Local and Global Existence of Multiple Waves Near Formal Approximations

Details Local and Global Existence of Multiple Waves Near Formal Approximations Local and Global Existence of Multiple Waves Near Formal Approximations

1996-01-26

arxiv.org/abs/patt-sol/9601003v1

Progress in Nonlinear Diff. Equations Their Appl. 19 (1996) 385-404

Nonlinear Sciences

Pattern Formation and Solitons

19 pages, in one dvi file

Scientific paper

Assuming that a formal approximation of multiple waves has been obtained by matched asymptotic methods, we derive a {\em Spatial Shadowing lemma} to construct exact solutions near the formal approximation. In Part I, we consider a general singularly perturbed parabolic system. $$ \epsilon u_t + (-\epsilon^2)^m D^{2m}_x u = f(u,\epsilon u_x,\cdots,(\epsilon D_x)^{2m-1} u,x,\epsilon). $$ We show that if the formal approximation is precise, there is always an exact solution nearby for at least a short time. Examples include Cahn-Hilliard equation and viscous profile of conservation laws. In Part II, we show under some more assumptions, the process in Part I can be repeated to obtain global solutions if the formal approximation is a global one. Examples include reaction-diffusion equations and phase field equations.

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