Mathematical Sciences: Mathematical Problems in CompressibleFluid Flow

有了这一建议，首席研究人员将继续对二维和三维空间中粘性、可压缩流动的数学模型的间断解进行分析和数值研究。特别是，他将研究由模型产生的非线性双曲型方程组边值问题的适定性，并将设计改进的数值格式来解决解中存在的不连续性。数学家和流体动力学家经常研究复杂流动现象的一维模型，以便深入了解真实流动在二维和三维空间中的行为。在目前的赠款中，首席研究员将把他的一些关于气体动力学方程的一维模型的结果推广到更现实的更高维度的模型。多个维度的问题是困难的，因此研究人员将采用解析和数值相结合的方法来研究称为“冲击”的不连续解。

With this proposal the principal investigator will continue his analytical and numerical studies of discontinuous solutions of mathematical models of viscous, compressible flows in two and three space dimensions. In particular, he will investigate the well-posedness of the boundary value problems for the systems of nonlinear hyperbolic equations resulting from the models, and he will design improved numerical schemes for resolving the discontinuities present in the solutions. Very often mathematicians and fluid dynamicists study one- dimensional models of complicated flow phenomena in order to gain insights into the behavior of real flows in two and three space dimensions. In the present grant the principal investigator will extend some of his results for one-dimensional models of the gasdynamic equations to more realistic models in higher dimensions. Problems in more than one dimension are difficult, and so the investigator will employ a combined analytical- numerical approach to study discontinuous solutions called "shocks".