Blow-Up Criteria and Weak--Strong Uniqueness for Compressible Fluid--Viscoelastic Shell Interactions
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arXiv
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| Format: | Preprint |
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2026
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| author | Ngougoue, Pierre Marie Ngougoue Mensah, Prince Romeo |
| author_facet | Ngougoue, Pierre Marie Ngougoue Mensah, Prince Romeo |
| contents | Existence and uniqueness of strong solutions to a barotropic compressible fluid--viscoelastic shell interaction system have recently been established on a finite time interval. A natural question is whether such solutions can be continued globally. In this work, we derive a continuation criterion for this coupled system. Our analysis is based on an energy estimate at the level of material acceleration, derived under Serrin-type and Beale--Kato--Majda-type control assumptions. While in the incompressible setting, such control is sufficient to prevent finite-time blow-up, in the compressible regime it does not by itself ensure propagation of the full regularity required for strong solutions. To obtain a genuine continuation criterion, we impose a Beale--Kato--Majda Lipschitz-type control on the density and velocity gradients with stronger time integrability. In combination with the control framework underlying the acceleration estimate, we close a higher-order energy estimate and thereby prevent loss of strong-solution regularity. Consequently, the solution can be extended beyond a potential blow-up time, provided that the corresponding control norms remain finite. We further establish a weak-strong uniqueness principle for the system under the above conditional regularity criterion. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04314 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Blow-Up Criteria and Weak--Strong Uniqueness for Compressible Fluid--Viscoelastic Shell Interactions Ngougoue, Pierre Marie Ngougoue Mensah, Prince Romeo Analysis of PDEs 35B65, 35Q74, 35R37, 76N10, 74F10, 74K25 Existence and uniqueness of strong solutions to a barotropic compressible fluid--viscoelastic shell interaction system have recently been established on a finite time interval. A natural question is whether such solutions can be continued globally. In this work, we derive a continuation criterion for this coupled system. Our analysis is based on an energy estimate at the level of material acceleration, derived under Serrin-type and Beale--Kato--Majda-type control assumptions. While in the incompressible setting, such control is sufficient to prevent finite-time blow-up, in the compressible regime it does not by itself ensure propagation of the full regularity required for strong solutions. To obtain a genuine continuation criterion, we impose a Beale--Kato--Majda Lipschitz-type control on the density and velocity gradients with stronger time integrability. In combination with the control framework underlying the acceleration estimate, we close a higher-order energy estimate and thereby prevent loss of strong-solution regularity. Consequently, the solution can be extended beyond a potential blow-up time, provided that the corresponding control norms remain finite. We further establish a weak-strong uniqueness principle for the system under the above conditional regularity criterion. |
| title | Blow-Up Criteria and Weak--Strong Uniqueness for Compressible Fluid--Viscoelastic Shell Interactions |
| topic | Analysis of PDEs 35B65, 35Q74, 35R37, 76N10, 74F10, 74K25 |
| url | https://arxiv.org/abs/2605.04314 |