scholarly journals Uniqueness of weak solutions to a prion equation with polymer joining

Analysis ◽  
2017 ◽  
Vol 37 (2) ◽  
Author(s):  
Elena Leis ◽  
Christoph Walker
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Differential Equation ◽  
Weak Solutions ◽  
Reaction Rates ◽  
Global Weak Solutions ◽  
Polymer Joining

AbstractWe consider a model for prion proliferation that includes prion polymerization, polymer splitting, and polymer joining. The model consists of an ordinary differential equation for the prion monomers and a hyperbolic nonlinear differential equation with integral terms for the prion polymers and was shown to possess global weak solutions for unbounded reaction rates [

scholarly journals Global classical and weak solutions of the prion proliferation model in the presence of chaperone in a banach space

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Kapil Kumar Choudhary ◽  
Rajiv Kumar ◽  
Rajesh Kumar
Keyword(s):  
Differential Equation ◽  
System Theory ◽  
Banach Space ◽  
Weak Solutions ◽  
Existence And Uniqueness ◽  
Weak Compactness ◽  
Evolution System ◽  
Global Classical Solution ◽  
Global Weak Solutions ◽  
Degradation Rates

<p style='text-indent:20px;'>The present work is based on the coupling of prion proliferation system together with chaperone which consists of two ODEs and a partial integro-differential equation. The existence and uniqueness of a positive global classical solution of the system is proved for the bounded degradation rates by the idea of evolution system theory in the state space <inline-formula><tex-math id="M1">\begin{document}$ \mathbb{R} \times \mathbb{R} \times L_{1}(Z,zdz). $\end{document}</tex-math></inline-formula> Moreover, the global weak solutions for unbounded degradation rates are discussed by weak compactness technique.</p>

scholarly journals Semi-discrete optimal transport methods for the semi-geostrophic equations

2022 ◽  
Vol 61 (1) ◽  
Author(s):  
David P. Bourne ◽  
Charlie P. Egan ◽  
Beatrice Pelloni ◽  
Mark Wilkinson
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Weak Solutions ◽  
Compact Support ◽  
Optimal Transport ◽  
Constructive Proof ◽  
3 Dimensional ◽  
Eulerian Coordinates ◽  
Numerical Solver ◽  
Explicit Relation

AbstractWe give a new and constructive proof of the existence of global-in-time weak solutions of the 3-dimensional incompressible semi-geostrophic equations (SG) in geostrophic coordinates, for arbitrary initial measures with compact support. This new proof, based on semi-discrete optimal transport techniques, works by characterising discrete solutions of SG in geostrophic coordinates in terms of trajectories satisfying an ordinary differential equation. It is advantageous in its simplicity and its explicit relation to Eulerian coordinates through the use of Laguerre tessellations. Using our method, we obtain improved time-regularity for a large class of discrete initial measures, and we compute explicitly two discrete solutions. The method naturally gives rise to an efficient numerical method, which we illustrate by presenting simulations of a 2-dimensional semi-geostrophic flow in geostrophic coordinates generated using a numerical solver for the semi-discrete optimal transport problem coupled with an ordinary differential equation solver.

Axisymmetric gravitational fields: a nonlinear differential equation that admits a series of exact eigenfunction solutions

1978 ◽  
Vol 362 (1711) ◽  
pp. 463-468 ◽  
Keyword(s):  
Differential Equation ◽  
Ordinary Differential Equation ◽  
Nonlinear Differential Equation ◽  
Nonlinear Ordinary Differential Equation ◽  
Einstein’S Equations ◽  
Einstein's Equations ◽  
Gravitational Fields

A nonlinear ordinary differential equation is presented. This equation is shown to occur as a particular integral of Einstein’s equations of gravitation, and may be used to generate a class of solutions which contains the Kerr, and the Tomimatsu–Sato metrics as particular cases.

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