Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model
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arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866918242225750016 |
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| author | Chaudhuri, Nilasis Rohde, Christian Wendt, Florian |
| author_facet | Chaudhuri, Nilasis Rohde, Christian Wendt, Florian |
| contents | We consider a parabolic relaxation model for the compressible Navier-Stokes-Korteweg equations in the isothermal framework. This system depends on the relaxation parameters $α,β>0$ and approximates formally solutions of the compressible Navier-Stokes-Korteweg equations in the relaxation limit $α\to \infty$ and $β\to 0$. Introducing the class of finite energy weak solutions for the initial-boundary value problem corresponding to the relaxation model in spatial dimension three, we show that the weak-strong uniqueness principle holds. It asserts that a weak solution and a strong solution emanating from the same initial data coincide as long as the strong solution exists. Furthermore, we contribute a rigorous convergence result for the relaxation limit $α\to \infty$ and $β\to 0$ and thus justify the relaxation model as an approximate model for the compressible Navier-Stokes-Korteweg equations from a mathematical point of view. Our results hold for general non-monotone pressure-density relations. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2512_09719 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model Chaudhuri, Nilasis Rohde, Christian Wendt, Florian Analysis of PDEs 35Q30, 76N06, 35B25, 76T10 We consider a parabolic relaxation model for the compressible Navier-Stokes-Korteweg equations in the isothermal framework. This system depends on the relaxation parameters $α,β>0$ and approximates formally solutions of the compressible Navier-Stokes-Korteweg equations in the relaxation limit $α\to \infty$ and $β\to 0$. Introducing the class of finite energy weak solutions for the initial-boundary value problem corresponding to the relaxation model in spatial dimension three, we show that the weak-strong uniqueness principle holds. It asserts that a weak solution and a strong solution emanating from the same initial data coincide as long as the strong solution exists. Furthermore, we contribute a rigorous convergence result for the relaxation limit $α\to \infty$ and $β\to 0$ and thus justify the relaxation model as an approximate model for the compressible Navier-Stokes-Korteweg equations from a mathematical point of view. Our results hold for general non-monotone pressure-density relations. |
| title | Weak-Strong Uniqueness and Relaxation Limit for a Navier-Stokes-Korteweg Model |
| topic | Analysis of PDEs 35Q30, 76N06, 35B25, 76T10 |
| url | https://arxiv.org/abs/2512.09719 |