In this paper we study the general diffusion equations with nonmonotone diffusion flux. We prove that certain geometric structures of a general nonmonotone flux function ensure the existence of infinitely many Lipschitz solutions to the diffusion equation. Relevant structures are characterized by certain bounded open sets satisfying the so-called chord condition and sufficient conditions for such sets are also given. Our method relies on a modification of the convex integration and Baire's category methods.
Convex integration with linear constraints and its applications
On the $Λ$-Convex Hull for Convex Integration Applied to the Isentropic Compressible Euler System
Convex integration for the Monge-Ampère equation in two dimensions
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