Optimal Time Decay of the Vlasov-Poisson-Boltzmann System in R3{\mathbb{R}}^3

The Vlasov-Poisson-Boltzmann System governs the time evolution of the distribution function for the dilute charged particles in the presence of a self-consistent electric potential force through the Poisson equation. In this paper, we are concerned with the rate of convergence of solutions to equilibrium for this system over ${\mathbb{R}}^3$. It is shown that the electric field which is indeed responsible for the lowest-order part in the energy space reduces the speed of convergence and hence the dispersion of this system over the full space is slower than that of the Boltzmann equation without forces, where the exact difference between both power indices in the algebraic rates of convergence is 1/4. For the proof, in the linearized case with a given non-homogeneous source, Fourier analysis is employed to obtain time-decay properties of the solution operator. In the nonlinear case, the combination of the linearized results and the nonlinear energy estimates with the help of the proper Lyapunov-type inequalities leads to the optimal time-decay rate of perturbed solutions under some conditions on initial data.Comment: 37 page

Allelopathic effects of Ulva pertusa, Corallina pilulifera and Sargassum thunbergii on the growth of the dinoflagellates Heterosigma akashiwo and Alexandrium tamarense

The allelopathic effects of fresh tissue, dry powder and aqueous extracts of three macroalgae, Ulva pertusa, Corallina pilulifera and Sargassum thunbergii, on the growth of the dinoflagellates Heterosigma akashiwo and Alexandrium tamarense were evaluated using coexistence culture systems in which concentrations of the three macroalga were varied. The results of the coexistence assay showed that the growth of the two microalgae was strongly inhibited by using fresh tissue, dry powder and aqueous extracts of the three macroalga; the allelochemicals were lethal to H. akashiwo at relatively higher concentrations of the three macroalga. The macroalgae showing the most allelopathic effect on H. akashiwo and A. tamarense using fresh tissue were U. pertusa and S. thunbergii, using dry powder were S. thunbergii and U. pertusa, and using aqueous extracts were U. pertusa and C. pilulifera. We also examined the potential allelopathic effect on the two microalgae of culture filtrate of the three macroalga; culture medium filtrate initially exhibited no inhibitory effects when first added but inhibitory effects became apparent under semi-continuous addition, which suggested that continuous release of small quantities of rapidly degradable allelochemicals from the fresh macroalgal tissue were essential to effectively inhibit the growth of the two microalgae