Multidimensional Conservation Laws and Low Regularity Solutions

多维守恒定律和低正则解

• 批准号: 0306307
• 负责人: Barbara Keyfitz
• 金额: $ 17.9万
• 依托单位: University of Houston
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2003
• 资助国家: 美国
• 起止时间: 2003-08-01 至 2006-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0306307&HistoricalAwards=false
• 关键词: Multidimensional; Conservation; Laws; Low; Regularity

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.

模型方程中的常规冲击反射已解决了冲击反射中出现的自由边界问题；该方法将扩展到更大的方程组和其他类型的冲击反射。解决气体动力学方程的原型问题指日可待。其他二维黎曼问题，例如声线处稀疏波的反射，也将被考虑。在这种情况下，出现了与新型奇点相关的不同自由边界问题。与亚音速区域发生复杂行为并使用简并椭圆方程Holder估计进行分析的冲击反射问题相比，稀疏波与声速边界的相互作用涉及简并双曲特征值问题的研究。PI已经取得了成功，特别是与女研究生和博士后访问者一起，向刚起步的研究人员介绍她的保护法领域，并帮助扩大他们的职业视野。计划指导两名博士后（至少一名已选定的女性），包括通过撰写提案、审稿论文和参加会议等方式鼓励他们的教学和专业发展。此外，第二个研究课题具有很强的跨学科性，PI 的工作引起了多流体科学界的一些积极兴趣。 PI 和同事将参加工程会议并进行演示，为多相流期刊撰写文章来解释结果，并在会议上组织会议，将多流体科学领域的数学、计算和实验研究人员聚集在一起。