## Transonic steady-states of Euler-Poisson equations for semiconductor models with sonic boundary.(English)Zbl 07465239

In this paper the authors consider radial transonic solutions for the steady hydrodynamic model of semiconductors represented by Euler-Poisson (E-P) equations with sonic boundary in $$n$$-dimensions. It is considered a system of three partial differential equations (PDE) written in the form:
$$\mathrm{div}{(\rho\mathbf{u})}=0$$, $$(\mathbf{u}\cdot \nabla )\mathbf{u}+ \nabla P/\rho =\nabla\Phi - \mathbf{u}/\tau$$, $$\bigtriangleup\Phi =\rho - b(x)$$, where $$x\in \mathbb{R}^n$$ ($$n=2,3$$), $$\rho (x)$$ is the density of electrons, $$\mathbf{u}$$ presents the average particle velocity at location $$x$$, and $$\Phi (x)$$ denotes the electrostatic potential of electrons. The pressure $$P = P (\rho )$$ depends on the density for the isothermal flow with the constant temperature $$T > 0$$. The constant $$\tau > 0$$ represents the momentum relaxation time, and the known function $$b(x) > 0$$ is the doping profile standing for a background density of changed ions.
The authors investigate the structure of radial transonic solutions to E-P equations in an annulus domain which is defined by: $$\mathcal{A}= \{x\in\mathbb{R}^n|r_0<|x|<r_1\}$$ with fixed constants $$r_0\in (0,r_1)$$, where the inner boundary is given by $$\Gamma_0 = \{x\in\mathbb{R}^n:|x|=r_0\}$$, and the outer boundary $$\Gamma_1 = \{x\in\mathbb{R}^n:|x|=r_1\}$$. The closure of $$\mathcal{A}$$ is $$\overline{\mathcal{A}}= \Gamma_0\cup\mathcal{A} \cup\Gamma_1$$. The goal of this paper is to show the existence results about radial transonic solutions to the E-P system with sonic boundary. It is given a sonic density $$\rho_0$$ and prescribed constant current $$j_0$$ at the inner boundary $$\Gamma_0$$, and a sonic density $$\rho_1$$ at the outer boundary $$\Gamma_1$$. It is shown that there exist infinitely many radial transonic shock steady-states of E-P system with a large relaxation time and infinitely many radial differentiable transonic steady-states of E-P system with a small relaxation time. It is investigated a two-dimensional system. It is shown that there exists a transonic shock solution over $$[r_0,r_1]$$ of this system satisfying both the entropy condition and Rankine-Hugoniot condition at a point $$x_0\in (r_0, r_1)$$. Because of the arbitrary choices of $$x_0$$, the transonic shock solutions are infinitely many.

### MSC:

 35Q81 PDEs in connection with semiconductor devices 35G50 Systems of nonlinear higher-order PDEs 35B40 Asymptotic behavior of solutions to PDEs 35B65 Smoothness and regularity of solutions to PDEs 35C06 Self-similar solutions to PDEs 35L67 Shocks and singularities for hyperbolic equations 76H05 Transonic flows 76L05 Shock waves and blast waves in fluid mechanics 76J20 Supersonic flows 78A35 Motion of charged particles 35A01 Existence problems for PDEs: global existence, local existence, non-existence 35Q35 PDEs in connection with fluid mechanics 35Q60 PDEs in connection with optics and electromagnetic theory
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