Multidimensional Conservation Laws and Low Regularity Solutions

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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.