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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Fractal Hamilton-Jacobi-KPZ equations


Authors: Grzegorz Karch and Wojbor A. Woyczynski
Journal: Trans. Amer. Math. Soc. 360 (2008), 2423-2442
MSC (2000): Primary 35K55, 35B40, 60H30
Published electronically: December 11, 2007
MathSciNet review: 2373320
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Abstract: Nonlinear and nonlocal evolution equations of the form $ u_t=\mathcal{L} u \pm\vert\nabla u\vert^q$, where $ \mathcal{L}$ is a pseudodifferential operator representing the infinitesimal generator of a Lévy stochastic process, have been derived as models for growing interfaces in the case when the continuous Brownian diffusion surface transport is augmented by a random hopping mechanism. The goal of this paper is to study properties of solutions to this equation resulting from the interplay between the strengths of the ``diffusive'' linear and ``hyperbolic'' nonlinear terms, posed in the whole space $ \mathbb{R}^N$, and supplemented with nonnegative, bounded, and sufficiently regular initial conditions.


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Additional Information

Grzegorz Karch
Affiliation: Instytut Matematyczny, Uniwersytet Wrocławski, pl. Grunwaldzki 2/4, 50-384 Wrocław, Poland
Email: karch@math.uni.wroc.pl

Wojbor A. Woyczynski
Affiliation: Department of Statistics and the Center for Stochastic and Chaotic Processes in Science and Technology, Case Western Reserve University, Cleveland, Ohio, 44106–7054
Email: waw@po.cwru.edu

DOI: http://dx.doi.org/10.1090/S0002-9947-07-04389-9
PII: S 0002-9947(07)04389-9
Keywords: Nonlinear evolution equation, L\'evy anomalous diffusion, self-similar asymptotics, surface transport
Received by editor(s): January 27, 2006
Published electronically: December 11, 2007
Additional Notes: The authors appreciate valuable comments of the referee which helped them to improve the original version of this paper. Also, they would like to thank J. Droniou for making his unpublished manuscript available to them. This paper was partially written while the first-named author enjoyed the hospitality and support of the Center for Stochastic and Chaotic Processes in Science and Technology at Case Western Reserve University, Cleveland, Ohio, sponsored by the U.S. National Science Foundation Grant INT-0310055, and of the Helsinki University of Technology, and the University of Helsinki, Finland, within the Finnish Mathematical Society Visitor Program in Mathematics 2005-2006, \sl Function Spaces and Differential Equations. The preparation of this paper was also partially supported by the KBN grant 2/P03A/002/24, and by the European Commission Marie Curie Host Fellowship for the Transfer of Knowledge “Harmonic Analysis, Nonlinear Analysis and Probability” MTKD-CT-2004-013389.
Dedicated: Dedicated to our friend and collaborator, Piotr Biler
Article copyright: © Copyright 2007 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.