  Let $u(t,x)$ be the solution to the Cauchy problem of a scalar conservation
law in one space dimension. It is well known that even for smooth initial data
the solution can become discontinuous in finite time and global entropy weak
solution can best lie in the space of bounded total variations. It is
impossible that the solutions belong to ,for example ,$H^1$ because by Sobolev
embedding theorem $H^1$ functions are H$\mathrm{\ddot{o}}$lder continuous.
However, we note that from any point $(t,x)$ we can draw a generalized
characteristic downward which meets the initial axis at $y=\alpha (t,x)$. if we
regard $u$ as a function of $(t,y)$, it indeed belongs to $H^1$ as a function
of $y$ if the initial data belongs to $H^1$. We may call this generalized
persistence (of high regularity) of the entropy weak solutions. The main
purpose of this paper is to prove some kinds of generalized persistence (of
high regularity) for the scalar and $2\times 2$ Temple system of hyperbolic
conservation laws in one space dimension .
