Analyticity of Essentially Bounded Solutions to Semilinear Parabolic Systems and Validity of the GinzburgLandau Equation Academic Article uri icon

abstract

  • Some analytic smoothing properties of a general strongly coupled, strongly parabolic semilinear system of order 2m in D (0, T) with analytic entries are investigated. These properties are expressed in terms of holomorphic continuation in space and time of essentially bounded global solutions to the system. Given 0 < T < T , it is proved that any weak, essentially bounded solution u = (u1, . . . ,uN) in D (0,T) possesses a bounded holomorphic continuation u(x + iy, + i) into a region in D defined by (x, ) D (T, T), |y| < A and || < B, where A and B are some positive constants depending upon T. The proof is based on analytic smoothing properties of a parabolic Green function combined with a contraction mapping argument in a Hardy space H. Applications include weakly coupled semilinear systems of complex reaction-diffusion equations such as the complex Ginzburg-Landau equations. Special attention is given to the problem concerning the validity of the derivation of amplitude equations which describe various instability phenomena in hydrodynamics.

published proceedings

  • SIAM Journal on Mathematical Analysis

author list (cited authors)

  • Tak, P., Bollerman, P., Doelman, A., van Harten, A., & Titi, E. S.

citation count

  • 32

complete list of authors

  • Takáč, P||Bollerman, P||Doelman, A||van Harten, A||Titi, ES

publication date

  • March 1996