On the Hamilton-Jacobi equation in the framework of generalized functions
The aim of this work is to obtain, under certain conditions, theorems on existence of solutions and in some cases uniqueness to the Hamilton-Jacobi equation \[ \frac {\partial u}{\partial t}+H\biggl(t,x_1,\dots,x_n,u,\frac {\partial u}{\partial x_1},\dots,\frac {\partial u}{\partial x_n}\biggr)=0, \] with the initial condition \[ u|_{\{0\}\times V}=f|_V, \] in the framework of Colombeau's generalized functions. Among the existing classical methods, the author searches to adapt the method of characteristics. This method consists in transforming a partial differential equation with a given initial condition in an ordinary differential equation problem with a certain given initial condition. To adapt the classical method of characteristics, the author defines the set \(S(I,\Omega,I',\Omega',H,f,J,W)\), in which the elements \((X,U,P)\) are so that a certain generalized mapping, defined from \(X\), is invertible and the derivatives in relation to the first variable of the generalized \(X, U\) and \(P\) satisfy a certain system of ODE evolving generalized functions. This leads him to obtain some results on invertible generalized functions (see Section 2) and on local solutions to ODE in the framework of generalized functions (see Section 4). In Section 3 the author proves that, under certain conditions, if \(S(I,\Omega, I',\Omega',H,f,J,W)\) is nonempty, then the HJ-problem has a generalized solution. On uniqueness of solutions to the HJ-problem, he has obtained partial answers to this question (see Section 5). The author finishes by presenting, under certain conditions, a theorem of existence and uniqueness of solution to the HJ-problem.
scientific article; zbMATH DE number 5925571