Nonlocal Flow Driven by the Radius of Curvature with Fixed Curvature Integral

Evolving convex curves by a generalized length-preserving flow

“…Tsai in [29] revealed the blow up phenomenon of a convex curve evolving according to (1.1) with F (κ) = −1/κ. The blow up phenomenon also occurs in some other nonlocal models with F < 0 [13,14]. Therefore, this flow may blow up in a finite time without the condition (ii).…”

“…The higher dimensional model for hypersurfaces in Euclidean space with α-homogeneity assumption of F has been treated by Andrews-McCoy [5]. Recently, Gao, Pan and Tsai [13,14] studied 1 κ α -type nonlocal flows for convex curves. A lot of blow-up phenomena of this kind of nonlocal flows have been found and all global flows are proved convergent as time tends to infinity.…”

Lagrangian, game theoretic, and PDE methods for averaging G-equations in turbulent combustion: existence and beyond

“…Let V : R 2 → R 2 be the steady two dimensional cellular flow, e.g., V (x) = A (DH(x)) ⊥ with stream function H(x 1 , x 2 ) = sin(x 1 ) sin(x 2 ) and flow intensity A > 0. Theorem 4.1 is the main theorem proved in [50].…”

“…With the help of the minimum principle and more delicate analysis, similar estimates hold over the cell boundary and in other quarter cells, and so key inequality (4.10) holds. Theorem 4.1 follows with a sub/supersolution argument [50].…”

“…The proof for the case of cellular flow in the last subsection can be modified to establish the existence of H(p) for a large class of 2D incompressible flows [50], for example, if V = D ⊥ H for a smooth periodic stream function H that has finitely many non-degenerate critical points whose flow struture is well described in [8]. Also, the constant local burning velocity can be replaced by a positive, continuous and periodic function a(x).…”