Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866929731858857984 |
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| author | Fuchs, Fabian Nendel, Max |
| author_facet | Fuchs, Fabian Nendel, Max |
| contents | In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called $K$-convexity, w.r.t. another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional Lévy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_18270 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups Fuchs, Fabian Nendel, Max Analysis of PDEs Optimization and Control 47H20, 35D40 (Primary) 49L25, 35K90 (Secondary) In this work, we provide conditions for nonlinear monotone semigroups on locally convex vector lattices to give rise to a generalized notion of viscosity solutions to a related nonlinear partial differential equation. The semigroup needs to satisfy a convexity estimate, so called $K$-convexity, w.r.t. another family of operators, defined on a potentially larger locally convex vector lattice. We then show that, under mild continuity requirements on the bounding family of operators, the semigroup yields viscosity solutions to the abstract Cauchy problem given in terms of its generator in the larger locally convex vector lattice. We apply our results to drift control problems for infinite-dimensional Lévy processes and robust optimal control problems for infinite-dimensional Ornstein-Uhlenbeck processes. |
| title | Existence of Viscosity Solutions to Abstract Cauchy Problems via Nonlinear Semigroups |
| topic | Analysis of PDEs Optimization and Control 47H20, 35D40 (Primary) 49L25, 35K90 (Secondary) |
| url | https://arxiv.org/abs/2502.18270 |