Fractal Hamilton-Jacobi-KPZ equations

Karch, Grzegorz; Woyczyński, Wojbor

doi:10.1090/S0002-9947-07-04389-9

Fractal Hamilton-Jacobi-KPZ equations
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by Grzegorz Karch and Wojbor A. Woyczyński PDF

Trans. Amer. Math. Soc. 360 (2008), 2423-2442 Request permission

Abstract:

Nonlinear and nonlocal evolution equations of the form $u_t=\mathcal {L} u \pm |\nabla u|^q$, 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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