Blowup Criterion for the Compressible Flows with Vacuum States

We prove that the maximum norm of the deformation tensor of velocity gradients controls the possible breakdown of smooth(strong) solutions for the 3-dimensional compressible Navier-Stokes equations, which will happen, for example, if the initial density is compactly supported \cite{X1}. More precisely, if a solution of the compressible Navier-Stokes equations is initially regular and loses its regularity at some later time, then the loss of regularity implies the growth without bound of the deformation tensor as the critical time approaches. Our result is the same as Ponce's criterion for 3-dimensional incompressible Euler equations (\cite{po}). Moreover, our method can be generalized to the full Compressible Navier-Stokes system which improve the previous results. In addition, initial vacuum states are allowed in our cases.Comment: 17 page

Modulating mitophagy in mitochondrial disease

Mitochondrial diseases may result from mutations in the maternally-inherited mitochondrial DNA (mtDNA) or from mutations in nuclear genes encoding mitochondrial proteins. Their bi-genomic nature makes mitochondrial diseases a very heterogeneous group of disorders that can present at any age and can affect any type of tissue. The autophagic-lysosomal degradation pathway plays an important role in clearing dysfunctional and redundant mitochondria through a specific quality control mechanism termed mitophagy. Mitochondria could be targeted for autophagic degradation for a variety of reasons including basal turnover for recycling, starvation induced degradation, and degradation due to damage. While the core autophagic machinery is highly conserved and common to most pathways, the signaling pathways leading to the selective degradation of damaged mitochondria are still not completely understood. Type 1 mitophagy due to nutrient starvation is dependent on PI3K (phosphoinositide 3-kinase) for autophagosome formation but independent of mitophagy proteins, PINK1 (PTEN-induced putative kinase 1) and Parkin. Whereas type 2 mitophagy that occurs due to damage is dependent on PINK1 and Parkin but does not require PI3K. Autophagy and mitophagy play an important role in human disease and hence could serve as therapeutic targets for the treatment of mitochondrial as well as neurodegenerative disorders. Therefore, we reviewed drugs that are known modulators of autophagy (AICAR and metformin) and may effect this by activating the AMP-activated protein kinase signaling pathways. Furthermore, we reviewed data available on supplements, such as Coenzyme Q and the quinone idebenone, that we assert rescue increased mitophagy in mitochondrial disease by benefiting mitochondrial function