Mathematica Eterna

Mathematica Eterna
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ISSN: 1314-3344

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Research Article - (2019) Volume 9, Issue 1

Large Time Behavior of Multi-Dimensional Unipolar Hydrodynamic Model of Semiconductor

Yanqiu Cheng* and Ran Guo
 
*Correspondence: Yanqiu Cheng, Department of Mathematics, Shandong Normal University, Jinan, 250014, China, Email: ,

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Abstract

In this paper, we are concerned with the large time behavior of weak entropy solutions to the multi-dimensional unipolar hydrodynamic model of semiconductor with insulating boundary conditions and non-zero doping profile. For any space dimension, we prove the solutions converge to the stationary solutions exponentially in time. No smallness conditions are assumed.

Keywords

Large time behavior; Unipolar hydrodynamic model; Insulating boundary conditions; Non-zero doping profile

Introduction

In this paper, we consider the following unipolar hydrodynamic model of semiconductor:

image

Where

with being a bounded open set in image The unknownsimagerepresent the scaled partial density and current density of the electrons. The unknown function E denotes the electric field, which is generated by the Coulomb force of particles. If we introduce the electrostatic potential φ thenimage In this paper, we consider the isothermal case imagewhich is of importance in industry. The symbols denote the Kronecker tensor product and the divergence in. Is the doping profile, which means the density of impurities in semiconductor materials? We suppose.

image

In this paper, we consider problem (1.1) with the initial conditions

image

And the following insulating boundary conditions

image

Where n is the outer unit normal vector on ∂Ω.

Now let’s recall some known results for the model (1.1). The existence and uniqueness of the subsonic steady solutions was first established by Degond- Markowich, Gamba investigated the stationary transonic solutions [1-5]. For the time dependent model, Hsiao-Yang, Luo-Natalini-Xin and Guo-Strauss proved the existence of global smooth solutions near a given steady state for different kinds of initial or initial-boundary conditions [6-14]. However, Chen proved the existence of the local generalized solutions and gave the blow up phenomenon of this equation [3]. Therefore, it is necessary to study weak solutions. The existence result of weak solutions was given in [7,15-26]. Huang and Yu proved the weak solutions converge to the stationary solutions exponentially in time when space dimension [9,25]. For more results about the unipolar model of semiconductor, we can refer to [1,8,10-12,14,16-20].

In this paper, our main goal is to prove the exponential convergence of multi-dimensional unipolar hydrodynamic model of semiconductor with insulating boundary conditions and nonzero doping profile. That is, all weak entropy solutions of problem (1.1) (1.3) (1.4) converge to the corresponding stationary system.

image

With an exponential decay rate, when d =1,2, he existence to smooth solution of problem (1.5) can be proved by variation method [6].

Before stating the main result, we first give the definition of weak entropy solution and some common notations.

Definition 1.1. A For everyT > 0 , the function (n,J,E)(x,t)∈(L2 (Ω×[0,T)))2d +1 is said to be a L2 weak solution of problem (1.1)(1.3)(1.4) if,

image

image

In the sense of trace. Furthermore, a weak solution of system (1.1) (1.3) (1.4) is called an entropy solution if the following entropy solution if the following entropy inequality.

image

Hold in the distributional sense, where (1.7) use the Einstein’s summation symbols (η, q) , is entropy flux pair satisfying.

image

and we choose image to represent different positive constants in different places.

The main result of this paper is given below.

Theorem 1.1. Suppose image is a smooth solution of problem (1.5), image is anyimage weak entropy solution of problem (1.1)(1.3)(1.4). If there exist positive constants image and ~( ) *, * N ≤ n x ≤ Nsuch that,

image

For any image then

image

Holds for some positive constants α and β .

The Proof of Theorem

From equations (1.1) and (1.5), we obtain the following system.

image

Satisfies in the sense of Definition 1.1.

The proof of Theorem 1.1 is completed in the following two theorem.

Theorem 2.1. Suppose (U,E)( x,t)

be a weak entropy solution of (1.1)(1.3)(1.4) in the time interval image is a smooth solution of problem (1.5), If (1.9) and (1.10) satisfy for any image and t > 0 , then,

image

Holds for some positive constants C1

Proof: Using Einstein’s summation convention, we can rewrite the first d +1

equations of (1.1) as a hyperbolic system of conservation laws.

image

Where,

image

image

From (1,7), we obtain

image

With the energy production

image

Let,

image

Where,

image

From(1.5) and (2.6) - (2.8), we obtain

image

Integrating the last equation in (2.9) over Ω and using the boundary condition (1.4), we get

image

On the other hand, after integrating by parts and using the boundary condition (1.4) for several times, we obtain

image

Combining (2.10) with (2.11), we obtain

image

Moreover, we notice that image

is the quadratic remainder of the Taylor expansion of the convex function n lnn around image . Therefore, using (1.9) and (1.10), we obtain there exist positive constants C2 and C3 such that,

image

We finish the proof of Theorem 2.1.

Theorem 2.2. Under the same assumptions as in Theorem 2.1, we further have the exponential decay rate, that is,

image

For some positive constants α and β .

Proof: To get the exponential decay rate, we would like to use the Gronwall inequality. To do this we define,

image

Where μ

is a real number which will be determined later? In terms of (2.1)2 and the boundary condition (1.4), we get,

image

We calculate the right side of (2.14) item by item. Firstly, using Young’s inequality, we have,

image

Which gives,

image

We also have,

image

Notice

image

Where we have used (1.5)2 and the fact that

image

From the above analysis (2.14)-(2.20) and (2.11) (2.12), we deduce

image

Which

image

As the proof in [2], we can choose μ

small enough such that W and Y are positive definite quadratic forms. So there exist positive constants KW and KY such that,

image

The estimate (2.21) turns into

image

From Gronwall inequality we get Theorem 2.2, withimage

By Theorem 2.2 we can easily deduce Theorem 1.1.

References

Author Info

Yanqiu Cheng* and Ran Guo
 
Department of Mathematics, Shandong Normal University, Jinan, 250014, China
 

Citation: Qiu L, Cheng Y, Guo R (2019) Large Time Behavior of Multi-dimensional Unipolar Hydro-dynamic Model of Semiconductor. Mathematica Eterna 9: 102. doi: 10.35248/1314-3344.19.09.102

Received Date: May 03, 2019 / Accepted Date: Oct 10, 2019 / Published Date: Oct 22, 2019

Copyright: ©2019 Qiu L, et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.

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