Well-posedness of parabolic KWC-systems with variable-dependent mobilities

In this paper, we deal with the parabolic KWC system, associated with the mathematical model of grain boundary motion. The goal of this paper is to guarantee the well-posedness of the parabolic KWC system. However, such results have not been reported under the setting where the mobility of grain boundary motion depends on the unknown. To overcome this difficulty, results for the pseudo-parabolic type KWC system in Antil et al., SIAM J. Math. Anal. $\mathbf{\text{56}}$(5), 6422–6445 suggest that the $H^1$-regularity of the time-derivative of the solution plays an essential role in verifying the uniqueness of the solution. In this light, we consider the pseudo-parabolic KWC system as an approximating system of the parabolic one, and focus on the improvements of regularity of solution to the parabolic system. By virtue of this regularity result, we establish the well-posedness theory on the parabolic KWC system.