Generalized solutions for inextensible string equations

Abstract

Abstract. We study the system of equations of motion for inextensible strings. This system possesses many internal symmetries, and is related to discontinuous systems of conservation laws and the total variation wave equation. We prove existence of generalized Young measure solutions with non-negative tension after transforming the problem into a system of conservation laws and approximating it with a regularized system for which we obtain uniform estimates of the energy and the tension. We also discuss sufficient conditions for non-negativity of the tension for strong solutions.

Uniformly Compressing Mean Curvature Flow

“…Indeed, by (ii) of Proposition 3.1σ ≤´1 0 |ξ| 2 , which in terms ofξ and c readsσ ≤´1 0 |ξ| 2 + c 2 4π 2 . By (53),σ ≤ 1 16π 2´1 0 |∂ sξ | 2 + c 2 4π 2 , which combined with (45) gives (56). With (56) at hand, one can bound ∂ t´1 0 |∂ sξ | 2 in terms of´1 0 |∂ sξ | 2 ds as…”

“…Indeed, the first inequality is immediate since by the definition of the projection ξ 0 L 2 (S 1 ) ≤ ξ − w 0 L 2 (S 1 ) . The second inequality follows from (45) having observed that…”

“…Substituting c 2 by 1 −´1 0 |∂ sξ | 2 ds (cf. (45)) and after some algebraic manipulations we arrive at (57).…”

“…The exponential decay of t → ∂ sξ (t, ·) L 2 immediately yields the convergence of the multiplicative factor c(t) due to (45). In order to show the global existence we need to control the norm ξ(t, s) 2+α,0 along the flow.…”

Propagating stress-pulses and wiggling transition revealed in string dynamics

“…Understanding the intricate dynamics of the filamentary string structure is important as they are ubiquitous in nature and industry covering length scales of several orders of magnitude [3,[16][17][18]. Much has been learnt about the string dynamics by analyzing the equations of motion of the string [4,15,[19][20][21][22]. However, the analytical solution to the coupled, nonlinear string equations is only limited to some special cases [13,23,24].…”