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

TL;DR

This paper proves the global well-posedness of the thermistor problem by establishing uniform BMO estimates for parabolic equations, leading to regularity and uniqueness results for solutions.

Contribution

It introduces a uniform-in-time BMO estimate for inhomogeneous parabolic equations, enabling the proof of global well-posedness of the thermistor problem.

Findings

Established a uniform BMO estimate for parabolic equations.

Proved the temperature's BMO bound and electric conductivity as an A_2 weight.

Demonstrated Hölder continuity and uniqueness of solutions.

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 $A_2$ weight. The H\"{o}lder continuity of the electric potential is then proved by applying the De Giorgi--Nash--Moser estimate for degenerate elliptic equations with $A_2$ 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.