We obtain the C α regularity for weak solutions of a class of nonhomogeneous ultraparabolic equation, with measurable coefficients. The result generalizes our recent C α regularity results of homogeneous ultraparabolic equations.

Consider a class of non-homogenous ultraparabolic differential equations with drift terms or lower order terms arising from some physical models, and we prove that weak solutions are Hölder continuous, which also generalizes the classic results of parabolic equations of second order. The main ingredients are a type of weak Poincaré inequality satisfied by non-negative weak sub-solutions and Moser iteration.

We establish some interior regularity criterions of suitable weak solutions for the 3-D Navier-Stokes equations, which allow the vertical part of the velocity to be large under the local scaling invariant norm. As an application, we improve Ladyzhenskaya-Prodi-Serrin's criterion and Escauriza-Seregin-Šverák's criterion. We also show that if weak solution u satisfiesfor some 3 < p < ∞, then the number of singular points is finite.

Microorganisms provide both beneficial and harmful effects to human beings. Beneficial effects come from the symbiotic relationship that exists between humans and microbiota, but then several human illnesses have turned some friendly microbes into opportunistic pathogens, causing several microbial-related diseases. Various efforts have been made to create and utilize antimicrobial agents in the treatment and prevention of these infections, but such efforts have been hampered by the emergence of antimicrobial resistance. Despite extensive studies on drug discovery to alleviate this problem, issues with the toxicity and tolerance of certain compounds and continuous microbial evolution have forced researchers to focus on screening various phytochemical dietary compounds for antimicrobial activity. Linolenic acid and its derivatives (eicosapentaenoic acid and docosahexaenoic acid) are omega-3 fatty acids that have been studied due to their role in human health, being important for the brain, the eye, the cardiovascular system, and general human growth. However, their utilization as antimicrobial agents has not been widely appreciated, perhaps due to a lack of understanding of antimicrobial mechanisms, toxicity, and route of administration. Therefore, this review focuses on the efficacy, mechanism, and toxicity of omega-3 fatty acids as alternative therapeutic agents for treating and preventing diseases associated with pathogenic microorganisms.