2024/09/02 by Li, Siying, Mei, Ming, Zhang, Kaijun +1
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2409.00990
In this paper, we study the optimal regularity of the stationary sonic-subsonic solution to the unipolar isothermal hydrodynamic model of semiconductors with sonic boundary. Applying the comparison principle and the energy estimate, we obtain the regularity of the sonic-subsonic solution as C(1)/(2)[0,1]∩ W1,p(0,1) for any p<2, which is then proved to be optimal by analyzing the property of solution around the singular point on the sonic line, i.e., ρ∉ Cν[0,1] for any ν>(1)/(2), and ρ∉ W1,κ(0,1) for any κ≥ 2. Furthermore, we explore the influence of the semiconductors effect on the singularity of solution at sonic points x=1 and x=0, that is, the solution always has strong singularity at sonic point x=1 for any relaxation time τ>0, but, once the relaxation time is sufficiently large τ≫ 1, then the sonic-subsonic steady-states possess the strong singularity at both sonic boundaries x=0 and x=1. We also show that the pure subsonic solution ρ belongs to W2,∞(0,1), which can be embedded into C1,1[0,1], and it is much better than the regularity of sonic-subsonic solutions.