On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization
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arXiv
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| Format: | Preprint |
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2026
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| author | Zlotnik, Alexander |
| author_facet | Zlotnik, Alexander |
| contents | We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(η,u,θ)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(η^0,u^0,e^0)$ in a norm of $L^2(Ω)\times H^{-1}(Ω)\times H^{-1}(Ω)$-type and also on the free terms in all the equations in some dual norms. Here $η$, $u$ and $θ$ are the specific volume, velocity and absolute temperature as well as $η^0$, $u^0$ and $e^0$ are the initial specific volume, velocity and specific total energy, and $Q=Ω\times (0,T)$. We also apply this result to the case of discontinuous rapidly oscillating, with the period $\varepsilon$, initial data and free terms and derive an estimate $O(\varepsilon)$ for the difference between the solutions to the Navier-Stokes equations and their Bakhvalov-Eglit two-scale homogenized version with averaged data. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2602_03481 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization Zlotnik, Alexander Analysis of PDEs 76N06, 35B30, 76M50 (Primary), 35M33, 35D30 (Secondary) We deal with the global in time weak solutions to the 1D compressible Navier-Stokes system of equations for large discontinuous initial data and nonhomogeneous boundary conditions of three standard types. We prove the Lipschitz-type continuous dependence of the solution $(η,u,θ)$, in a norm slightly stronger than $L^{2,\infty}(Q)\times L^2(Q)\times L^2(Q)$, on the initial data $(η^0,u^0,e^0)$ in a norm of $L^2(Ω)\times H^{-1}(Ω)\times H^{-1}(Ω)$-type and also on the free terms in all the equations in some dual norms. Here $η$, $u$ and $θ$ are the specific volume, velocity and absolute temperature as well as $η^0$, $u^0$ and $e^0$ are the initial specific volume, velocity and specific total energy, and $Q=Ω\times (0,T)$. We also apply this result to the case of discontinuous rapidly oscillating, with the period $\varepsilon$, initial data and free terms and derive an estimate $O(\varepsilon)$ for the difference between the solutions to the Navier-Stokes equations and their Bakhvalov-Eglit two-scale homogenized version with averaged data. |
| title | On weak solutions to the 1d compressible Navier-Stokes equations: a Lipschitz continuous dependence on data in weaker norms and an error of their homogenization |
| topic | Analysis of PDEs 76N06, 35B30, 76M50 (Primary), 35M33, 35D30 (Secondary) |
| url | https://arxiv.org/abs/2602.03481 |