2002/03/05 by O Costin, S Tanveer
Mathematics · #math.AP #msc:35G20 #msc:35G25
published as Communications on Pure and Applied Mathematics, Vol. LIII, 0001- 0026 (2000))
arxiv created 2002/03/05 · arxiv updated 2009/11/30
We prove existence and uniqueness results for nonlinear third order partial differential equations of the form ft - fyyy = ∑j=03 bj (y, t; f) ~f(j) + r(y, t) where superscript j denotes the j-th partial derivative with respect to y. The inhomogeneous term r, the coefficients bj and the initial condition f(y,0) are required to vanish algebraically for large |y| in a wide enough sector in the complex y-plane. Using methods related to Borel summation, a unique solution is shown to exist that is analytic in y for all large |y| in a sector. Three partial differential equations arising in the context of Hele-Shaw fingering and dendritic crystal growth are shown to be of this form after appropriate transformation, and then precise results are obtained for them. The implications of the rigorous analysis on some similarity solutions, formerly hypothesized in two of these examples, are examined.