When an airplane's flight speed is close to the speed of sound (the so-called transonic regime), the airflow around the plane has very unusual properties: in some areas the flow is supersonic, and in other areas the flow is subsonic; these two are separated by a surface where the flow is exactly sonic (i.e., has the speed of sound). The sonic surface is attached to the airfoil, and characteristics of the flow, such as drag and lift, change dramatically along the airfoil when the sonic surface is crossed. This causes a serious strain experienced by the airfoil. Mathematical description of this phenomenon is given by partial differential equations (PDE) of mixed type that are the main subject of this project. Other applications of similar equations are found in fluid and quantum mechanics, general relativity, bio-sciences, and plasma physics. Although these equations are widely used and serve as a foundation of some engineering tasks, for example, of the computer-aided design in aerodynamics, among others, and despite recent extensive studies of mixed type PDE, many fundamental mathematical questions concerning the behavior of their solutions are still unresolved. In this project, the Principal Investigator will study simpler, frequently used systems and classes of initial data that play a paramount role in understanding the wave interaction, asymptotic behavior of solutions and their stability. This project will also serve as a training ground for graduate and undergraduate students who will contribute to this research.
Several model systems of PDE will be studied whose solutions involve the so-called singular shocks, where at least one state variable develops an extreme concentration in the form of a weighted Dirac delta function. These can be used as building blocks for gaining a broader knowledge, insight and perspective on global in time existence of large solutions to systems of conservation laws in one spatial dimension. Differential equations in several spatial dimensions will also be considered. Simpler cases of solutions with spherical symmetry are among the principal topics of this project. Diverse tools from dynamical systems, geometry, harmonic and Fourier analysis will be used in this project. The ultimate goal of this research is to find building blocks that provide information on models of compressible fluid flow and can be used in a well-defined construction scheme to approximate solutions for any initial data.
