On Lipschitz solutions for some forward–backward parabolic equations. II: the case against Fourier

Abstract

Abstract. As a sequel to the paper [9], we study the existence and properties of Lipschitz solutions to the initial-boundary value problem of some forward-backward parabolic equations with diffusion fluxes violating Fourier's inequality.

On weak closure of some diffusion equations

“…The following result shows that some special structures in the set Σ(p) defined above have a variational description; this result also shows that the subsolutionsū used in [9,10] automatically satisfyū t ∈ div Σ(Dū).…”

“…In the one spatial dimension, two special functions of diffusion function σ(p) are given as shown in Figures 1 and 2. (b) Some interesting special structures in the set Σ(p) can be characterized by a variational principle (see Proposition 3.3), which may have some relevance to the existence results in the recent work [8,9,10].…”

“…Recently, for certain non-monotone functions σ, exact Lipschitz solutions have been constructed for the initial-Neumann problem of (1.1) in [8,9,10]; in such cases, solutions can converge weakly to a function that is not a solution of (1.1).…”

“…In what follows, we denote by g(p, β) the convex hull of function |σ(p)−β| 2 on R n × R n . For p ∈ R n , define 3), which may have some relevance to the existence results in the recent work [8,9,10]. Theorem 1.2.…”

Convex integration with linear constraints and its applications

“…We begin this section by introducing a pivotal approximation result, Theorem 3.1, for proving the main results of the paper, Theorems 1.1 and 1.3. Its special cases have been successfully applied to some nonstandard evolution problems [15,16,17,14]. Although the proof of Theorem 3.1 already appeared in [14], we include it in Section 6 for the sake of completeness as we make use of the general version of the theorem for the first time in this paper.…”