Sharp Strichartz estimates for water waves systems

Water waves are well-known to be dispersive at the linearization level. Considering the fully nonlinear systems, we prove for reasonably smooth solutions the optimal Strichartz estimates for pure gravity waves and the semi-classical Strichartz estimates for gravity-capillary waves; for both 2D and 3D waves. Here, by optimal we mean the gains of regularity (over the Sobolev embedding from Sobolev spaces to H\"older spaces) obtained for the linearized systems. Our proofs combine the paradifferential reductions of Alazard-Burq-Zuily with a dispersive estimate using a localized wave package type parametrix of Koch-Tataru.Comment: Some typos fixe

Electron microscopy analysis of alpha-synuclein and LRRK2 transgenic C. elegans

Thesis (M.A.)--Boston UniversityMutations in alpha-synuclein and leucine-rich repeat kinase 2 (LRRK2) have been implicated in the cause of Parkinson’s disease (PD). These two proteins have been the targets of a great deal of recent research that has transformed our understanding of this disorder. Recent research using C. elegans as a model species has shown that alpha- synuclein expression and the LRRK2-G2019S mutation potentiate neurodegeneration similar to that seen in cases PD. Further exploration revealed that defects in autophagy of dopaminergic neurons may be the cause for the observed pathology. In the current study, the confirmation of autophagy as a possible cause of pathology due to the expression of alpha-synuclein and the LRRK2-G2019S mutation is completed through the use of electron microscopy. We observed that large vacuoles had formed in the cephalic dopaminergic neurons of alpha-synuclein + LRRK2 transgenic samples not seen in wild-type samples. Further, large morphological changes in the nerve ring area of the transgenic nematodes were also observed that may implicate that alpha- synuclein expression in conjunction with the LRRK2-G2019S mutation may have a widespread effect on many neurons that was not previously expected