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arXiv · 2503.05692

Global dissipative solutions of the 3D Naiver-Stokes and MHD equations

Alexey Cheskidov·Zirong Zeng·Deng Zhang
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Abstract

For any divergence free initial data in $H^\frac12$, we prove the existence of infinitely many dissipative solutions to both the 3D Navier-Stokes and MHD equations, whose energy profiles are continuous and decreasing on $[0,\infty)$. If the initial data is only $L^2$, our construction yields infinitely many solutions with continuous energy, but not necessarily decreasing. Our theorem does not hold in the case of zero viscosity as this would violate the weak-strong uniqueness principle due to Lions. This was achieved by designing a convex integration scheme that takes advantage of the dissipative term.

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Alexey Cheskidov, Zirong Zeng, Deng Zhang. 2025-03-07. Global dissipative solutions of the 3D Naiver-Stokes and MHD equations. https://arxiv.org/abs/2503.05692

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