Auxiliary Statements

Abstract

In this chapter we consider the universal toolbar required for the investigation of evolutional Earth Data Processes which contains in particular partial differential equations with discontinuous or multivalued relationship between determinative parameters of the problem. In Sect. 1.1 we introduce classes of phase spaces and extended phase spaces of the generalized solutions for differential-operator inclusions and evolutional multivariational inequalities. We study the structured properties of such spaces, give the theorems of embedding for non-reflexive classes of spaces of distributions with integrable derivatives. We consider also the basis theory for such spaces. In the second section we study classes of energetic extensions and the Nemytskii operators for differential operators from evolutional mathematical models for geophysical processes and fields. In particular, we consider a class of multivalued variational calculus operators. We prove the completeness with respect to the sum for quasibounded ${w}_{{\lambda }_{0}}$-pseudomonotone weakly coercive maps. The introduced in this chapter results are used for the qualitative and constructive investigation of differential-operator inclusions and evolutional multi-variational inequalities in the next chapters more than once. Corresponding theorems contains these results. The convergence of the Faedo–Galerkin method for differential-operators inclusions with maps of pseudomonotone type is proved by the help of the obtained properties. The compact embedding theorems in non-reflexive spaces together with the results of Sect. 1.2 allow us to sweep a substantially wider class of evolutional problems and validate the penalty method for evolutional multivariational inequalities with maps of ${w}_{{\lambda }_{0}}$-pseudomonotone type. Since in the majority of cases the introduced properties of classes of infinite-dimensional distribution spaces have not been considered yet, so the exposition of results are made with detail proofs. The obtained in this section results have an independent value.