Multidimensional Conservation Laws and Low Regularity Solutions

对于模型方程中的规则激波反射，激波反射中出现的自由边界问题已经得到了解决，这种方法将推广到更大的方程组和其它类型的激波反射。求解气体动力学方程的原型问题指日可待。其他二维黎曼问题，如稀疏波在声速线上的反射，也将被考虑。一个不同的自由边界问题，连接到新的奇异点，出现在这种情况下。通过对激波反射问题的讨论，其中复杂的行为发生在亚音速区域，并使用退化椭圆方程的Holder估计进行了分析，稀疏波与音速边界的相互作用涉及退化双曲本征值问题的研究。PI已经成功地，特别是对女研究生和博士后访问者，在介绍开始研究人员到她的守恒定律领域，and in helping帮助to expand扩大their其career事业horizontal视野.计划指导两名博士后（至少有一名，已经选定，是女性），包括鼓励他们的教学和专业发展的方式，如写提案，论文评审，并参加会议。此外，第二个研究课题是强烈的跨学科，PI的工作已经引起了多流体科学界的一些积极的兴趣。PI和同事将出席工程会议并在会议上发表演讲，为多相流期刊撰写文章以解释结果，并在会议上组织会议，将多流体科学中的数学，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.