On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866908731043741696 |
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| author | Vorotnikov, Dmitry Acharya, Amit |
| author_facet | Vorotnikov, Dmitry Acharya, Amit |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_12878 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs Vorotnikov, Dmitry Acharya, Amit Analysis of PDEs Numerical Analysis Optimization and Control 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. |
| title | On the variational dual formulation of the Nash system and an adaptive convex gradient-flow approach to nonlinear PDEs |
| topic | Analysis of PDEs Numerical Analysis Optimization and Control |
| url | https://arxiv.org/abs/2512.12878 |