Uniform Bmo Estimate of Parabolic Equations and Global Well-Posedness of the Thermistor Problem

Abstract

We prove global well-posedness of the time-dependent degenerate thermistor problem by establishing a uniform-in-time bounded mean ocsillation (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 an A 2 weight. The Hölder continuity of the electric potential is then proved by applying the De GiorgiNash-Moser estimate for degenerate elliptic equations with an A 2 coefficient. The uniqueness of the 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.

Maximal Regularity of Fully Discrete Finite Element Solutions of Parabolic Equations

“…It is known that the elliptic partial differential operator A = d i,j=1 ∂ ∂x i a ij (x) ∂ ∂x j , under the Dirichlet boundary condition, generates a bounded analytic semigroup on the space L q := L q (Ω) for all 1 < q < ∞; see [33,Theorem 2.4]. When u 0 = 0, the solution of (1.1) possesses maximal L p -regularity on L q , namely, ∂ t u L p (R + ;L q ) + Au L p (R + ;L q ) ≤ C p,q f L p (R + ;L q ) ∀ 1 < p, q < ∞, (1.4) which is an important tool in studying well-posedness and regularity of solutions to nonlinear parabolic PDEs; see [3,7,31]. In the numerical solution of parabolic equations, it is also desirable to have a discrete analogue of this estimate, which has a number of applications in analysis of stability and convergence of numerical methods for nonlinear parabolic problems, including semilinear parabolic equations with strong nonlinearities [11], quasi-linear parabolic equations with nonsmooth coefficients [28], optimal control and inverse problems [23,25], and so on.…”

Finite element convergence analysis for the thermoviscoelastic Joule heating problem

“…For discussions on existence and uniqueness, see e.g. [2,5,6,8,9,17,18,19,24,31] and the references therein. For the fully coupled, deformable problem the literature is less extensive.…”

The theta‐Galerkin finite element method for coupled systems resulting from microsensor thermistor problems

“…Cimatti proved the existence of the weak solution of to for an arbitrary large interval of time using the Faedo‐Galerkin method. Recently, Li and Yang proved the existence and uniqueness for the thermistor Equations to without nondegenerate assumptions. For the special case where the temperature‐dependent thermal conductivity κ ( u )=1 (which is not of particular interest for microsensor applications), Yuan and Liu proved the existence and uniqueness of a C α solution in 3‐dimensional space, from which further regularity of the solution can be derived with suitable assumptions on the initial and boundary conditions.…”