Harmonic Analysis and Hyperbolic Partial Differential Equations

DMS-9970407摘要研究人员继续研究波在波速为“粗”的非均匀介质中的传播行为。目标是在系数的最小可微性假设下获得变系数波动方程的Strichartz解和零型解的估计。在当前的NSF资助下，研究人员已经证明，如果波速度规的曲率张量是有界和可测的，则Strichartz估计成立。所提出的研究包括推广这些结果以获得解的双线性零型的估计，以及有界曲率空间上拉普拉斯算子的特征函数的估计。该研究人员还参与了联合研究凸障碍物反射波的Strichartz估计和双线性零形估计的工作。以前的联合研究已经在时空的有界区域获得了这样的估计。建议的研究包括将这些局部方法与已知的能量衰减估计相结合，以获得在空间和时间上全局有效的估计。这些结果被应用于建立凸障碍物外区域上某些非线性波动方程的长期存在性。发展具有低正则系数的偏微分方程解的方法具有重要的理论和实际意义。理论上的兴趣来自对数学和物理中的非线性方程的研究，如爱因斯坦的引力场方程，或流体动力学方程。这类方程的一个中心问题是解的“长时间”存在；也就是说，表明解不会在未来某个时候爆炸。这样做的主要技术是证明人们可以根据已知保持受控的量来控制方程的非线性部分的大小，例如解的能量。对于爱因斯坦方程，这需要理解具有粗糙波速的波动方程。这类研究的实际意义在于，从理论上理解波在粗糙介质中的传播应该导致更有效的算法来数值计算解。我们的工作包括将解决方案分成几个基本部分，每个部分都在一个特别简单的管理器中运行。这类似于将函数分解成小波，其中许多现代理论都是为了理解稳态现象的非线性方程而发展起来的。小波集反过来又导致了求解此类方程的简单而稳定的算法，以及图像和信号处理领域的强大算法。

DMS-9970407AbstractThe investigator is continuing his research into the behavior ofwave propagation in nonhomogeneous media with "rough" wave speeds.The goal is to obtain estimates of Strichartz and nullform typeon solutions to variable coefficient wave equations underminimal differentiability assumptions on the coefficients.Under current NSF funding, the investigator has shown thatthe Strichartz estimates hold if the curvature tensor of the wavespeed metric is bounded and measurable. The proposed researchincludes extending these results to obtain estimates for bilinearnull-forms of solutions, as well as estimates for eigenfunctionsof the Laplacian on spaces with bounded curvature. The investigatoris also involved in joint work studying the Strichartz and bilinearnullform estimates for waves which reflect from convex obstacles.Previous joint research has obtained such estimates in bounded regions of space-time. The proposed research involves combining these local methods with known energy decay estimates to obtain suchestimates valid globally in space and time. These results are thenbeing applied to establish long time existence for certain nonlinearwave equations on the region exterior to a convex obstacle.The development of techniques to handle partial differential equationswith low regularity coefficients is of both theoretical and practicalinterest. The theoretical interest comes from the study ofnonlinear equations in math and physics, such as Einstein's equationsfor the gravitational field, or the equations of fluid dynamics. A central question for such equations is the "long time" existenceof solutions; that is, showing that solutions do not blow upat some future time. The main technique for doing this is to showthat one can control the size of the nonlinear part of the equationin terms of quantities that are known to remain controlled, suchas the energy of the solution. For Einstein's equations, thisrequires understanding the wave equation with rough wave speeds.The practical importance of such studies is that a theoreticalunderstanding of the propagation of waves in rough media shouldlead to more efficient algorithms for the numerical computationof solutions. Our work involves splitting the solution intoelementary pieces, each of which behaves in a particularly simplemanner. This is analogous to the decomposition of functions intowavelets, much of the modern theory of which was developed tounderstood nonlinear equations of stable-state phenomena. Waveletshave in turn led to simple and stable algorithms for solving such equations, as well as to powerful algorithms in the fields of imageand signal processing.