{"ID":2829426,"CreatedAt":"2026-06-01T04:54:23.091178241Z","UpdatedAt":"2026-06-01T04:54:23.091178241Z","DeletedAt":null,"paper_url":"https://arxiv.org/abs/2512.12878","arxiv_id":"2512.12878","title":"On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear 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.","short_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...","url_abs":"https://arxiv.org/abs/2512.12878","url_pdf":"https://arxiv.org/pdf/2512.12878v2","authors":"[\"Dmitry Vorotnikov\",\"Amit Acharya\"]","published":"2025-12-14T23:28:40Z","proceeding":"math.AP","tasks":"[\"math.AP\",\"math.NA\",\"math.OC\"]","methods":"[]","has_code":false}