Variational statements and generalized solutions of transport problems

Abstract

In this chapter we establish the existence of solutions of some typical transport problems, and for the reader’s convenience we rewrite them again at the beginning of the chapter. The basis for investigations consists of three variational formulations of the original problems. In the first and the second formulations the function giving the minimum of a corresponding functional is a generalized (weak) solution of the original problem. In the first problem we minimize the functional of the residual, and in the second problem - a quadratic functional. In the third variational problem the symmetrized (with the aid of the adjoint problem) equation is used and an existence theorem for that equation is proved. After that we construct generalized solutions of the original and adjoint problems from that solution. If generalized solutions of transport equation belong to the class H ¹ _p (Ω х D), then they satisfy the equation a.e. in Ω х D and the boundary condition a.e. in Ω х ∂D- (or, for periodic problems, they satisfy periodic conditions a.e.). As we shall see later, generalized solutions can sometimes belong only to the space L²(Ω х D), and weakened restrictions are imposed on the original data. The solution satisfies only some integral correlations, and we do not imply that the solution should satisfy a.e. the equation and boundary conditions. Therefore we use the term “generalized (or weak) solutions” and do not consider the solutions in the classical sense.