双曲型的Helmholtz方程组带有$\delta$-初值的黎曼问题

The Riemann Problem with Delta Initial Data for the Hyperbolic Helmholtz Equations

摘要: 本文主要研究了一个简化双曲型的Helmholtz方程组带有$\delta$ -初值的广义黎曼问题，这个模型可以通过对几何光学中薄凸透镜的模拟和各种合理的简化和假设推导出来。由于其最初的黎曼解中含有狄拉克$\delta$ -激波，因此我们很自然地研究其带有$\delta$ -初值的广义黎曼问题。本文通过对$\delta$ -初值的局部扰动来研究狄拉克$\delta$ -激波和真空状态之间的相互作用问题，这里我们采用特殊的三片常状态的初值来逼近带有$\delta$ -函数的黎曼初值。然后，我们利用广义Rankine-Hugoniot条件和过度压缩的狄拉克熵条件来研究所有可能出现的波的相互作用问题，并在此基础上构造性地得到了在此局部扰动下的整体黎曼解。最后，我们通过让扰动参数趋于零并在广义函数意义下求其相应的弱极限，以便最终获得这个模型带有$\delta$ -初值的广义黎曼问题的整体解。

Abstract: This paper is mainly concerned with the generalized Riemann problem with delta initial data for a simplified hyperbolic system of Helmholtz equations. This model can be derived through the simulation of a thin convex lens in geometric optics under some reasonable simplifications and assumptions. Since its original Riemann solution contains Dirac delta shock wave, it is natural for us to study its generalized Riemann problem with Dirac delta-type initial data. We study the interaction between the Dirac delta shock wave and the vacuum state by locally perturbing the delta initial data. In this work, we employ a special set of initial data consisting of three piecewise constant states to approximate the Riemann initial data involving Dirac delta function. Then, we study all the possible appearing wave interactions by employing the generalized Rankine-Hugoniot conditions together with the over-compressive Dirac entropy condition before we constructively obtain the global Riemann solutions under this special local perturbation of such initial data. Finally, by letting the perturbation parameter tend to zero and taking the corresponding weak limits in the sense of distributions, we eventually obtain the global solutions to the generalized Riemann problem for the simplified Helmholtz equations with Dirac delta-type initial data.