The Degasperis-Procesi equation as a non-metric Euler equation

In this paper we present a geometric interpretation of the periodic Degasperis-Procesi equation as the geodesic flow of a right invariant symmetric linear connection on the diffeomorphism group of the circle. We also show that for any evolution in the family of $b$-equations there is neither gain nor loss of the spatial regularity of solutions. This in turn allows us to view the Degasperis-Procesi and the Camassa-Holm equation as an ODE on the Fr\'echet space of all smooth functions on the circle.Comment: 17 page

Global generalized solutions for Maxwell-alpha and Euler-alpha equations

We study initial-boundary value problems for the Lagrangian averaged alpha models for the equations of motion for the corotational Maxwell and inviscid fluids in 2D and 3D. We show existence of (global in time) dissipative solutions to these problems. We also discuss the idea of dissipative solution in an abstract Hilbert space framework.Comment: 27 pages, to appear in Nonlinearit

Self-Consistent Multiscale Theory of Internal Wave, Mean-Flow Interactions

This is the final report of a three-year, Laboratory Directed Research and Development (LDRD) project at Los Alamos National Laboratory (LANL). The research reported here produced new effective ways to solve multiscale problems in nonlinear fluid dynamics, such as turbulent flow and global ocean circulation. This was accomplished by first developing new methods for averaging over random or rapidly varying phases in nonlinear systems at multiple scales. We then used these methods to derive new equations for analyzing the mean behavior of fluctuation processes coupled self consistently to nonlinear fluid dynamics. This project extends a technology base relevant to a variety of multiscale problems in fluid dynamics of interest to the Laboratory and applies this technology to those problems. The project's theoretical and mathematical developments also help advance our understanding of the scientific principles underlying the control of complex behavior in fluid dynamical systems with strong spatial and temporal internal variability

High-precision molecular dynamics simulation of UO2-PuO2: superionic transition in uranium dioxide

Our series of articles is devoted to high-precision molecular dynamics simulation of mixed actinide-oxide (MOX) fuel in the rigid ions approximation using high-performance graphics processors (GPU). In this article we assess the 10 most relevant interatomic sets of pair potential (SPP) by reproduction of the Bredig superionic phase transition (anion sublattice premelting) in uranium dioxide. The measurements carried out in a wide temperature range from 300K up to melting point with 1K accuracy allowed reliable detection of this phase transition with each SPP. The {\lambda}-peaks obtained are smoother and wider than it was assumed previously. In addition, for the first time a pressure dependence of the {\lambda}-peak characteristics was measured, in a range from -5 GPa to 5 GPa its amplitudes had parabolic plot and temperatures had linear (that is similar to the Clausius-Clapeyron equation for melting temperature).Comment: 7 pages, 6 figures, 1 tabl

Fractional Sobolev Metrics on Spaces of Immersed Curves

Motivated by applications in the field of shape analysis, we study reparametrization invariant, fractional order Sobolev-type metrics on the space of smooth regular curves Imm(S1 , R ) and on its Sobolev completions ℐ (S1 , R ). We prove local well-posedness of the geodesic equations both on the Banach manifold ℐ (S1 , R ) and on the Fr´echetmanifold Imm(S1 , R ) provided the order of the metric is greater or equal to one. In addition we show that the -metric induces a strong Riemannian metric on the Banach manifold ℐ (S1 , R ) of the same order , provided > 3 2 . These investigations can be also interpreted as a generalization of the analysis for right invariant metrics on the diffeomorphism group