Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations
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| Format: | Preprint |
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2024
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| _version_ | 1866913444012228608 |
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| author | Cheskidov, Alexey Zeng, Zirong Zhang, Deng |
| author_facet | Cheskidov, Alexey Zeng, Zirong Zhang, Deng |
| contents | The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with smooth energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong.
Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2407_17463 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations Cheskidov, Alexey Zeng, Zirong Zhang, Deng Analysis of PDEs Fluid Dynamics 35A02 (Primary) 35D30, 35Q30, 60H15 (Secondary) The continuity of the kinetic energy is an important property of incompressible viscous fluid flows. We show that for any prescribed finite energy divergence-free initial data there exist infinitely many global in time weak solutions with smooth energy profiles to both the 3D deterministic and stochastic incompressible Navier-Stokes equations. In the stochastic case the constructed solutions are probabilistically strong. Our proof introduces a new backward convex integration scheme with delicate selections of initial relaxed solutions, backward time intervals, and energy profiles. Our initial relaxed solutions satisfy a new time-dependent frequency truncated NSE, different from the usual approximations as it decreases the large Reynolds error near the initial time, which plays a key role in the construction. |
| title | Existence and non-uniqueness of weak solutions with continuous energy to the 3D deterministic and stochastic Navier-Stokes equations |
| topic | Analysis of PDEs Fluid Dynamics 35A02 (Primary) 35D30, 35Q30, 60H15 (Secondary) |
| url | https://arxiv.org/abs/2407.17463 |