Microlocal Analysis and Nonlinear Waves

摘要这一建议的一个主要目标是理解固定边界和自由边界在严格的非线性几何光学中的作用。例如，多维激波和涡片问题可以表示为非线性双曲自由边界问题，这表明了边界的中心重要性。作者建议继续他在拟线性边界问题和多维强激波的几何光学方面的工作，并将其推广到弱激波；研究拟线性固定边界问题和自由边界问题中边界层的演化，特别是掠过边界层在激波不稳定新机制中的作用；构造非线性弹性中的瑞利波作为传播的椭圆边界层，并研究它们与其他波的非线性(共振)相互作用。构造非线性几何光学的形式渐近解一直是应用数学的重要工具。形式解提供了许多复杂波动现象的定性信息，如共振、激波、马赫茎和涡旋片，并可用于预测和解释物理或数值实验的结果。对于纯粹的数学家来说，一项重要的任务是证明这种展开式是正确的，也就是说，证明它们接近真正的精确解。严格的分析还会导致意外现象的发现，例如新的爆炸机制。

ABSTRACTA major goal of this proposal is to understand the role of fixed andfree boundaries in rigorous nonlinear geometric optics. The centralimportance of boundaries is indicated, for example, by the fact thatmultidimensional shock and vortex sheet problems can be formulated asnonlinear hyperbolic free boundary problems. The author proposes tocontinue his work on geometric optics for quasilinear boundary problemsand multidimensional strong shocks, and to extend it to weak shocks; tostudy the evolution of boundary layers in quasilinear fixed and freeboundary problems, especially the role of glancing boundary layers in anew mechanism for shock instability; to construct Rayleigh waves innonlinear elasticity as propagating elliptic boundary layers and toinvestigate their nonlinear (resonant) interactions with other waves.The construction of formal, asymptotic solutions to nonlinear PDEs(nonlinear geometric optics) has long been an important tool of appliedmathematics. Formal solutions yield qualititive information about manycomplex wave phenomena such as resonance, shocks, Mach stems, and vortexsheets, and can be used to predict and explain the results of physical ornumerical experiments. An important task for pure mathematicians is torigorously justify such expansions, that is, to show that they are closeto genuine exact solutions. Rigorous analysis also leads to the discoveryof unexpected phenomena, such as new blow-up mechanisms, for example.