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[**Dois Problemas em Equações Diferenciais Parciais**]{}\
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[Paulo R. Zingano]{}\
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[Departamento de Matemática Pura e Aplicada]{}\
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[Universidade Federal do Rio Grande do Sul]{}\
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[Porto Alegre, RS 91509-900, Brazil]{}\
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*Trabalho inédito submetido para avaliação*\
*para fins de progressão funcional à Classe E*\
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[**Abstract**]{}\
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[ In this work, we examine two important problems in the theory of nonlinear partial differential equations. In [Part I]{}, we propose and solve a more general and complete version of the celebrated Leray’s problem for the incompressible Navier-Stokes equations in $ \mathbb{R}^{3} \!$, which in its simplest form was suggested by J.$\;$Leray in 1934 (and solved only in the 1980s by T.$\;$ Kato, K.$\;$Masuda and other authors). A number of related new results of clear interest to the theory of Leray’s solutions are also given here.\
In [Part II]{}, which is independent of [Part I]{} and can be read separately, we introduce an important new collection of problems concerning global existence results and blow-up phenomena for solutions of conservative advection-diffusion equations in $ \mathbb{R}^{n} $ where heterogeneities in the lower order terms tend to destabilize the solution (everywhere or in certain regions), strongly competing with the viscous 