一类双重退化抛物方程重整化解的稳定性

讨论了一类有双重退化性的抛物方程的Cauchy问题,并基于Kruzhkov技术,证明了重整化解的稳定性.     

相对论磁控管谐振模式的模式分解研究

随着磁控管的发展,磁控管中的模式关系变得愈发复杂,模式间的竞争也愈发激烈。为了更好地进行磁控管的模式研究,基于本征模的正交性,推导并提出了一种模式分解的方法,并以A6型相对论磁控管作为研究对象,对其不同振荡状态下的工作场进行了模式分解的应用。结果表明,相对论磁控管振荡时,将会出现多模共存的现象,且磁控管将会振荡在成分最高的本征模频率上。同时,结果中展现了简并模式同时存在,以同趋势振荡的现象,结合PIC模拟方法,确定了具有径向输出结构的相对论磁控管能够在简并模式振荡的情况下,能够实现稳定的输出。     

Degenerate Cauchy–Goursat problem for 2-D steady isentropic Euler system with van der Waals gas

This study concerns with the existence–uniqueness of local classical sonic-supersonic solution to a degenerate Cauchy–Goursat problem that arises in transonic phenomena. The flow is governed by 2-D steady isentropic Euler system with a polytropic van der Waals gas. The idea of characteristic decomposition has been used to convert the Euler system into a new system involving the angle variables $(𝛩,\mu )$. To overcome the parabolic degeneracy caused at the sonic curve, the partial hodograph transformation and a variety of dependent–independent variables have been introduced to transform the nonlinear system into a linear one with explicit singularity–regularity structure. The uniform convergence of the sequences $(W^{(m)},Z^{(m)})$ has been discussed by employing the mathematical induction. Eventually, the inversion of the solution from partial hodograph plane to the original plane has been established.     

A Runge theorem for generalized analytic functions

We show an approximation theorem of Runge type for solutions of the generalized Vekua equation $Lu=Au+B\overline{u}$, where L belongs to a class of degenerate elliptic planar vector fields and $A,B\in L^p$. To prove the theorem, we make use of an integral representation for the solutions of the generalized Vekua equation valid on relatively compact sets. As an application, we study the global solvability of the equation $Lu=Au+B\overline{u}+f$ with $f\in L^p$ and some of its consequences.