×

zbMATH — the first resource for mathematics

Existence and uniqueness for a class of nonlinear higher-order partial differential equations in the complex plane. (English) Zbl 1069.35003
Summary: We prove existence and uniqueness results for nonlinear third-order partial differential equations of the form \[ f_t-f_{yyy}= \sum^3_{j=0}b_j (y,t; f)f^{(j)}+r(y,t) \] where superscript \(j\) denotes the \(j\)th partial derivative with respect to \(y\). The inhomogeneous term \(r\), the coefficients \(b_j\), 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. By 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.

MSC:
35A05 General existence and uniqueness theorems (PDE) (MSC2000)
76D27 Other free boundary flows; Hele-Shaw flows
Software:
YYY
PDF BibTeX XML Cite
Full Text: DOI arXiv
References:
[1] Caflisch, Comm Pure Appl Math 39 pp 807– (1986) · Zbl 0603.76039
[2] Caflisch, SIAM J Math Anal 20 pp 293– (1989) · Zbl 0697.76029
[3] Caflisch, SIAM J Appl Math 50 pp 1517– (1990) · Zbl 0712.76026
[4] Caflisch, Phys D 41 pp 197– (1990) · Zbl 0692.76017
[5] Costin, Duke Math J 93 pp 289– (1998) · Zbl 0948.34068
[6] Fokas, Math Proc Cambridge Philos Soc 124 pp 169– (1998) · Zbl 0918.76020
[7] Garabedian, J Math Mech 9 pp 905– (1960)
[8] Howison, SIAM J Appl Math 46 pp 20– (1986) · Zbl 0592.76042
[9] Howison, European J Appl Math 3 pp 209– (1992) · Zbl 0759.76022
[10] Krasny, J Fluid Mech 167 pp 65– (1986) · Zbl 0601.76038
[11] Kunka, Phys Rev E (3) 59 pp 673– (1999)
[12] Moore, Proc Roy Soc London Ser A 365 pp 105– (1979) · Zbl 0404.76040
[13] Numerical and analytical aspects of Helmholtz instability. Theoretical and applied mechanics (Lyngby, 1984), 263-274. North-Holland, Amsterdam-New York, 1985.
[14] Sammartino, Comm Math Phys 192 pp 433– (1998) · Zbl 0913.35102
[15] Sammartino, Comm Math Phys 192 pp 463– (1998) · Zbl 0913.35103
[16] Siegel, J Fluid Mech 323 pp 201– (1996) · Zbl 0885.76022
[17] Tanveer, Philos Trans Roy Soc London Ser A 343 pp 155– (1993) · Zbl 0778.76029
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.