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Global solvability for differential complexes and converse to the theorem of the existence of Lyapunov function to gradient-like evolution process

Abstract

This project has, basically, three goals. One of them is concerned to the theory of the linear PDE´s and the other one is related to the study of the dynamics properties of a evolution process which allow us to define a Lyapunov function, non autonomous for it. Besides it, we want to look for connections which might have between some features of these two theories, such as, to use techniques from the evolution equations in the study of a class of differential complexes.In the first case, one of the goals is to continue the study of the theory concerned to the problem of the global solvability, in top degree, and the global hypoellipticity, in the first degree, to the differential complex difined by the operator L:=dt - w(t,A)A,where A : D(A) ‚c H -- ’ H is a self-adjoint linear operator positive with bounded inverse in a Hillbert space H, w a power serie of the negative powers of A with coefficients in the space of the smooth 1-formas closeds over a compact manifold M of dimension m and dt stands for the exterior derivative in M. Also, we want to consider, perhaps, scales of fractional power spaces more general than one which came from the Hilbert spaces framework. In the second case, we want to study the possibility for a converse of a result that we have got in my PHD teses, which says that a gradient-like evolution process has a Lyapunov function, in non autonomous sense. Besides that, we want to see what happen if we don not assume that the set of the isolated invariants is not finite, when we try to build a Lyapunov function for the semigroup (or process) gradient-like. (AU)

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Scientific publications (4)
(References retrieved automatically from Web of Science and SciELO through information on FAPESP grants and their corresponding numbers as mentioned in the publications by the authors)
ARAGAO-COSTA, E. R.; DA SILVA, A. P. Strongly compatible generators of groups on Frechet spaces. Journal of Mathematical Analysis and Applications, v. 484, n. 2 APR 15 2020. Web of Science Citations: 0.
ARAGAO-COSTA, E. R. PARTIAL HYPOELLIPTICITY FOR A CLASS OF ABSTRACT DIFFERENTIAL COMPLEXES ON BANACH SPACE SCALES. ANNALS OF FUNCTIONAL ANALYSIS, v. 10, n. 2, p. 262-276, MAY 2019. Web of Science Citations: 0.
ARAGAO COSTA, EDER RITAS. AN EXTENSION OF THE CONCEPT OF EXPONENTIAL DICHOTOMY IN FRECHET SPACES WHICH IS STABLE UNDER PERTURBATION. COMMUNICATIONS ON PURE AND APPLIED ANALYSIS, v. 18, n. 2, p. 845-868, MAR 2019. Web of Science Citations: 0.
ARAGAO-COSTA, E. R.; FIGUEROA-LOPEZ, R. N.; LANGA, J. A.; LOZADA-CRUZ, G. Topological Structural Stability of Partial Differential Equations on Projected Spaces. Journal of Dynamics and Differential Equations, v. 30, n. 2, p. 687-718, JUN 2018. Web of Science Citations: 0.

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