Convex integration for diffusion equations and Lipschitz solutions of polyconvex gradient flows

Abstract

In this sequel to the paper [22], we construct certain smooth strongly polyconvex functions F on M 2×2 such that σ = DF satisfies the Condition (OC) in that paper. As a result, we show that the initialboundary value problem for the gradient flow of such polyconvex energy functionals is highly ill-posed even for some smooth initial-boundary data in the sense that the problem possesses a weakly* convergent sequence of Lipschitz weak solutions whose limit is not a weak solution.

Convex Integration for Diffusion Equations

“…Since the weak* limit is not a weak solution, the Lipschitz weak solutions in the sequence of the theorem will eventually be all distinct and different from the stationary solution u 0 (x). This shows that under the Condition (OC) the initial-boundary value problem (1.4) is highly ill-posed; this turns out to be the case even for certain strongly polyconvex gradient flows, as will be proved in [27].…”

“…However, if σ = DF and F is strongly polyconvex, then a result of [16,Proposition 3.11] shows that it is impossible for any τ 4 -configuration to be embedded on K 4 . In Yan [27], motivated by [24], we show that, for certain polyconvex functions F , the embedding of even certain special τ 5 -configurations in K 5 is possible.…”

“…Thus for such a function σ one can always find sets ∆ ± ; so, by Proposition 3.2, σ satisfies the Condition (OC) with N = 2. Such ideas will be greatly explored and generalized in [27] for constructing certain polyconvex gradient flows.…”

“…The main purpose of the present paper is to study problem (1.4) for certain general nonmonotone flux functions σ that may satisfy the parabolicity condition (1.2) in the case of m, n ≥ 2. Our main results will apply to the special case of gradient flows for certain polyconvex functionals; this will be studied in the sequel paper Yan [27].…”

On nonuniqueness and nonregularity for gradient flows of polyconvex functionals

“…The polyconvex functions F in the theorem will be the same as constructed in Yan [32], and the theorem will be proved as a corollary of a more general result proved later (see Theorem 5.1). Some aspects of the theorem in regard to certain known results will be discussed below, along with the main strategies of the proof.…”