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The Effect of a Nonlinear Viscous Regularization on the Solution of a Cauchy–Riemann Equation

✍ Scribed by Anne C. Morlet

Publisher
Elsevier Science
Year
1996
Tongue
English
Weight
418 KB
Volume
204
Category
Article
ISSN
0022-247X

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✦ Synopsis

Here we consider a variant of the Cauchy᎐Riemann equation, in which the Cauchy᎐Riemann equation has been regularized with a nonlinear second-order wŽ < < 2 . x viscous term ⑀ q u u . The equation is degenerate of parabolic type when

⑀s0 and has a weak solution for all time. We use an embedding process to analyze the properties of the solution. We also show that the smallest scale of the solution is ⑀ , the coefficient of the regularizing second-order term. We numerically confirm the result. We show that from the sequence of regularized solutions, we can extract a converging subsequence. A limit of the subsequence u belongs to 1 Žw x w x. H 0, 1 = 0, T . For each fixed t, the first-order spatial derivative of u also ϱ Žw x. 0 Žw x w x. belongs to L 0, 1 . Therefore u belongs to C 0, 1 = 0, T and, for fixed t, is Holder continuous with exponent ␣, ␣ -1. We numerically see that the solution ḧas some of the features of the solution of the porous medium equation even though the equation is complex. We also numerically see the influence of the regularizing second-order diffusion term on the solution.

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