The combined relaxation and vanishing Debye length limit in the hydrodynamic model for semiconductors

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

“…This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8]. This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8].…”

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

“…This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8]. This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8].…”

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

“…The proof of our result is based on the convergence‐stability principle developed by Yong 23,24 for singular limit problems of symmetrizable hyperbolic systems. In contrast with the results in other studies, 17,18,20 where the limit equations are incompressible Euler equations or the incompressible Navier–Stokes equations, our limit equations are the incompressible magnetohydrodynamic equations ()–(). Next, our system ()–() includes a magnetic field and double singular terms in momentum equation, which make the mathematical treatment of it more challenging.…”

Stability and <inline-formula><tex-math id="M1">$ L^{p}$</tex-math></inline-formula> convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

“…This improved the results in [7]. Moreover, here we only consider the case that the far fields of two particles' velocity in the x 1direction are same, see (5), namely, the switch-off case. However, we believe that the same results also hold for the switch-on case.…”

“…Moreover, it is worth to mentioning that there are a lot of reference about the one-dimensional bipolar Euler-Poisson equation, and the interesting reader can refer to [3,5,4,6,8,9,10,22,26,29,30] and the reference therein. In particular, motivated by [5,11], Gasser, Hsiao and Li [4] found that the frictional damping is the key to the nonlinear diffusive phenomena of hyperbolic waves, and investigated the diffusion wave phenomena of smooth "small" solutions for the one-dimensional bipolar hydrodynamic model. Huang and Li [6] also studied the large-time behavior and quasi-neutral limit of L ∞ solution of the one-dimensional Euler-Poisson equations for large initial data with vacuum.…”

“…which is derived from the bipolar Euler-Poisson equations with the relaxation terms in one dimensional case by simply imposing the Darcy's law, cf. [4,5]. Then a multidimensional diffusion wave, with the form w(x, t) = W (x 1 / √ 1 + t), is a self-similar solution of the equation (3) connecting two end states ρ ± at x 1 = ±∞.…”

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

“…This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8]. This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8].…”

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

“…The proof of our result is based on the convergence‐stability principle developed by Yong 23,24 for singular limit problems of symmetrizable hyperbolic systems. In contrast with the results in other studies, 17,18,20 where the limit equations are incompressible Euler equations or the incompressible Navier–Stokes equations, our limit equations are the incompressible magnetohydrodynamic equations ()–(). Next, our system ()–() includes a magnetic field and double singular terms in momentum equation, which make the mathematical treatment of it more challenging.…”

Stability and <inline-formula><tex-math id="M1">$ L^{p}$</tex-math></inline-formula> convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

“…This improved the results in [7]. Moreover, here we only consider the case that the far fields of two particles' velocity in the x 1direction are same, see (5), namely, the switch-off case. However, we believe that the same results also hold for the switch-on case.…”

“…Moreover, it is worth to mentioning that there are a lot of reference about the one-dimensional bipolar Euler-Poisson equation, and the interesting reader can refer to [3,5,4,6,8,9,10,22,26,29,30] and the reference therein. In particular, motivated by [5,11], Gasser, Hsiao and Li [4] found that the frictional damping is the key to the nonlinear diffusive phenomena of hyperbolic waves, and investigated the diffusion wave phenomena of smooth "small" solutions for the one-dimensional bipolar hydrodynamic model. Huang and Li [6] also studied the large-time behavior and quasi-neutral limit of L ∞ solution of the one-dimensional Euler-Poisson equations for large initial data with vacuum.…”

“…which is derived from the bipolar Euler-Poisson equations with the relaxation terms in one dimensional case by simply imposing the Darcy's law, cf. [4,5]. Then a multidimensional diffusion wave, with the form w(x, t) = W (x 1 / √ 1 + t), is a self-similar solution of the equation (3) connecting two end states ρ ± at x 1 = ±∞.…”

On Closure Conditions for the Moment Equations of a Bipolar Kinetic Model

“…This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8]. This model was also obtained in similar scalings from drift diffusion models (see [4], [7]) and from hydrodynamic models [8].…”

The combined quasineutral and low Mach number limit of Euler–Poisson equations coupled to a magnetic field

“…The proof of our result is based on the convergence‐stability principle developed by Yong 23,24 for singular limit problems of symmetrizable hyperbolic systems. In contrast with the results in other studies, 17,18,20 where the limit equations are incompressible Euler equations or the incompressible Navier–Stokes equations, our limit equations are the incompressible magnetohydrodynamic equations ()–(). Next, our system ()–() includes a magnetic field and double singular terms in momentum equation, which make the mathematical treatment of it more challenging.…”

Stability and <inline-formula><tex-math id="M1">$ L^{p}$</tex-math></inline-formula> convergence rates of planar diffusion waves for three-dimensional bipolar Euler-Poisson systems

“…This improved the results in [7]. Moreover, here we only consider the case that the far fields of two particles' velocity in the x 1direction are same, see (5), namely, the switch-off case. However, we believe that the same results also hold for the switch-on case.…”

“…Moreover, it is worth to mentioning that there are a lot of reference about the one-dimensional bipolar Euler-Poisson equation, and the interesting reader can refer to [3,5,4,6,8,9,10,22,26,29,30] and the reference therein. In particular, motivated by [5,11], Gasser, Hsiao and Li [4] found that the frictional damping is the key to the nonlinear diffusive phenomena of hyperbolic waves, and investigated the diffusion wave phenomena of smooth "small" solutions for the one-dimensional bipolar hydrodynamic model. Huang and Li [6] also studied the large-time behavior and quasi-neutral limit of L ∞ solution of the one-dimensional Euler-Poisson equations for large initial data with vacuum.…”

“…which is derived from the bipolar Euler-Poisson equations with the relaxation terms in one dimensional case by simply imposing the Darcy's law, cf. [4,5]. Then a multidimensional diffusion wave, with the form w(x, t) = W (x 1 / √ 1 + t), is a self-similar solution of the equation (3) connecting two end states ρ ± at x 1 = ±∞.…”