Functional-valued solutions to conservation laws and difference schemes

A new class of generalized solutions to system of conservation laws proposed lately – functional-valued and solutions in the mean – is considered. Some existence and convergence theorems are proved. Definition of $\mathscr{A}$-system, whose Lax's difference solutions converge to functional-valued solutions for polydimensional Cauchy problem, is proposed. Convergence of Lax's difference solution for isentropic gas dynamics to a functional-valued one is based. A new family of floating net implicit difference schemes for isentropic gas dynamics is proposed and convergence of its solutions to functional-valued one for original problem is proved.