Abstract
本論文包含兩個部份。 第一部分針對定義在S¹上一種特定型式的擬線性拋物方程式(a quasilinear parabolic equation)初值問題來探討其解的正則性(regularity)。當我們透過拋物曲率流(parabolic curvature flow)來演化平面上封閉曲線時，對應的演化方程式(evolution equation)為此種型式的擬線性拋物方程，所以我們探討這種型式的拋物方程式。我們將證明當解是有界時，其對時間和空間的高階導函數也會保持有界。另外當解存留的時間為無窮大時，我們可近ㄧ步證明解沿著空間方向的各階導函數將隨著時間以指數形式的速度衰減至零(exponentially decay)。證明的方法主要是選取適當的函數，運用簡單極大值原理(maximum principle)的技巧以及數學歸納法，便可以得到解的正則性。正則性主要的應用是從解的正則性，我們可知當解在演化的過程中，高階導函數一旦開始變壞其導因於解本身的C0 norm趨近無窮大；若解在存留的時間範圍內有界，藉由正則性的結果，解則依C∞的方式往這個時間範圍外延伸，即得到解的長時間存留性(long time existence)並進而研究解的漸近行為(asymptotical behavior)。 第二部份探討某種形式的曲率流：在平面上給定兩條長度相同、封閉、沒有自我相交的凸曲線 (simple convex closed curves) γ1、γ2，其中γ2固定不動，希望γ1透過定義的拋物曲率流演化變成那條固定住的曲線 (兩者中間最後可能差一個平移向量(a translation vector))。這條曲率流對應的the evolution of support function 是線性的，此種有趣的性質使得我們可利用富氏分析(Fourier analysis)的分析技巧來探討這個曲率流的收斂性。我們將證明兩條有相同長度的簡單封閉的凸曲線(simple convex closed curves)γ1、γ2，其中γ2 固定，γ1透過定義的曲率流演化，只要在任何有限的時間裡，演化曲線的曲率不會blow up，那麼曲線就可以一直演化下去，終究演化至γ2。 This dissertation consists of two parts. The first part (Part I) is about regularity estimate for a certain class of quasilinear equations on the unit circle S¹ and exponential decay of the derivatives of solutions. The curvature-driven evolution of closed plane curves in R² are always governed by certain types of nonlinear parabolic equations on the unit circle. Hence it is important to establish regularity estimate of the solutions to these nonlinear equations. A major application of these regularity estimate is to obtain long time existence of plane curves evolution. One can see the classic paper by Gage and Hamilton[GH] for this. The key point is that we can use a simple maximum principle to achieve these estimate. For the exponential decay of the derivatives of solutions, we are motivated by observation based on linear equations. The second part (Part II) is about a special type of curvature flow. Motivated by a recent curvature flow introduced by Professor S.-T. Yau [Y], we use a simple curvature flow to evolve a convex closed curve to another one (under the assumption that both curves have the same length). This kind of flow is very interesting since it produces a linear equation on the support functions. Therefore it can be analyzed using Fourier analysis, and this is a huge advantage over other types of nonlinear flow. We show that, under the evolution, the length is preserved and the center is fixed and if the curvature is bounded above during the evolution, then an initial convex closed curve can be evolved to another given one.