Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows
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arXiv
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| Autori principali: | , , |
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| Natura: | Preprint |
| Pubblicazione: |
2026
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| _version_ | 1866912801448001536 |
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| author | Guo, Bin Kim, Seonghak Yan, Baisheng |
| author_facet | Guo, Bin Kim, Seonghak Yan, Baisheng |
| contents | We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction scheme based on new geometric structures, referred to as $\mathcal{T}_N$-configurations, together with a simplified structural hypothesis on the diffusion functions, termed Condition $O_N$. Under this condition, we show that the associated initial and boundary value problems with certain smooth initial-boundary data admit infinitely many Lipschitz weak solutions that are nowhere $C^1$. We further analyze specific $\mathcal{T}_N$-configurations and establish nondegeneracy conditions that are essential for verifying Condition $O_N$. As an application, we construct examples of strongly polyconvex energy functionals whose gradient flows generate irregular diffusion equations, thereby revealing a failure of regularity and uniqueness even within the class of polyconvex gradient flows. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_01035 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows Guo, Bin Kim, Seonghak Yan, Baisheng Analysis of PDEs 35K40, 35K51, 35D30, 35F50, 49A20 We investigate diffusion-type partial differential equations that are irregular in the sense that they admit weak solutions which are nowhere smooth, even for prescribed smooth data. By reformulating these equations as first-order partial differential relations and adapting the method of convex integration, we develop a construction scheme based on new geometric structures, referred to as $\mathcal{T}_N$-configurations, together with a simplified structural hypothesis on the diffusion functions, termed Condition $O_N$. Under this condition, we show that the associated initial and boundary value problems with certain smooth initial-boundary data admit infinitely many Lipschitz weak solutions that are nowhere $C^1$. We further analyze specific $\mathcal{T}_N$-configurations and establish nondegeneracy conditions that are essential for verifying Condition $O_N$. As an application, we construct examples of strongly polyconvex energy functionals whose gradient flows generate irregular diffusion equations, thereby revealing a failure of regularity and uniqueness even within the class of polyconvex gradient flows. |
| title | Irregular Diffusions and Loss of Regularity in Polyconvex Gradient Flows |
| topic | Analysis of PDEs 35K40, 35K51, 35D30, 35F50, 49A20 |
| url | https://arxiv.org/abs/2601.01035 |