scholarly journals GEODESIC COMPLETENESS FOR SOBOLEV METRICS ON THE SPACE OF IMMERSED PLANE CURVES

2014 ◽  
Vol 2 ◽  
Author(s):  
MARTINS BRUVERIS ◽  
PETER W. MICHOR ◽  
DAVID MUMFORD
Keyword(s):  
Initial Data ◽  
Geodesic Distance ◽  
Geodesic Equation ◽  
Plane Curves ◽  
Sobolev Type ◽  
Geodesic Completeness ◽  
Constant Coefficients ◽  
Geodesically Complete ◽  
Well Posed ◽  
Closed Plane

AbstractWe study properties of Sobolev-type metrics on the space of immersed plane curves. We show that the geodesic equation for Sobolev-type metrics with constant coefficients of order 2 and higher is globally well-posed for smooth initial data as well as for initial data in certain Sobolev spaces. Thus the space of closed plane curves equipped with such a metric is geodesically complete. We find lower bounds for the geodesic distance in terms of curvature and its derivatives.

On an area-preserving inverse curvature flow of convex closed plane curves

2021 ◽  
Vol 280 (8) ◽  
pp. 108931
Author(s):  
Laiyuan Gao ◽  
Shengliang Pan ◽  
Dong-Ho Tsai
Keyword(s):  
Curvature Flow ◽  
Plane Curves ◽  
Area Preserving ◽  
Closed Plane

scholarly journals Global theorems for closed plane curves

1970 ◽  
Vol 76 (1) ◽  
pp. 96-101 ◽  
Author(s):  
Benjamin Halpern
Keyword(s):  
Plane Curves ◽  
Closed Plane

scholarly journals Almost Sure Global Well-posedness for the Fractional Cubic Schrödinger Equation on the Torus

2015 ◽  
Vol 58 (3) ◽  
pp. 471-485 ◽  
Author(s):  
Seckin Demirbas
Keyword(s):  
Probability Measure ◽  
Initial Data ◽  
Schrödinger Equation ◽  
Schrodinger Equation ◽  
Invariant Probability Measure ◽  
Well Posedness ◽  
Cubic Nonlinearity ◽  
Cubic Schrödinger Equation ◽  
Well Posed ◽  
Fractional Schrodinger Equation

AbstractIn a previous paper, we proved that the 1-d periodic fractional Schrödinger equation with cubic nonlinearity is locally well-posed inHsfors> 1 −α/2 and globally well-posed fors> 10α− 1/12. In this paper we define an invariant probability measureμonHsfors<α− 1/2, so that for any ∊ > 0 there is a set Ω ⊂Hssuch thatμ(Ωc) <∊and the equation is globally well-posed for initial data in Ω. We see that this fills the gap between the local well-posedness and the global well-posedness range in an almost sure sense forin an almost sure sense.

Global and exponential attractor of the repulsive Keller–Segel model with logarithmic sensitivity

2020 ◽  
pp. 1-19
Author(s):  
LIN CHEN ◽  
FANZE KONG ◽  
QI WANG
Keyword(s):  
Initial Data ◽  
Steady States ◽  
Sensitivity Function ◽  
Exponential Attractor ◽  
Physical Stimulus ◽  
Chemotaxis Model ◽  
Constant Solution ◽  
Cellular Aggregation ◽  
Logarithmic Sensitivity ◽  
Well Posed

We consider a Keller–Segel model that describes the cellular chemotactic movement away from repulsive chemical subject to logarithmic sensitivity function over a confined region in ${{\mathbb{R}}^n},\,n \le 2$ . This sensitivity function describes the empirically tested Weber–Fecher’s law of living organism’s perception of a physical stimulus. We prove that, regardless of chemotaxis strength and initial data, this repulsive system is globally well-posed and the constant solution is the global and exponential in time attractor. Our results confirm the ‘folklore’ that chemorepulsion inhibits the formation of non-trivial steady states within the logarithmic chemotaxis model, hence preventing cellular aggregation therein.

2. Applications of the Theorem that Two Closed Plane Curves intersect an even number of times

1878 ◽  
Vol 9 ◽  
pp. 237-246 ◽  
Author(s):  
Tait
Keyword(s):  
Special Class ◽  
Double Point ◽  
Closed Curve ◽  
British Association ◽  
Plane Curves ◽  
Double Points ◽  
The Wire ◽  
Closed Plane

The theorem itself may be considered obvious, and is easily applied, as I showed at the late meeting of the British Association, to prove that in passing from any one double point of a plane closed curve continuously along the curve to the same point again, an even number of intersections must be passed through. Hence, if we suppose the curve to be constructed of cord or wire, and restrict the crossings to double points, we may arrange them throughout so that, in following the wire continuously, it goes alternately over and under each branch it meets. When this is done it is obviously as completely knotted as its scheme (defined below) will admit of, and except in a special class of cases cannot have the number of crossings reduced by any possible deformation.

scholarly journals Vanishing geodesic distance for the Riemannian metric with geodesic equation the KdV-equation

2011 ◽  
Vol 41 (4) ◽  
pp. 461-472 ◽  
Author(s):  
Martin Bauer ◽  
Martins Bruveris ◽  
Philipp Harms ◽  
Peter W. Michor
Keyword(s):  
Kdv Equation ◽  
Geodesic Distance ◽  
Riemannian Metric ◽  
Geodesic Equation ◽  
The Kdv Equation ◽  
Vanishing Geodesic Distance

Parity conditions for realizability of Gauss diagrams

2019 ◽  
Vol 28 (01) ◽  
pp. 1950015
Author(s):  
Oleg N. Biryukov
Keyword(s):  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Plane Curves ◽  
Double Points ◽  
Gauss Diagram ◽  
Necessary And Sufficient ◽  
Closed Plane

We consider a problem of realizability of Gauss diagrams by closed plane curves where the plane curves have only double points of transversal self-intersection. We formulate the necessary and sufficient conditions for realizability. These conditions are based only on the parity of double and triple intersections of the chords in the Gauss diagram.

scholarly journals Contracting convex immersed closed plane curves with slow speed of curvature

2012 ◽  
Vol 364 (11) ◽  
pp. 5735-5763 ◽  
Author(s):  
Yu-Chu Lin ◽  
Chi-Cheung Poon ◽  
Dong-Ho Tsai
Keyword(s):  
Plane Curves ◽  
Slow Speed ◽  
Closed Plane

scholarly journals A novel method of solution for the fluid-loaded plate

2009 ◽  
Vol 465 (2112) ◽  
pp. 3667-3685 ◽  
Author(s):  
A. C. L. Ashton ◽  
A. S. Fokas
Keyword(s):  
Initial Data ◽  
Fourier Transforms ◽  
A Priori ◽  
Technical Difficulty ◽  
Half Line ◽  
Loaded Plate ◽  
Method Of Solution ◽  
The Laplace Transform ◽  
Solution Representation ◽  
Well Posed

We study the equations governing a fluid-loaded plate. We first reformulate these equations as a system of two equations, one of which is an explicit non-local equation for the wave height and the velocity potential on the free surface. We then concentrate on the linearized equations and show that the problems formulated either on the full or the half-line can be solved by employing the unified approach to boundary value problems introduced by one of the authors in the late 1990s. The problem on the full line was analysed by Crighton & Oswell (Crighton & Oswell 1991 Phil. Trans. R. Soc. Lond. A 335 , 557–592 (doi:10.1098/rsta.1991.0060)) using a combination of Laplace and Fourier transforms. The new approach avoids the technical difficulty of the a priori assumption that the amplitude of the plate is in L d t 1 ( R + ) and furthermore yields a simpler solution representation that immediately implies that the problem is well-posed. For the problem on the half-line, a similar analysis yields a solution representation, which, however, involves two unknown functions. The main difficulty with the half-line problem is the characterization of these two functions. By employing the so-called global relation, we show that the two functions can be obtained via the solution of a complex-valued integral equation of the convolution type. This equation can be solved in a closed form using the Laplace transform. By prescribing the initial data η 0 to be in H 〈5〉 5 ( R + ), or equivalently four times continuously differentiable with sufficient decay at infinity, we show that the solution depends continuously on the initial data, and, hence, the problem is well-posed.

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