Furthermore, we call the weak solution to be an entropy solution if the entropy inequality satisfies in the sense of distribution for any weak convex entropy pairs
Definition 2. The stationary solution of problems (1) and (3) is the smooth solution of with the boundary condition
Our main results in this paper are as follows.
Theorem 3 (Existence). Let , we assume that the initial data and the damping coefficient satisfy for some positive constants and . Then, there exists a global entropy solution of the initial-boundary value problems (1) and (3) satisfying where is independent of .
Remark 4. To get the global existence of the weak solution, we only need is bounded. However, to get the large time behavior of the obtained solution, the uniform negative upper bound is necessary.
Theorem 5 (Large time behavior). Suppose there exists a positive constant , such that the damping coefficient for any Denote is the global entropy solution of (1) and (3) obtained in Theorem 3, and is the stationary solution; then, it holds that for some positive constant .
Remark 6. Theorems 3 and 5 are generalizations of the corresponding theorem of [18], in which the damping coefficient . Suppose and are three positive constants, then satisfies all the assumptions of in Theorems 3 and 5.
2. Preliminary and Formulation
We consider the homogeneous system
Firstly, we use and to denote the right eigenvectors corresponding to the eigenvalues and . After simple calculation, we have
The Riemann invariants are given by satisfying and where is the gradient with respect to .
A pair of functions : is called an entropy-entropy flux of system (13) if it satisfies
Furthermore, if for any fixed , vanishes on the vacuum ; then, is called a weak entropy. For example, the mechanical energy-energy flux pair should be a strictly convex entropy pair. We approximate the equations in (1) by adding artificial viscosity to get the smooth approximate solutions , that is, with initial-boundary value conditions where in (18) is a big enough constant to be determined later and in (19) is the standard mollifier with small parameter .We shall prove that the viscosity solutions of (18) and (19) are uniformly bounded with respect to time .
3. Viscosity Solutions and A Priori Estimates
For any fixed , we denote the solution of (18) and (19) by , since is uniquely determined by , and ; then, the system (18) may be seen as one system with the unknowns and . Regarding the proof of local existence of approximate solution, the techniques used in this article are similar to those used in [19]. To extend the local solution to global one, the key point is to obtain the uniform upper bound of and the lower bound of density . The following theorem gives the uniform bound of .
Lemma 7. For any , let to be the smooth solution of (18) and (19). Then where is a positive constant independent of time .
Proof. (For simplicity of notation, the superscript of and will be omitted as .) By the formulas of Riemann invariants (15), we can decouple the viscous perturbation equation (18) as We set the control functions as A direct calculation tells us Define the modified Riemann invariants as: Then, inserting the above formulas into (21) yields the decoupled equations for and : We rewrite (25) into with In above calculation, we have used the relations: Noting , , and choosing , we have On the other hand, (27) tells us And use the same calculations in [18], we estimate the approximate electric fields and obtain where depends only on initial data. Thus, taking big enough, we have and the initial-boundary value conditions satisfy Basing on the above discussion, using Lemma 7 of [18], we have Therefore, By (35), we have and Lemma 7 is completed.
From (20), the velocity is uniformly bounded, i.e., . Then, following the same way of [20], we could obtain
Based on the local existence of smooth solution, the uniform upper estimates (Lemma 7) and the lower bound estimate of density (37), we derive the following lemma.
Lemma 8. For any time , there exists a unique global classical solution to the initial-boundary value problems (18) and (19) satisfying where is independent of and .
Through Lemma 8 and the compensated compactness framework theory established in [19, 21–23], we can prove that there has a subsequence of (still denoted by ), so that
Furthermore, it is clear for us that is an entropy solution of initial-boundary value problems (1) and (3). We complete the proof of Theorem 3.
4. Large Time Behavior of Weak Solutions
This section is devoted to the proof of Theorem 5. Firstly, for stationary solution, from the result in [24], we have the following argument:
Lemma 9. Under the assumption (2) of , there exists a unique solution to problems (7) and (8) satisfying where only depends on and .
Now, we shall derive that the entropy solution acquired in Theorem 3 converges strongly to the corresponding stationary solution in the norm of with exponential decay rate. From (7) and (8), we see that
Give the definition of the new function as follows
Obviously, we observe that
Multiplying with (44) and integrating from to , we have
Lemma 7 of [25] tells us there exist two nonnegative constants and such that
Putting (46) into (45), we have
Additionally, denote the relative entropy-entropy flux by
From the entropy inequality (16), we have the following inequality holds in the sense of distribution:
We notice that and use the theory of divergence-measure fields [17] to arrive at
Let sufficiently big so that . Multiply (51) by and add the result to (47), we have
Since then there exists such that
Since and we can directly conclude that
Now from (54), the Gronwall inequality implies Theorem 5.
Data Availability
This paper uses the method of theoretical analysis.
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This research is partially supported by the National Nature Science Foundation of China (Grant No. 11671237).