New minimal bounds for the derivatives of rational Bézier paths and rational rectangular Bézier surfaces

New minimal bounds are derived for the magnitudes of the derivatives of the rational Bézier paths and the rational rectangular Bézier surface patches of arbitrary degree, which improve previous work of this type in many cases. Moreover, our new bounds are explicitly given by simple and closed-form expressions. An important advantage of the closed-form expressions is that they allow us to prove that our bounds are sharp under certain well- defined conditions. Some numerical examples, highlighting the potential of the new bounds in providing improved estimates, are given in an appendix

On sharp bilinear Strichartz estimates of Ozawa-Tsutsumi type

We provide a comprehensive analysis of sharp bilinear estimates of Ozawa-Tsutsumi type for solutions u of the free Schr\"odinger equation, which give sharp control on $|u|^2$ in classical Sobolev spaces. In particular, we provide a generalization of their estimates in such a way that provides a unification with some sharp bilinear estimates proved by Carneiro and Planchon-Vega, via entirely different methods, by seeing them all as special cases of a one parameter family of sharp estimates. We show that the extremal functions are solutions of the Maxwell-Boltzmann functional equation and provide a new proof that this equation admits only Gaussian solutions. We also make a connection to certain sharp estimates on $u^2$ involving certain dispersive Sobolev norms.Comment: 17 pages, references update

Heat-flow monotonicity of Strichartz norms

Most notably we prove that for $d=1,2$ the classical Strichartz norm $$\|e^{i s\Delta}f\|_{L^{2+4/d}_{s,x}(\mathbb{R}\times\mathbb{R}^d)}$$ associated to the free Schr\"{o}dinger equation is nondecreasing as the initial datum $f$ evolves under a certain quadratic heat-flow.Comment: 11 page

On the Strichartz estimates for the kinetic transport equation

We show that the endpoint Strichartz estimate for the kinetic transport equation is false in all dimensions. We also present a new approach to proving the non-endpoint cases using multilinear analysis.Comment: 5 page