Verifying Well-Posedness of Linear PDEs using Convex Optimization
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arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| _version_ | 1866910098845073408 |
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| author | Jagt, Declan S. Peet, Matthew M. |
| author_facet | Jagt, Declan S. Peet, Matthew M. |
| contents | Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of $L_{2}$ -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2604_00573 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Verifying Well-Posedness of Linear PDEs using Convex Optimization Jagt, Declan S. Peet, Matthew M. Analysis of PDEs Systems and Control Ensuring that a PDE model is well-posed is a necessary precursor to any form of analysis, control, or numerical simulation. Although the Lumer-Phillips theorem provides necessary and sufficient conditions for well-posedness of dissipative PDEs, these conditions must hold only on the domain of the PDE -- a proper subspace of $L_{2}$ -- which can make them difficult to verify in practice. In this paper, we show how the Lumer-Phillips conditions for PDEs can be tested more conveniently using the equivalent Partial Integral Equation (PIE) representation. This representation introduces a fundamental state in the Hilbert space $L_{2}$ and provides a bijection between this state space and the PDE domain. Using this bijection, we reformulate the Lumer-Phillips conditions as operator inequalities on $L_{2}$. We show how these inequalities can be tested using convex optimization methods, establishing a least upper bound on the exponential growth rate of solutions. We demonstrate the effectiveness of the proposed approach by verifying well-posedness for several classical examples of parabolic and hyperbolic PDEs. |
| title | Verifying Well-Posedness of Linear PDEs using Convex Optimization |
| topic | Analysis of PDEs Systems and Control |
| url | https://arxiv.org/abs/2604.00573 |