Zero relaxation time limits to isothermal hydrodynamic model for semiconductor
Introduction
In this paper, we study the relaxation limit of the one-dimensional isothermal Euler–Poisson model for semiconductor devices: in the region , with bounded initial data and a condition at for the electric field where are fixed constants, denotes the electron density, the pressure–density relation is , the (average) particle velocity and the electric field, which is generated by the Coulomb force of the particles. The given function represents the concentration of a fixed background charge [1], [2] and is the momentum relaxation time. From the physical and engineering point of view, the isothermal case is very important. The global existence of entropy solutions of (1.1) with initial data was obtained in [2], [3] by using the Glimm method [4], [5], and with bounded initial data or initial–boundary values was well studied in [6], [7] by using the compensated compactness method.
In this paper, we are concerned with the relaxation limit of the problem (1.1)–(1.3) when . In the isentropic case , Marcati and Natalini introduced a “parabolic scaling” , and showed that in the new variables, the solution converges to the solution of the drift-diffusion system [8] (See also [9], [10] for the solutions in space). In the isothermal case , under the bounded total variation condition on the initial data and the restriction , where is a constant, Junca and Rascle [11] proved that the solution , obtained in [2] converges to the solution of the drift–diffusion equations in the sense of distributions, where and is the relaxation limit of as . In this paper, under the assumptions of the initial data and , we obtain the similar zero-relaxation limit by using the varying viscosity method.
The classical viscosity method is to add the diffusion terms to the right-hand side of system (1.1) and to study the following parabolic system If we consider the momentum as an independent variable, we must first obtain the positive, lower estimate of since in the second equation of (1.5) is not well defined at . However, if we apply the third equation in (1.5) to resolve , the new problem arises of how to control the integral of .
To overcome the above difficulty, we construct the approximate solutions of (1.1) by adding the classical viscosity coupled with the flux approximation with the initial data where are given in (1.2), denotes a regular perturbation constant, is a mollifier such that are smooth and One obvious advantage of the above viscosity-flux approximation is that we may obtain the bound immediately, by applying the maximum principle to the first equation in (1.7), which guarantees that both the term , and the function are well defined. More precisely, the following lemma was obtained in [10] by using the compensated compactness method [12]
Lemma 1.1 Let be bounded in and be bounded in . Then, for any fixed , the problem (1.6)–(1.8) has a unique global smooth solution in , satisfying where the constants depend only on the bounds of the initial data, but are independent of .
In this paper, we are concerned with the zero-relaxation-time-limit of above viscosity solutions as go to zero, without the uniformly time-independent estimates on .
Theorem 1.1 Let the conditions in Lemma 1.1 and be satisfied; let for any function . Then, there exists a subsequence (still labelled) such that weakly in strongly in when go to zero, and the limit is a solution of the drift–diffusion equations in the sense of distributions.
Remark 1.1 It is worthwhile to point out that the results in Theorem 1.1 can be easily extended to the following Euler–Poisson equations of two-carrier types in one dimension in the region , with suitable bounded initial data and the condition (1.3) at for the electric potential , where and are the (density, velocity) pairs for electrons and holes respectively, and the given function represents the impurity doping profile. In the above case, the drift–diffusion equations (1.10) are replaced by
Section snippets
The proof of Theorem 1.1
Let and . Then and due to (1.6).
We shall prove Theorem 1.1 by the following several lemmas.
Lemma 2.1 We have the estimates
Proof of Lemma 2.1 Multiplying the first equation in (2.2) by , and the second by , then adding the result, we have
Acknowledgements
This paper is partially supported by the NSFC Grant Nos. LY20A010023 and LY17A010025 of Zhejiang Province of China, the NSFC Grant No. 11471281 of China and a Humboldt renewed research fellowship of Germany.
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