Short-Time Existence of the Möbius-Invariant Willmore Flow

2017 ◽  
Vol 28 (2) ◽  
pp. 1151-1181
Author(s):  
Ruben Jakob
Keyword(s):  
Willmore Flow ◽  
Short Time ◽  
Time Existence

scholarly journals Ricci flow on manifolds with boundary with arbitrary initial metric

2021 ◽  
Vol 0 (0) ◽  
Author(s):  
Tsz-Kiu Aaron Chow
Keyword(s):  
Ricci Flow ◽  
Positive Curvature ◽  
Curvature Operator ◽  
Manifolds With Boundary ◽  
Convex Boundary ◽  
Uniqueness Of The Solution ◽  
Positive Time ◽  
Natural Boundary Conditions ◽  
Short Time ◽  
Time Existence

Abstract In this paper, we study the Ricci flow on manifolds with boundary. In the paper, we substantially improve Shen’s result [Y. Shen, On Ricci deformation of a Riemannian metric on manifold with boundary, Pacific J. Math. 173 1996, 1, 203–221] to manifolds with arbitrary initial metric. We prove short-time existence and uniqueness of the solution, in which the boundary becomes instantaneously totally geodesic for positive time. Moreover, we prove that the flow we constructed preserves natural boundary conditions. More specifically, if the initial metric has a convex boundary, then the flow preserves positive curvature operator and the PIC1, PIC2 conditions. Moreover, if the initial metric has a two-convex boundary, then the flow preserves the PIC condition.

scholarly journals Dislocation Dynamics: Short-time Existence and Uniqueness of the Solution

2006 ◽  
Vol 181 (3) ◽  
pp. 449-504 ◽  
Author(s):  
Olivier Alvarez ◽  
Philippe Hoch ◽  
Yann Le Bouar ◽  
Régis Monneau
Keyword(s):  
Existence And Uniqueness ◽  
Dislocation Dynamics ◽  
Uniqueness Of The Solution ◽  
Short Time ◽  
Time Existence

The elastic flow of curves on the sphere

2018 ◽  
Vol 3 (1) ◽  
pp. 1-13 ◽  
Author(s):  
Anna Dall’Acqua ◽  
Tim Laux ◽  
Lin ◽  
Paola Pozzi ◽  
Adrian Spener
Keyword(s):  
Critical Points ◽  
Elastic Energy ◽  
Gradient Flow ◽  
Closed Curves ◽  
Long Time Existence ◽  
Long Time ◽  
Short Time ◽  
Time Existence

Abstract We consider closed curves on the sphere moving by the L2-gradient flow of the elastic energy both with and without penalisation of the length and show short-time and long-time existence of the flow. Moreover, when the length is penalised, we prove sub-convergence to critical points.

scholarly journals SHORT-TIME EXISTENCE FOR SCALE-INVARIANT HAMILTONIAN WAVES

2006 ◽  
Vol 03 (02) ◽  
pp. 247-267 ◽  
Author(s):  
JOHN K. HUNTER
Keyword(s):  
Rayleigh Waves ◽  
Evolution Equations ◽  
Hyperbolic Surface ◽  
Smooth Solutions ◽  
Scale Invariant ◽  
Weakly Nonlinear ◽  
Hamiltonian Evolution ◽  
Short Time ◽  
Nonlinear Scale ◽  
Time Existence

We prove short-time existence of smooth solutions for a class of nonlinear, and in general spatially nonlocal, Hamiltonian evolution equations that describe the self-interaction of weakly nonlinear scale-invariant waves. These equations include ones that describe weakly nonlinear hyperbolic surface waves, such as nonlinear Rayleigh waves in elasticity.

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