Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed
We consider the compressible Navier-Stokes-Korteweg equations describing the dynamics of a liquid-vapor mixture with diffuse interphase in Rd{{\mathbb{R}}}^{d} with d≥3d\ge 3 when the initial perturbation is suitably small. In particular, when the base sound speed P′(ρ¯)=0\sqrt{P^{\prime} \left(\bar...
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De Gruyter
2025-04-01
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| Series: | Advances in Nonlinear Analysis |
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| Online Access: | https://doi.org/10.1515/anona-2025-0078 |
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| author | Liu Mengqian Wu Zhigang |
| author_facet | Liu Mengqian Wu Zhigang |
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| description | We consider the compressible Navier-Stokes-Korteweg equations describing the dynamics of a liquid-vapor mixture with diffuse interphase in Rd{{\mathbb{R}}}^{d} with d≥3d\ge 3 when the initial perturbation is suitably small. In particular, when the base sound speed P′(ρ¯)=0\sqrt{P^{\prime} \left(\bar{\rho })}=0, we first give the global existence and optimal L2{L}^{2}-decay rate of the smooth solution, where the optimality means that the decay rate of the solution is the same as that for the corresponding linearized system, and there is no decay loss for the highest-order spatial derivatives of the solution. Then, we establish space-time behavior of the solution based on Green’s function method. It is obviously different from the case P′(ρ¯)>0\sqrt{P^{\prime} \left(\bar{\rho })}\gt 0, which obeys the generalized Huygens’ principle as the compressible Navier-Stokes equations. |
| format | Article |
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| institution | OA Journals |
| issn | 2191-950X |
| language | English |
| publishDate | 2025-04-01 |
| publisher | De Gruyter |
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| series | Advances in Nonlinear Analysis |
| spelling | doaj-art-a3f5445d99884f46a5eee388e13262162025-08-20T02:12:33ZengDe GruyterAdvances in Nonlinear Analysis2191-950X2025-04-0114184386810.1515/anona-2025-0078Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speedLiu Mengqian0Wu Zhigang1College of Mathematics and Statistics, Donghua University, Shanghai 201620, ChinaDepartment of Applied Mathematics, Hubei University of Automotive Technology, Shiyan, 442002, ChinaWe consider the compressible Navier-Stokes-Korteweg equations describing the dynamics of a liquid-vapor mixture with diffuse interphase in Rd{{\mathbb{R}}}^{d} with d≥3d\ge 3 when the initial perturbation is suitably small. In particular, when the base sound speed P′(ρ¯)=0\sqrt{P^{\prime} \left(\bar{\rho })}=0, we first give the global existence and optimal L2{L}^{2}-decay rate of the smooth solution, where the optimality means that the decay rate of the solution is the same as that for the corresponding linearized system, and there is no decay loss for the highest-order spatial derivatives of the solution. Then, we establish space-time behavior of the solution based on Green’s function method. It is obviously different from the case P′(ρ¯)>0\sqrt{P^{\prime} \left(\bar{\rho })}\gt 0, which obeys the generalized Huygens’ principle as the compressible Navier-Stokes equations.https://doi.org/10.1515/anona-2025-0078navier-stokes-korteweg equationzero sound speedclassical solution35a0935b4035q35 |
| spellingShingle | Liu Mengqian Wu Zhigang Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed Advances in Nonlinear Analysis navier-stokes-korteweg equation zero sound speed classical solution 35a09 35b40 35q35 |
| title | Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed |
| title_full | Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed |
| title_fullStr | Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed |
| title_full_unstemmed | Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed |
| title_short | Classical solution for compressible Navier-Stokes-Korteweg equations with zero sound speed |
| title_sort | classical solution for compressible navier stokes korteweg equations with zero sound speed |
| topic | navier-stokes-korteweg equation zero sound speed classical solution 35a09 35b40 35q35 |
| url | https://doi.org/10.1515/anona-2025-0078 |
| work_keys_str_mv | AT liumengqian classicalsolutionforcompressiblenavierstokeskortewegequationswithzerosoundspeed AT wuzhigang classicalsolutionforcompressiblenavierstokeskortewegequationswithzerosoundspeed |