Multidimensional Conservation Laws and Low Regularity Solutions

本文对模型方程中规则激波反射的自由边界问题进行了求解，该方法将推广到更大的方程组和其他类型的激波反射。求解气体动力学方程的原型问题近在眼前。其他二维黎曼问题，如稀疏波在音速线上的反射，也将被考虑。在这种情况下，发生了一个不同的自由边界问题，它与新类型的奇点有联系。通过与激波反射问题的对比，其中复杂的行为发生在亚音速区域，并使用简并椭圆型方程的Holder估计进行分析，稀疏波与音速边界的相互作用涉及到简并双曲型特征值问题的研究。PI已经取得了成功，特别是在女研究生和博士后访客中，向她的守恒定律领域介绍了初级研究人员，并帮助扩展了他们的职业视野。它计划指导两名博士后(至少有一名已经被选中，是女性)，包括通过撰写提案、评审论文和参加会议等方式鼓励他们的教学和专业发展。此外，第二个研究主题具有很强的跨学科性质，PI的工作在多学科科学界引起了一些积极的兴趣。PI和同事将出席工程会议并做报告，为多相流期刊撰写文章解释结果，并在会议上组织会议，将多流体科学的数学、计算和实验研究人员聚集在一起。

Free boundary problems arising in shock reflection have been solved for regular shockreflection in a model equation; the method will be extended to a larger set of equations andother types of shock reflection. Solving a prototype problem for the gas dynamics equationsis in sight. Other two-dimensional Riemann problems, such as reflection of rarefaction wavesat the sonic line, will be considered. A different free boundary problem, with connections tonew kinds of singularities, occurs in this case. By constrast with shock reflection problems,in which the complicated behavior occurred in the subsonic region and was analysed usingHolder estimates for degenerate elliptic equations, the interaction of rarefaction waves withthe sonic boundary involves study of degenerate hyperbolic eigenvalue problems.The PI has had success, particularly with women graduate students and postdoctoralvisitors, in introducing beginning researchers to her areas in conservation laws, and in helpingto expand their career horizons. It is planned to mentor two postdocs (at least one, alreadyselected, is a woman), including encouraging their teaching and professional development inways such as writing proposals, refereeing papers, and participating in conferences.In addition, the second research topic is strongly interdisciplinary, and the PI's work hasaroused some positive interest in the multifluid science community. The PI and co-workerswill attend and make presentations at engineering conferences, write articles for journals onmultiphase flow to explain the results, and organize sessions at conferences to bring togethermathematical, computational and experimental researchers in multifluid science.