On a simple maximum principle technique applied to equations on the circle

Abstract

For a class of general quasilinear equations on S 1 , we show that, by a very simple maximum principle technique, as long as the solution stays finite, all of its derivatives also remain finite. Some specific examples are given. Under suitable assumptions, we also derive exponential decay of the derivatives of the solution.

Length-preserving evolution of non-simple symmetric plane curves

“…Some references can also be found in . Here, we apply the maximum principle to do this, and the main idea is from . For convenience of notation, we let Step We set u = k 1 + αk 2 with any fixed α > 0.…”

“…Some references can also be found in [10,23]. Here, we apply the maximum principle to do this, and the main idea is from [24]. For convenience of notation, we let…”

Evolution of highly symmetric curves under the shrinking curvature flow

“…This implies that we have a uniform upper bound and lower bound for , which guarantees that the equation for is uniformly parabolic. Then, we can estimate the higher order derivatives of in θ and τ by a routine method, see . By the Arzela–Ascoli Theorem, along any time sequence, there is a subsequence along which converges to a limit function, which must be zero in view of the uniqueness of limit.…”

More Mixed Volume Preserving Curvature Flows

“…Completion of the proof of Theorem 1.2 in this case: Our argument for long time existence in this case is simplified by axial symmetry, which reduces our problem to the setting of a scalar parabolic PDE with one spatial direction. While there are various results particular to such parabolic PDEs (see, for example, [14,24] and the references therein), here we use an argument more closely related to that in the previous section; when we fix time the resulting evolution equation is an ODE. Specifically, let us parametrise the evolving hypersurface as a radial graph by X :…”