Mathematics – Analysis of PDEs
Scientific paper
2000-10-20
Mathematics
Analysis of PDEs
Scientific paper
In Part I of our study on 2D Euler equation, we established the spectral theorem for a linearized 2D Euler equation. We also computed the point spectrum through continued fractions, and identified the eigenvalues with nonzero real parts. In this Part II of our study, first we discuss the Lax pairs for both 2D and 3D Euler equations. The existence of Lax pairs suggests that the hyperbolic foliations of 2D and 3D Euler equations may be degenerate, i.e., there exist homoclinic structures. Then we investigate the question on the degeneracy v.s. nondegeneracy of the hyperbolic foliations for Galerkin truncations of 2D Euler equation. In particular, for a Galerkin truncation, we have computed the explicit representation of the hyperbolic foliation which is of the degenerate case, i.e., figure-eight case. We also study the robustness of this degeneracy for a so-called dashed-line model through higher order Melnikov functions. The first order and second order Melnikov functions are all identically zero, which indicates that the degeneracy is relatively robust. The study in this paper serves a clue in searching for homoclinic structures for 2D Euler equation. The recent breakthrough result of mine on the existence of a Lax pair for 2D Euler equation, strongly supports the possible existence of homoclinic structures for 2D Euler equation.
No associations
LandOfFree
On 2D Euler Equations: Part II. Lax Pairs and Homoclinic Structures does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.
If you have personal experience with On 2D Euler Equations: Part II. Lax Pairs and Homoclinic Structures, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and On 2D Euler Equations: Part II. Lax Pairs and Homoclinic Structures will most certainly appreciate the feedback.
Profile ID: LFWR-SCP-O-667269