非线性几何光学理论研究及其应用

In this project, we have established the nonlinear geometric optics for.generalized solutions, and the interaction theory between nonlinear highly oscillatory waves, shock waves and δ-waves. We have also studied the reflection and transmisson of highly oscillatory waves. For hyperbolic-parabolic coupled systems, we introduced the microlocal method for decoupling the hyperbolic and the parabolic operators, and obtained the hyperbolic-parabolic coupled systems have the finite speeds for the propagation of singularities. We have established the theory of propagation of singularities of solutions to thermoelastic systems and viscous fluid equations. Furthermore, we obtained the decay rates of solutions to linear thermoelastic systems with time-dependent coefficients in one space variable. We have obtained the uniform.stability and existence of ultidimensional phase boundaries, and the interaction theory of boundary layers and highly oscillatory waves for linearized applications in image denoising and impainting. These results have enriched and developed some theory and applications of nonlinear partial differential equations. We have given invited talks in international conferences for several times. In 2001, Wang Yaguang obtained the ISAAC (The International Society for Analysis, Applications and Computation) Award for Young Scientists in Berlin, ermany.compressible .Navier-Stokes equations. We obtained the viscosity solutions and Colombeau.generalized solutions to nonlinear degenerate parabolic equations, and the

建立非线性高频振荡波在边界上的反射、折射理论，以及高维激波、中心波等与高频振荡波的干扰理论；用非线性几何光学方法研究高维激波反射问题，揭示由正则反射向Ma反射转换的一些本质。此研究将极大地丰富与发展非线性几何光学理论、拟线性双曲守恒律方程组间断解理论，促进空气动力学、弹性力学的发展，为应用领域提供数学理论与数值模拟。