Uniform BMO estimate of parabolic equations and global well-posedness of the thermistor problem

2013/04/13 by Li, Buyang, Yang, Chaoxia
#Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.1304.3797

Abstract

Global well-posedness of the time-dependent (degenerate) thermistor problem remains open for many years. In this paper, we solve the problem by establishing a uniform-in-time BMO estimate of inhomogeneous parabolic equations. Applying this estimate to the temperature equation, we derive a BMO bound of the temperature uniform with respect to time, which implies that the electric conductivity is a A₂ weight. The Hölder continuity of the electric potential is then proved by applying the De Giorgi--Nash--Moser estimate for degenerate elliptic equations with A₂ coefficient. Uniqueness of solution is proved based on the established regularity of the weak solution. Our results also imply the existence of a global classical solution when the initial and boundary data are smooth.

Related

• Regularity results for a nonlinear elliptic-parabolic system with oscillating coefficients
• Stable Determination of the Discontinuous Conductivity Coefficient of a Parabolic Equation
• Global existence analysis for degenerate energy-transport models for semiconductors
• Existence of weak solutions of an unsteady thermistor problem with p-Laplacian type equation
• A note on uniqueness in the identification of a spacewise dependent source and diffusion coefficient for the heat equation