|Budget Amount *help
¥2,600,000 (Direct Cost : ¥2,600,000)
Fiscal Year 2000 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1999 : ¥700,000 (Direct Cost : ¥700,000)
Fiscal Year 1998 : ¥1,200,000 (Direct Cost : ¥1,200,000)
Our subject is to study the Cauchy problem for nonlinear hyperbolic equations. We are interested in the global theory for this Cauchy problem. But it is not complete at today's point. One of the reasons is that classical solutions do not exist in the large, that is to say that singularities appear in their solutions. Moreover we see that "singularities" cause many interesting phenomena. To establish the global theory, we must study the following two problems. The first problem is "to describe the domain where classical solutions exist", and the second one is "to extend the solutions beyond the singularities". The problem of "singularities" has been one of fundamental problems of mathematics, especially in "Algebraic geometry". The method used very often in algebraic geometry is the "resolution of singularities", that is to say to lift the surfaces into higher dimensional space so that the singularities would disappear. We will apply this idea to our problems. That is to say, we lift t
he solution surface to the cotangent space so that the singularities would disappear, and construct the global solution there. Next we will project it to the base space and get the solution as a function defined on the base space.
First we have studied single first order nonlinear partial differential equations from the above point of view. Our achievements on this subject are summerized in our monograph published in 1999, USA.See the list of publications in this report. Next we have considered nonlinear second order single hyperbolic equations, and also hyperbolic systems of conservation laws. In this case also, singularities generally appear in finite time. We are interested in the global theory. Therefore our problem is how to extend the solution after the appearance of singularities. For this purpose, we have lifted the solution surface into cotangent space so that the singularities would disappear. Though this has been a difficult problem, we could do so for a certain case. Then, as the lifted solution, called a "geometric solution", have no singularity, we can extend it to the whole cotangent space. Finally we project it to the base space for getting a weak solution. We have written this result in our manuscript which would be published in "Acta Mathematica Vietnamica".
Our problem has been partially solved. What we must do from now is to complete the geometric theory for second order nonlinear hyperbolic equations and to generalize the equations. The equations which we have treated until now are a little too special. Less