On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs

TL;DR

This paper explores the dual variational formulation of quadratic PDE systems like the Nash system, establishing conditions for consistency, existence of solutions, and proposing an adaptive gradient-flow scheme for convergence.

Contribution

It introduces a sufficient condition for dual formulation consistency, proves existence of dual solutions for arbitrary base states, and develops an adaptive gradient-flow method for solving nonlinear PDEs.

Findings

Consistency holds over large time intervals under certain conditions.

Existence of dual solutions is proven for any base state.

The proposed gradient-flow scheme converges to solutions of the PDEs.

Abstract

We investigate the influence of base states on the consistency of the dual variational formulation for quadratic systems of PDEs, which are not necessarily conservative (typical examples include the noise-free Nash system with a quadratic Hamiltonian and multiple players). We identify a sufficient condition under which consistency holds over large time intervals. In particular, in the single-player case, there exists a sequence of base states (each exhibiting full consistency) that converges in mean to zero. We also prove existence of variational dual solutions to the noise-free Nash system for arbitrary base states. Furthermore, we propose a scheme based on Hilbertian gradient flows that, starting from an arbitrary base state, generates a sequence of new base states that is expected to converge to a solution of the original PDE.