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Full-Text Articles in Physical Sciences and Mathematics
An H1 Model For Inextensible Strings, Stephen Preston, Ralph Saxton
An H1 Model For Inextensible Strings, Stephen Preston, Ralph Saxton
Ralph Saxton
We study geodesics of the H1 Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L2 metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H1 geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C∞ in the Banach topology C1 ([0,1], R2), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, we have global-in-time solutions. We conclude with …
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Blow-Up Of Solutions To The Generalized Inviscid Proudman-Johnson Equation, Alejandro Sarria, Ralph Saxton
Ralph Saxton
For arbitrary values of a parameter --- finite-time blowup of solutions to the generalized, inviscid Proudman Johnson equation is studied via a direct approach which involves the derivation of representation formulae for solutions to the problem.
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Influence Of Damping On Hyperbolic Equations With Parabolic Degeneracy, Katarzyna Saxton, Ralph Saxton
Ralph Saxton
This paper examines the effect of damping on a nonstrictly hyperbolic 2 x 2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.
Phase Transitions And Change Of Type In Low-Temperature Heat, Ralph A. Saxton, Katarzyna Saxton
Phase Transitions And Change Of Type In Low-Temperature Heat, Ralph A. Saxton, Katarzyna Saxton
Ralph Saxton
Classical heat pulse experiments have shown heat to propagate in waves through crystalline materials at temperatures close to absolute zero. With increasing temperature, these waves slow down and finally disappear, to be replaced by diffusive heat propagation. Several features surrounding this phenomenon are examined in this work. The model used switches between an internal parameter (or extended thermodynamics) description and a classical (linear or nonlinear) Fourier law setting. This leads to a hyperbolic-parabolic change of type, which allows wavelike features to appear beneath the transition temperature and diffusion above. We examine the region around and immediately below the transition temperature, …
Global Existence Of Some Infinite Energy Solutions For A Perfect Incompressible Fluid, Ralph Saxton, Feride Tiğlay
Global Existence Of Some Infinite Energy Solutions For A Perfect Incompressible Fluid, Ralph Saxton, Feride Tiğlay
Ralph Saxton
This paper provides results on local and global existence for a class of solutions to the Euler equations for an incompressible, inviscid fluid. By considering a class of solutions which exhibits a characteristic growth at infinity we obtain an initial value problem for a nonlocal equation. We establish local well-posedness in all dimensions and persistence in time of these solutions for three and higher dimensions. We also examine a weaker class of global solutions.
On The Influence Of Damping In Hyperbolic Equations With Parabolic Degeneracy, Ralph Saxton, Katarzyna Saxton
On The Influence Of Damping In Hyperbolic Equations With Parabolic Degeneracy, Ralph Saxton, Katarzyna Saxton
Ralph Saxton
This paper examines the effect of damping on a nonstrictly hyperbolic 2x2 system. It is shown that the growth of singularities is not restricted as in the strictly hyperbolic case where dissipation can be strong enough to preserve the smoothness of solutions globally in time. Here, irrespective of the stabilizing properties of damping, solutions are found to break down in finite time on a line where two eigenvalues coincide in state space.