Compactness Framework of Lp Approximate Solutions for Scalar Conservation Laws

In this paper, we study the strong convergence of a sequence of uniform Llpoc R=Rq.bounded approximate solutions ue x,t.4 to the following scalar conservation laws utqf x,t,u.xqg x,t,u.s0, xgR, t)0, with initial data u x, 0.su0 x.gLp R.lL2 R., xgR, 1-pFq`. Without the convexity assumption and growth condition at infinity for f x, t,u., we prove strong convergence of a subsequence of u« x,t.4. Under a more general growth condition than those in the previous work, we prove the existence of weak solution for the equation. The result obtained here generalizes those in earlier work. Some applications of the results are also given at the end of this paper