Global well-posedness of a nonlinear Fokker-Planck type model of grain growth

Batuhan Bayir; Yekaterina Epshteyn; William M Feldman

doi:10.3934/dcds.2025146

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2026, Volume 48: 404-422. Doi: 10.3934/dcds.2025146

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Global well-posedness of a nonlinear Fokker-Planck type model of grain growth

Department of Mathematics, University of Utah, Salt Lake City, UT 84112, USA

^*Corresponding author: Batuhan Bayir
^*Corresponding author: Batuhan Bayir

Received: February 2025

Revised: August 2025

Early access: September 9, 2025

Published online: September 9, 2025

The work of B. Bayir and Y. Epshteyn was partially supported by the NSF grant DMS-2118172, and the work of Y. Epshteyn was also partially supported by the Simons Foundation Fellowship SFI-MPS-SFM-00010667. The work of W. M. Feldman was partially supported by the NSF grant DMS-2407235.

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Abstract

Most technologically useful materials spanning multiple length scales are polycrystalline. Polycrystalline microstructures are composed of a myriad of small crystals or grains with different lattice orientations which are separated by interfaces or grain boundaries. The changes in the grain and grain boundary structure of polycrystals highly influence the material's properties, including, but not limited to, electrical, mechanical, and thermal. Thus, an understanding of how microstructures evolve is essential for the engineering of new materials. In this paper, we consider a recently introduced nonlinear Fokker–Planck-type system and establish a global well-posedness result for it. Such systems under specific energy laws emerge in the modeling of the grain boundary dynamics in polycrystals.

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Mathematics Subject Classification: Primary: 74A25, 74N15, 35Q84, 35A01.

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Figure 1. Left figure: A schematic plot of three grain boundaries that meet at a triple junction point $ \boldsymbol{a}(t) $. For each $ i, j \in \{1, 2, 3\} $, $ \alpha^{(j)} = \alpha^{(j)}(t) $ represents the lattice orientation that corresponds to grid lines on the figure, and the differences $ \alpha^{(i)}-\alpha^{(j)} $ represent the lattice misorientations. Right figure: An instance of the simulation of the two-dimensional grain network that is a collection of grain boundaries and triple junction points [15,4]

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Table 1. Summary of the functions that appear in (nFP). Assumptions on these functions can be found in Section 2.2

Function	Description
$f(x, t)$	Unknown probability density function
$f_0(x)$	Initial condition
$f^ \rm{eq}(x)$	Energy minimizing equilibrium state defined in (3)
$\pi(x, t)$	Mobility
$D(x)$	Inhomogeneous absolute temperature coefficient
$\phi(x)$	Grain boundary energy density
$V(x, t)$	Coefficient of the nonlinear part defined by $V(x, t) := \frac{ \nabla D(x)}{\pi(x, t)}$

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